WEBVTT
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Let’s look at the graphs of some basic functions.
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The first function is y equals x. This is a linear function.
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It is called the identity function.
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It has its x - intercept and y - intercept both at zero, at the origin.
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It is an odd function since it is symmetric with respect to the origin,
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and its domain is all real numbers. x can be anything.
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Its range is also all real numbers.
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The second function is y equals x squared.
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This is a basic quadratic function. It is called the square function.
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Notice this one is symmetric with respect to the y - axis.
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This one is said to be an even function.
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The x - intercept is at 0. The y - intercept is at 0.
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The domain here is the set of all real numbers.
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Notice how the y - values are never lower than 0,
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so the range will be the set of real numbers from 0 to infinity.
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The third function is y equals x cubed.
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Notice that this one has its x - intercept
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and it y - intercept both at 0, at the origin.
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The domain of this one is the set of all real numbers.
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The range is also the set of all real numbers.
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This one is symmetric with respect to the origin, so it is an odd function.
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The fourth function we want to look at is the absolute value function,
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y equals the absolute value of x.
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This one also has its x and y - intercepts at the origin.
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This one is symmetric with respect to the y - axis, so it is an even function.
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Its domain, the set of allowed x - values, is all real numbers.
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Again, here the y - values are never smaller than 0,
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so the range is the set of real numbers from 0 to positive infinity.
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The next function is y equals the square root of x.
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First thing I notice on this one is that it has no symmetry.
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The next thing you want to notice is that
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the domain is all real numbers from 0 to infinity.
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Alright, you know you cannot take the square root of a negative number,
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not to get a real number anyway, so that is the domain.
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The range would be the set of values from 0 to positive infinity again.
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The sixth function we want to look at is y equals the cube root of x.
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This one is also an odd function since it is
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symmetric with respect the origin.
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The domain is the set of all real numbers.
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The range is the set of all real numbers,
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and the x and y - intercepts are both at the origin.
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The last function is called the reciprocal function.
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Notice first of all this one has no x or y - intercepts.
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x cannot equal 0, that would give us division by 0.
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So the domain is the set of all real numbers except 0.
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The range is also the set of all real numbers except 0,
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because you will notice that the graph never touches the x - axis or crosses.
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Finally, this is an odd function since it is
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symmetric with respect to the origin.
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These are very important basic functions.
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You should be familiar with both the characteristics,
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and the graphs of these functions.
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Piecewise Functions are functions that are defined by
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different rules in different parts of the domain.
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Let’s examine a piecewise function closely.
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The first thing you will notice is that it looks different from our usual functions.
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This is a function f of x, and it is defined by the rule
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y is equal to x plus 1 if x is greater than or equal to 0,
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but for values of x less than 0, we have the same function but a different rule.
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The first thing we would like to do is look at some specific function values.
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This is asking us to find f of 0.
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That means find the value of the function when x is equal to 0.
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Well I need to decide which part of the rule for this function applies.
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x equals 0 so that is here, so I want to use the first part of the rule.
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Substitute 0 in place of the x, so we find that f of 0 is equal to 1.
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The next one is f of 3. This is asking to find the value of the function when x is equal to 3.
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Well again I need to decide which part of this rule applies.
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3 is greater than or equal to 0, so we are going to use the first rule again.
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Substitute 3 in place of the x, and you find that f of 3 is equal to 4.
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One more. This is asking for f of negative 1.
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We want to know the value of the function when x is equal to negative 1.
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Well negative 1 is less than 0, so we want to use this rule.
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Substitute negative 1 for the x, and we find that x of negative 1 is equal to negative 3.
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The next thing we would like to find is the domain of this function.
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The domain is the values that x is allowed to have.
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So let’s look at these inequalities.
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This tells me x can be any real number less than 0.
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This tells me x could equal 0 or anything greater than 0.
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That is the set of all real numbers.
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The domain we could write in interval notation as negative infinity to infinity,
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or we could just say it is just the set of all real numbers.
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The next thing it asks us for is the range, but it will be easier
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to answer this once we have drawn the graph so let’s stop and draw the graph.
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Okay, well we need to examine the pieces of our piecewise function again.
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Notice that both of them are linear pieces, so what we need to do is
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graph this piece of a line for x greater than or equal to 0,
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and graph the piece of this line for x less than 0.
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Okay, well to graph a line we need just need to find two points,
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maybe specific two points, but two points will do.
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Alright, let’s look at the first one.
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We want to start with x equal to 0 because that is the endpoint,
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that is where something changes in this piecewise function.
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I want to find the value of this piece of the function when x is equal to 0.
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Okay, but we did that right? f of 0 is 1. Let’s plot this point.
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Now we are still working on the first piece.
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This rule is used for values of x greater than or equal to 0,
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so I want to pick another value of x greater than 0
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and use that to graph the rest of the line
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or the portion of the line that I need. Alright, well look at our work.
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We have used x equal 3, 3 is greater than 0.
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We know that when x is 3, the value of the function is 4.
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We can plot the point (3, 4).
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Now we have already said that this piece of the graph is linear,
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and we use this rule for values of x greater than or equal to 0.
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Let’s draw that portion of our line. That takes care of the first piece.
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Now let’s move to the second piece. It is also linear, and again I need to plot two points.
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Alright, the first thing I am going to do is look at this endpoint.
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This is where the graph changes, this is where strange things can happen if they are going to.
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We want to substitute x equals 0 and
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see what the value of the function would be there.
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Okay well it would be 0 minus 2, that is negative 2.
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But 0 is not included there, so to show this on my graph I want to use an open circle.
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I am using the endpoint, and I want to make sure
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my graph is complete but 0 is not included.
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So this point cannot be included. Make sure you use an open circle there.
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Now I need one more point. It needs to have an x - coordinate that is less than 0,
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so let’s use this one we found earlier.
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When x is negative 1, the value of the function is negative 3. So (negative 1, negative 3).
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Again this portion of this piecewise function is a line, and we use this
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rule for x less than 0. So we are going to draw the portion of the line for x less than 0.
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Now that we have the graph for this piecewise function, let’s answer the last two questions.
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This one asked us for the range. The range is the set of values y is allowed to have.
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The first thing I want you to notice is that the graph has two pieces,
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and the y - values are different in those two pieces.
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For instance, here the y - values can be anything up to but not including negative 2.
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So to write that in interval notation is going to be negative infinity to negative 2.
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Use parentheses to signify the negative 2 is not included, by the open circle again.
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Now, the other piece of the graph has y - values that start at 1
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and they increase without bound.
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So we would say this is the interval from 1 to infinity,
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and we want to include the endpoint 1. Use a square bracket there.
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The last question is for the intercepts, and we are lucky on this one.
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We can sort of look at the graph and pick them out.
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Never does this piecewise function touch, never does its graph touch the x - axis,
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so there are no x - intercept. It touches the y - axis up here at positive 1,
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so the y - intercept is 1, and as we said there are no x - intercepts.
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Now be careful with the hole that is right here.
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There is no point where that hole is, so there is no y - intercept there.
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Be careful of that. Let’s look at one more example.
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Let’s look at another piecewise function.
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Notice that this one actually has three rules.
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So we are going to use the square root of x as our rule for x greater than or equal to 0.
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We are going to just use f of x equals four for the piece where x is just equal to 0.
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The third piece is negative 2x. This is going to be a linear piece to the graph.
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Okay, again let’s start by finding some specific function values.
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When x is 0, we are going to use the middle piece of the graph.
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It says the value of the function is 4 when x is 0.
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The next one is when x is 1, find the value of the function.
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1 is greater than 0, so I want to use this part of the function. This rule.
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f of 1 is going to be the square root of 1, which is 1.
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Finally let’s see what happens when x is negative 1.
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We want to find the value of the function here.
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Alright, negative 1 is between negative 3 and 0, so we want to use this rule.
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So negative 2 times negative 1 gives me positive 2. Now let’s look at the domain.
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Again to find the domain we always want to look at what values x is allowed to have.
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x can be anything bigger than 0, any real number bigger than 0. It could be 0,
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and it could be negative 3, and any real number between negative 3 and 0.
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The important thing here is that x is never anything smaller than negative 3.
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So our domain, our allowed x-values start at negative 3, include that using a bracket,
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and then they increase without bound so to positive infinity. Now let’s look at the graph.
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The first portion of the graph is the square root of x
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but only for values of x greater than 0.
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Well the square root of x is a basic function that you should know by now,
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and let me just sketch what that might look like down here.
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Okay, the square root of x has a point (0,0).
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It has a point at (1,1) since that is the square root of 1.
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If we go way out here at x equals 4, the positive square root of 4 is 2.
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It looks like this, and its domain is all the real numbers from 0 to infinity
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which is almost exactly what we needed. What is the difference?
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Well, zero. This does not allow me to include the 0,
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whereas our basic square root of x graph would.
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How can I change that to work for my piecewise function?
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I am just going to change this to a big open circle.
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That point should not be included since x equal to 0 is not used in this rule.
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Let me transfer that to the graph of my piecewise function.
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Now let’s look at the middle portion.
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This simply says the value of the function is 4, if x is equal to 0.
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This is a single point that has coordinates (0,4).
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The third portion of this function is defined by a linear rule.
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Notice this one has two different endpoints,
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so I want to make sure that I get the whole portion of the line
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between those endpoints and including this one.
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So let’s see. Let’s take x equal to negative 3.
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That is going to be the leftmost point on our function on our graph.
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If x is equal to negative 3, we are going to get negative 2 times negative 3.
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That is positive 6. Okay, so let’s plot this point.
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Alright, now I have decided that is the leftmost point on the graph.
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I know that the line or the portion of the line is over here,
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and I even know that it has a negative slope. But I need to know where it goes.
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Exactly what does it do over there?
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So what I am going to do is look at what happens at the endpoint.
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Now I know that the endpoint is not included, so I am going to
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take care of that on my graph by using an open circle again.
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If x were equal to 0, this would be 0. That means I would have the point (0,0),
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again it is not included so it is still going to stay this open circle,
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and the portion of the line I have will be right here.
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Now once we have the graph, we should be able to find the range and the intercepts.
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Okay so let’s look at the range. This is values that y is allowed to have.
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We are going to look for the smallest y - value. Well it is very close to 0
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but not equal to 0, and then y can be anything bigger up to 6 on this side.
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But look at this one. This one goes up forever and ever, so actually the range would be
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the interval that goes from 0 to infinity, without including the 0.
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Finally, the intercepts for this graph.
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There is a y - intercept at 4 since there is a point that touches the y - axis there,
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and the graph actually never touches or crosses the x - axis,
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There is an open circle there, that point is not included so there is no x - intercept.
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Continue to practice with piecewise functions until you can graph them comfortably
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and answer all of these questions.
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Now let's look some more at our basic functions.
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We would like to be able to graph more complicated functions based on our basic functions.
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We do this using transformations of the graphs. Let’s try a couple of examples.
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Notice this one. Let’s read it first. y equal x plus 1 cubed, minus 2.
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I do see one of our basic functions in there. It is a cubed function,
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and the first thing I would like to do is draw y equals x cubed.
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You remember what it looks like, right? Okay.
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It has the point (0,0). It contains the point (1,1), the point (negative 1, negative 1).
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Then I am going to go ahead and label this y equals x cubed
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to remind us that that is our basic function,
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and we are going to be working with this one
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and doing some transformations. What kind of transformations?
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Well let’s start right here. This is x plus 1 cubed, so you will need to think back.
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Yes that moves this horizontally. In fact it is going to go to the left one unit.
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Okay let’s remember that. We need to move this to the left one unit.
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If you look at this other number, this minus 2 is sort of outside of that basic function.
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This tells us whether to move the graph up or down.
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Because it is negative 2, we need to move the graph down two units.
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If you are very careful you can go ahead and do these at the same time in this problem.
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So we are going to take this graph y equals x cubed,
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take each point we have located and move it to the left one unit and down two.
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Let's start with this one that is at the origin.
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To the left one and down two will put that here.
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Let’s take this one, to the left one unit and down two will put that one here.
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Then finally the third point started out here.
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If we go to the left one unit and down two.
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Now we have to keep the shape of the same
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because all we are doing is sort of sliding this one around.
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This one is y equals x plus 1 cubed, minus 2.
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Our second example is y equals 3x squared,
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and the first thing we want to notice is that
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it does involve one of our basic functions x squared.
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Do you remember what x squared looks like? Right.
00:20:06.910 --> 00:20:13.450
It is the parabola. It is going to have the points (0,0), (1,1),
00:20:14.850 --> 00:20:18.900
and (negative 1,1), and remember it is this parabola shape.
00:20:22.300 --> 00:20:27.950
Let’s label this y equals x squared. This is the basic function we are dealing with.
00:20:28.050 --> 00:20:33.000
Now, what about this 3 that is multiplied in front of the x squared?
00:20:33.001 --> 00:20:37.400
What does that do to the graph? Right, it stretches it.
00:20:37.410 --> 00:20:41.900
It is going to make the y - coordinates grow faster for a given x - value.
00:20:43.400 --> 00:20:47.600
So if I want to graph y equals 3x squared and I know this,
00:20:47.610 --> 00:20:52.900
one way to approach that is to look at the y - coordinates of the points that I know
00:20:52.910 --> 00:20:57.000
and multiply them by 3 and get our new y - coordinates. Let’s try that.
00:20:57.450 --> 00:20:59.960
Remember, the x - coordinates are going to stay the same,
00:20:59.970 --> 00:21:02.850
because for a given x - value we are just stretching it.
00:21:02.860 --> 00:21:04.900
It is growing faster in this case.
00:21:05.500 --> 00:21:10.900
Again I want to start with this point at the bottom of the parabola.
00:21:12.850 --> 00:21:18.850
It has a y-coordinate of 0, and when I multiply that y - coordinate 0 by 3. It stays 0,
00:21:18.860 --> 00:21:23.750
so this point does not move. I like to think of that as kind of the anchor in this case.
00:21:26.150 --> 00:21:29.950
Now let’s look at this point. The coordinates of this point are (1,1).
00:21:29.960 --> 00:21:32.800
The y-coordinate is 1, that is what is important,
00:21:32.801 --> 00:21:35.250
and when I multiply that y-coordinate by 3
00:21:35.251 --> 00:21:39.950
I get the new value for 3x squared. 1 times 3 is 3.
00:21:42.350 --> 00:21:45.250
I have a new point, keeping the x - coordinate 1
00:21:45.260 --> 00:21:48.950
but I have made the y - value three times bigger.
00:21:49.750 --> 00:21:54.800
Okay, do that one more time. The y - coordinate for this point is at negative 1.
00:21:54.801 --> 00:21:59.960
No it is not, the y - coordinate is 1. If I take that y - coordinate 1,
00:21:59.970 --> 00:22:08.950
multiply it by the 3, I get 3 again. Keep the x the same. Plot the point (negative 1,3).
00:22:08.960 --> 00:22:13.950
What I want you to notice is that it is still parabola shaped,
00:22:13.960 --> 00:22:19.000
but it looks like someone stood at the top and pulled. It is stretched.
00:22:19.980 --> 00:22:21.970
Let’s try a few more examples.
00:22:23.900 --> 00:22:28.600
The next pair of examples deals with reflections of our basic graphs.
00:22:28.601 --> 00:22:33.930
Let's look at this first one. Alright, notice how it's out in front, it is
00:22:33.931 --> 00:22:36.190
like a negative 1 multiplied by our basic
00:22:36.191 --> 00:22:39.200
function: the absolute value of x.
00:22:39.201 --> 00:22:41.959
So, what effects should that have on the
00:22:41.960 --> 00:22:43.463
graph of the absolute value of x?
00:22:43.463 --> 00:22:47.950
Well, it's going to be negative 1 times the value of this basic function.
00:22:47.951 --> 00:22:49.466
That's the y - value, so we're going to
00:22:49.467 --> 00:22:52.430
multiply the y - values by negative 1 to get our new graph.
00:22:53.212 --> 00:22:55.219
What does that look like when you do it?
00:22:55.220 --> 00:22:58.469
Right, it just reflects it across the x - axis.
00:22:58.470 --> 00:23:01.211
So, let's go through the steps and graph this one.
00:23:01.212 --> 00:23:05.980
The basic function is the absolute value of x.
00:23:05.981 --> 00:23:07.750
Do you remember what it looks like?
00:23:07.751 --> 00:23:12.983
Good, it has the point (0,0), the point (1,1), and the point (negative 1,1).
00:23:15.734 --> 00:23:19.150
Now the absolute value of x, that's the one
00:23:19.151 --> 00:23:19.998
that's more V-shaped.
00:23:19.999 --> 00:23:27.250
It's made of pieces of two straight lines.
00:23:27.251 --> 00:23:30.257
So, there's your absolute value of x.
00:23:30.258 --> 00:23:34.000
Okay now, we talked about how the negative
00:23:34.001 --> 00:23:35.256
in the front is multiplying the
00:23:35.257 --> 00:23:37.500
y - coordinates by negative 1.
00:23:37.501 --> 00:23:40.266
It's going to reflect this across the x - axis.
00:23:40.267 --> 00:23:44.500
What's going to happen to this point at the bottom of the V?
00:23:44.501 --> 00:23:47.750
Right, nothing because the y - coordinate is 0,
00:23:47.751 --> 00:23:51.250
and multiplying that by negative 1 is still 0.
00:23:51.251 --> 00:23:53.028
Not only that, if you're reflecting
00:23:53.029 --> 00:23:57.777
it across the x - axis, it's on the x - axis. It doesn't move.
00:23:57.778 --> 00:24:02.200
What about this point here, it has the coordinates (1,1).
00:24:02.201 --> 00:24:05.293
It's got to come down here, so the x doesn't change.
00:24:05.294 --> 00:24:07.542
I want it to just be a reflection.
00:24:07.543 --> 00:24:11.047
The y - coordinate needs to be the negative of what it was.
00:24:11.048 --> 00:24:13.800
Okay and let's draw this piece.
00:24:13.801 --> 00:24:18.557
Notice how it looks just like this reflected across the x - axis.
00:24:18.558 --> 00:24:21.057
Same thing is going to happen on the other side.
00:24:21.058 --> 00:24:23.400
This has a y - coordinate of 1.
00:24:23.401 --> 00:24:26.307
We're going to make it negative because of this.
00:24:26.308 --> 00:24:31.400
It's going to give you a new point at (negative 1, negative 1).
00:24:31.401 --> 00:24:39.000
You can draw the other piece. This is y equals negative absolute value of x.
00:24:39.001 --> 00:24:43.577
It's just a reflection of our basic function across the x - axis.
00:24:43.578 --> 00:24:50.577
One more: I bet you're worried about the negative under the radical.
00:24:50.578 --> 00:24:53.577
Well don't worry, we can take care of that.
00:24:53.578 --> 00:24:58.846
It's not nearly as big a problem as you might think. What does it do?
00:24:58.847 --> 00:25:02.400
If you think about the square root of x,
00:25:02.401 --> 00:25:06.900
our basic function, we needed to use x - values that are positive.
00:25:06.901 --> 00:25:08.856
This changes all of that because now, if I
00:25:08.857 --> 00:25:13.856
put in a positive number for x, let's try 2,
00:25:13.857 --> 00:25:16.600
I'd get the square root of negative 2. We can't do that.
00:25:16.601 --> 00:25:18.104
It's not a real number, right.
00:25:18.105 --> 00:25:20.868
But on the other hand, if I pick a
00:25:20.869 --> 00:25:22.618
negative number now, like maybe I don't know
00:25:22.619 --> 00:25:26.350
negative 2, you get the square root of a negative, negative 2.
00:25:26.351 --> 00:25:28.400
That's the square root of positive 2, and this is good.
00:25:28.401 --> 00:25:31.875
This is a real number, something we can handle okay.
00:25:31.876 --> 00:25:34.850
If you think about the domain of this,
00:25:34.851 --> 00:25:39.000
instead of 0 and positive numbers, I need to
00:25:39.001 --> 00:25:45.124
use negative numbers and maybe 0. So what does all that mean?
00:25:45.125 --> 00:25:47.374
It means that this negative in front of the
00:25:47.375 --> 00:25:50.875
x takes the graph of the square root of x,
00:25:50.876 --> 00:25:54.150
and it's going to reflect it across the y - axis.
00:25:54.151 --> 00:25:58.150
Let's make this clear by actually drawing those graphs.
00:25:58.151 --> 00:26:01.000
What does the square root of x look like?
00:26:01.001 --> 00:26:08.661
Okay, it has a point at (0,0). It has a point at (1,1), and I'll put a third
00:26:08.662 --> 00:26:20.500
point: the square root of 4 is 2.
00:26:20.501 --> 00:26:26.661
Okay alright, so now we want to take care of this reflection.
00:26:26.662 --> 00:26:30.160
As I said, it's a reflection across the
00:26:30.161 --> 00:26:34.300
y - axis, so let's start with this point.
00:26:34.301 --> 00:26:36.693
It doesn't move, it's already on the y - axis.
00:26:36.694 --> 00:26:39.943
So when you reflect it, it stays where it is.
00:26:39.944 --> 00:26:43.450
This point needs to come reflect over to here,
00:26:43.451 --> 00:26:45.450
so where will its coordinates be?
00:26:45.451 --> 00:26:46.800
Well, it's going to still have a
00:26:46.801 --> 00:26:49.400
y - coordinate at 1, but it's x - coordinate
00:26:49.401 --> 00:26:54.460
needs to be a negative 1, and then one more point.
00:26:54.461 --> 00:26:59.500
This has an x - coordinate of 4, a y - coordinate of 2.
00:26:59.501 --> 00:27:03.450
The y - coordinate doesn't change when it's reflected across the y - axis.
00:27:03.451 --> 00:27:15.650
The x - coordinate does. It becomes negative, so this is now (negative 4,2).
00:27:16.961 --> 00:27:20.700
Take a look at the difference in these two examples.
00:27:20.701 --> 00:27:24.235
Here the negative is in front of, outside
00:27:24.236 --> 00:27:25.985
of our basic function, and that shows you
00:27:25.986 --> 00:27:28.235
it is a reflection across the x - axis.
00:27:28.236 --> 00:27:33.220
Here, the x itself inside is negative, and
00:27:33.221 --> 00:27:37.200
that is the one that reflects across the y - axis.
00:27:38.751 --> 00:27:41.750
This next pair of examples asks us a totally
00:27:41.751 --> 00:27:45.250
different question, though very much related to what we have just done.
00:27:45.251 --> 00:27:49.000
Here you are given the graph, and the
00:27:49.001 --> 00:27:52.000
instructions are to choose which of these is
00:27:52.001 --> 00:27:54.500
the function that matches the graph.
00:27:54.501 --> 00:27:58.271
Okay so we are going to do this by process of elimination.
00:27:58.272 --> 00:28:02.278
The first thing I need to do is figure out which basic function is involved.
00:28:02.279 --> 00:28:04.528
Okay so let's look at this one.
00:28:04.5289 --> 00:28:10.528
Which basic function looks like this, has this basic shape?
00:28:10.529 --> 00:28:14.291
It is a tricky one because you actually have two possibilities.
00:28:14.292 --> 00:28:26.302
Is it x cubed which looks roughly like this, or is it the cube root of x?
00:28:26.303 --> 00:28:28.801
Remember that's the one that looks like it's
00:28:28.802 --> 00:28:35.100
kind of turned on its side compared to that.
00:28:35.101 --> 00:28:37.550
You're right, so this is the one we want to use.
00:28:37.551 --> 00:28:40.551
This is definitely a cube root of x function.
00:28:40.552 --> 00:28:44.552
Okay so we can eliminate some things right away.
00:28:44.553 --> 00:28:48.900
Let's just cross these out. These have x cubed in them. They're not right.
00:28:48.901 --> 00:28:52.300
Now, let's look for something else.
00:28:52.301 --> 00:28:56.575
Alright it doesn't look like it's reflected, right?
00:28:56.576 --> 00:29:00.075
It still has the same, it's still increasing,
00:29:00.076 --> 00:29:04.574
still increasing, so it's not reflected.
00:29:04.575 --> 00:29:09.275
That doesn't really help me any, so let's look for something else.
00:29:09.276 --> 00:29:15.825
This one, right, has sort of this anchor point at (0,0), at the origin.
00:29:15.826 --> 00:29:17.855
This one does not so. Let's see if we can figure out where
00:29:17.856 --> 00:29:20.355
this point, how has it been moved?
00:29:20.356 --> 00:29:26.500
If we started out at the origin, then we go to the right 2 units and up 1.
00:29:26.501 --> 00:29:30.105
Okay the up 1 is the easier part to use, so let's look at these.
00:29:30.106 --> 00:29:33.873
Up 1, I need to see the plus 1 out here, and I do.
00:29:33.874 --> 00:29:38.122
Again it doesn't help me much, so I'm forced to think about is it moved left or
00:29:38.123 --> 00:29:44.128
right, and how does that look in the equation or in the function?
00:29:44.129 --> 00:29:48.128
Okay, so we said it's moved to the right 2.
00:29:48.129 --> 00:29:54.300
Would I see a plus 2 in the function or a minus 2 in the function?
00:29:54.301 --> 00:30:00.650
What do you think? Right, minus 2.
00:30:00.652 --> 00:30:04.651
You want to be careful with that. It's kind of tricky.
00:30:04.652 --> 00:30:12.200
Let's try another one. What basic function do you see here?
00:30:12.201 --> 00:30:18.200
The overall shape looks to me like it's a parabola. It is y equals x squared
00:30:18.201 --> 00:30:23.415
So, the basic function is y equals x squared.
00:30:23.416 --> 00:30:27.675
All of these have an x squared in them, so that doesn't help me eliminate anything,
00:30:27.676 --> 00:30:32.180
but it's a good start. One thing I notice right away is it's flipped over.
00:30:32.181 --> 00:30:37.679
It looks like it's been reflected across the x - axis.
00:30:37.680 --> 00:30:47.750
Well all of those also were reflected across the x - axis, so I need to keep looking.
00:30:47.751 --> 00:30:51.430
This point is now at the top of the parabola since it's been reflected.
00:30:51.431 --> 00:31:00.750
Usually it would be at (0, 0), so this one's been moved up 3 units.
00:31:00.751 --> 00:31:05.000
Which of these looks like it might have been moved up 3 units?
00:31:05.001 --> 00:31:12.220
This one? No, no, no. It's inside, so this is actually moved to the left 3.
00:31:12.221 --> 00:31:16.500
That's wrong. Good, we can eliminate something.
00:31:16.501 --> 00:31:21.727
This one has this plus 3. That's a possibility.
00:31:21.728 --> 00:31:23.229
Let's skip C for just a minute.
00:31:23.230 --> 00:31:27.400
Let's look at D: plus 3 out there, that's good.
00:31:27.401 --> 00:31:29.900
That's been moved up 3. These are both good choices.
00:31:29.901 --> 00:31:33.741
Now, what about C? Ah yes, you're paying attention.
00:31:33.742 --> 00:31:37.000
This isn't really written in the right form, is it?
00:31:37.001 --> 00:31:41.741
This is really negative 2 x squared minus 6.
00:31:41.742 --> 00:31:47.255
That should go down 6 units, not up 3. C is out.
00:31:47.256 --> 00:31:51.755
Okay, I've narrowed it down to B and D.
00:31:51.756 --> 00:31:57.765
They look remarkably similar, except what? Except this 2.
00:31:57.766 --> 00:32:01.018
There is no 2 over here, and what is it that 2 does?
00:32:01.019 --> 00:32:03.018
Right, it stretches the graph.
00:32:03.019 --> 00:32:05.515
Okay, so all we have to decide now is:
00:32:05.516 --> 00:32:11.523
is this graph stretched, or is it a regular x squared graph? What do you think?
00:32:11.524 --> 00:32:19.000
I agree. I think it's stretched too because if we look at y equals x squared,
00:32:19.001 --> 00:32:25.042
look at how these other points are positioned relative to the bottom of the vertex.
00:32:25.043 --> 00:32:29.298
It's 1 unit to the right and 1 unit up. Now I know it's reflected,
00:32:29.299 --> 00:32:33.500
but here it's 1 unit to the right and 2 down.
00:32:33.501 --> 00:32:42.000
That's stretched. So my answer has to be B.