WEBVTT
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In this section, we're going to learn how to graph piecewise-defined functions, and we're also going to discuss
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shifting and reflecting graphs of functions. Let's begin with piecewise-defined functions. A piecewise-defined
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function is a function defined by two or more expressions. This is an example of a piecewise function.
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It says that f of x is defined as three when x is less than negative two and f of x is defined as negative one when x
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is greater than or equal to negative two. Let's graph this piecewise-defined function on our coordinate plane.
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Let's think for a moment about the ordered pairs that each of these pieces of this function represent.
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If I choose an x-value, for instance, let's say negative five. If x is negative five, then the piece of this function that I
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must use to define the function is f of x equals three because this is the piece of the function
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that is defined when the x-value is less than negative two. Thus, the y-value, or the function value here, is three.
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If I choose another x-value that is less than negative two, the function value, or the y-value, will also again be three.
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So negative four, three. Negative three, three. These are ordered pairs that are on this function that are represented
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by this top piece. Now once x gets to negative two, this top piece is no longer the definition of the function
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because this top piece is only defined as the function value when x is less than negative two.
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When x is greater than or equal to negative two, the function is defined as negative one.
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So when x is negative two, the y-value, or the function value, is negative one.
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And for any x-value that is greater than negative two, the function is so defined as
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f of x equals negative one. So I can choose any other x-value greater than negative two,
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and the y-value will be negative one. Let's plot these points. Here we have negative five, three.
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We have negative four, three. We have negative three, three. We can see that these ordered pairs are lying on a horizontal line.
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We should also recognize the equation of a horizontal line is of the form y equals a number. In this case,
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f of x is equal to three represents a horizontal line with the y-values of three. So this top piece
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is a horizontal line for all x's that are less than negative two. So for all of these
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x values that are less than negative two, the y-value is three. But when I get to negative two,
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I'm going to put an open circle there because the function value is not defined as three when x is negative two.
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Once x becomes negative two, the function now is defined as the y-value equal to negative one.
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And again, this is a horizontal line y equals a number. All of the function values are negative one
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when the x's are greater than or equal to negative two. So I can see these ordered pairs and
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an infinite number more ordered pairs all who have the same y-value of negative one,
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beginning with x equals negative two. We'll put a closed dot at negative two to show that when
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x is equal to negative two, this is the value of the function. And for all of these other x-values,
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the y-value is negative one. This is the graph of the piecewise function defined here.
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Let's take a look at another piecewise-defined function. This function is defined as three x plus
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one, when x is less than or equal to zero, and it is defined as five minus x when x is greater than
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zero. So let's begin by thinking about this first definition of this piecewise-defined function.
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You may recognize it in the form of m x plus b. So this is a linear function or a graph
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of a linear function for this first piece. It has a slope of three and a y-intercept of one.
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We're only going to graph this line for x's that are less than or equal to zero.
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The y-intercept of one is where we'll start, which is at an x-value of zero. And because zero
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is included in this function, less than or equal to, we put a closed dot at the y-intercept of one.
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The slope here is three. Now recall when we graph a line, and we have a slope, we can
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go up three for this and to the right one. Or we can go down three and to the left one
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because negative three divided by negative one is equivalent to three over one, which is
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our rise over run. Because I want to graph the values where x is less than or equal to zero,
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we'll go down three and left one from our y-intercept, giving us the second point here.
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I can graph a third point to make drawing the line a little simpler,
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and this would be the third point on our graph. I'll connect these points with a straight line.
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And now we'll discuss this bottom portion of our piecewise-defined function.
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This is telling us that the function is defined as five minus x when x's are greater than zero.
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Again, recognize this is in the form or can be in put in the form m x plus b,
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where m would be negative one and b would be five. So we have a y-intercept of five
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and a slope of negative one for this bottom piece of our piecewise-defined function.
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My y-intercept is five, but recall that this function is only valid for x is greater than zero.
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So at zero we will have an open circle as opposed to a closed circle at five. Our slope again here is negative one,
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so we can have a rise of negative one, meaning down one, and to the right one.
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Or we can have a rise of positive one and to the left one. These are equivalent.
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Because I want to graph to the right of this open circle, I'm going to choose to go down one and
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to the right one because to the right of x equals zero are the values where x is greater than zero.
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Down one and to the right one. Down one and to the right one. And now I have three points,
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making it easy to draw the line that goes through these points, and this is the graph of the piecewise-defined function.
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Now let's consider the function f of x equal to the absolute value of x.
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We are going to complete the table and then use the data in the table to graph the function.
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Let's begin with the x-value negative two. When x is negative two, f of x is equal to
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the absolute value of negative two, and the absolute value of negative two is positive two.
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When x is negative one, f of x is the absolute value of negative one, which is positive one.
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When x is zero, f of x is the absolute value of zero, which is zero. When x is one, f of x is the
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absolute value of one, which is one. And when x is two, f of x is the absolute value of two,
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which is two. Let's plot these ordered pairs on our graph. We'll begin with negative two, two.
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Next, we have negative one, one. Zero, zero which is here at the origin.
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We have one, one, and two, two. Now as we think about the shape of this graph,
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let's think about what would happen if we extended this table in both directions. For the x-
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values that continue greater than two, these are all positive values, and the absolute value
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of a positive value is a positive value. In this case, the x- and the y-values will be the same.
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So this right-hand side of the graph will continue in a linear format with a slope of positive one,
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where we go up one and over one. All of the x-values here have the same y-value
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for their function value. And so the graph of this side of the absolute value of x
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looks like this. It looks like a linear function with a slope of one. Now on this side where we have negative x-values,
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notice that the function value has the opposite sign of the x-values. So where the x-values are negative,
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but when I take the absolute value of those negative numbers, I get a positive value. And so these y-values
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are the opposite sign of the x-values, and they continue in that same pattern on the left side of zero.
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So as I graph this side of the absolute value of x, I can see that this side, left of zero,
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looks like a linear function that has a slope of negative one, down one and to the right one for
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each of these ordered pairs. The overall shape of this absolute value function is the shape of a v.
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However, we can define this absolute value function as a piecewise-defined function.
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Since we notice that the right side looks like a linear function with a slope of
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one and the left side looks like a linear function with a slope of negative one,
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then we can define f of x which is the absolute value of x as a
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piecewise-defined function where f of x is equal to x when x is greater than zero
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and f of x is equal to negative x when x is less than or equal to zero.
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Both of these definitions the absolute value of x and this piecewise-defined
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function represent the same function and the same graph, the absolute value of x.
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Consider the two functions f of x equals the absolute value of x and g of x equals the absolute
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value of x plus three. Complete the table, and then graph the functions on the same axes.
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Then we'll describe the relationship between the graph of f and the graph of g. Well, we just
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recently graphed f of x equals the absolute value of x. So here are those function values completed.
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And here's the graph of f of x equals the absolute value of x. Recall that that was the shape of a v
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that could be also written as a piecewise-defined function. Now let's do the table for g of x.
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If x is negative two, then g of negative two is the absolute value of negative two,
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which is two, plus three, giving us five. If x is negative one, g of negative one is the absolute value of negative one,
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which is one, plus three, or four. When x is zero, g of x is the absolute value of zero, which is zero, plus three.
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When x is one, g of one is the absolute value of one, which is one, plus three. And if x is two, g of two is the
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absolute value of two, which is two, plus three which is five. Let's plot these ordered pairs. Negative two, five.
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Negative one, four. Zero, three. One, four. And two, five. We notice the same type of pattern showing up
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on this red graph of g of x as we see on the green graph f of x. In other words, the right side appears to be
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linear with a slope of positive one. And the left side also appears to be linear. It has a slope of negative one.
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These two graphs, f of x and the graph of g of x, have the same shape. They both still
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have the shape of the v, the absolute value of x shape. We're going to describe the relationship
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a little more precisely now. While they have the same shape, notice that the red graph is higher
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or vertically shifted from the green graph. In fact, every ordered pair is one, two, three units
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higher than the corresponding ordered pair of the green graph. Every ordered pair is three
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units higher. So we say that the red graph g of x has been vertically shifted three units.
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This was the graph of g, or g of x, is a vertical shift of three units from the graph of f.
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As we noticed in the last problem, g of x was a vertical shift three units up from f of x,
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and the difference in the equations of f of x and g of x was this plus three.
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Let's look at our rules for vertical shifts. Let c be a positive number.
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The graph of g of x equals f of x plus c is the graph of y equals f of x shifted c units upward.
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This makes sense based on what we saw here. Our c value was three.
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Therefore, since we had a plus three, this graph shifted up three units.
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The graph of g of x equals f of x minus c is the graph of y equals f of x shifted c units downward.
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So if this problem had been the absolute value of x minus three,
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we would expect to see the absolute value of x graph shifted down three units.
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Let's do one more example. Let's graph h of x equals the absolute value of x minus five. Here
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we know that f of x is the absolute value of x. That is our green graph. And here we
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are subtracting five from f of x, so h of x can be defined as f of x minus
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five. Meaning that all of the ordered pairs on f of x are going to be shifted downward five units.
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I can choose any of the ordered pairs to make this shift. I'll start here at negative two,
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two. If I shift this down five, this puts me at the ordered pair negative two, negative three.
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I can shift this ordered pair, which is at the origin, down five, and I will be at negative five
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on the y-axis. And if I shift this ordered pair, which was at two, two,
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down five units, then I will be at two, negative three. And it's going to maintain the same shape
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as the graph of f of x. So it will still have the v shape. So on the right side, we will expect this
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linear shape with a slope of one. And on the left side, we will have this linear shape on the graph with a
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slope of negative one. Every ordered pair from the green graph has been shifted down five units
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to obtain the graph of h of x which is the absolute value of x minus five.
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Consider the functions f of x equal the absolute value of x and g of x equals the absolute value
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of x minus two. We're going to complete the table and then graph f and g on the same axes.
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And we will describe the relationship between the graph of f and the graph of g. We've
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already graphed the values for f of x equal to the absolute value of x. And here is the
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graph that we've seen in previous examples. Now let's complete the table for g of x.
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When x is negative two, g of negative two will be the absolute value of negative two minus two,
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which is the absolute value of negative four. The absolute value of negative four is positive four.
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g of negative one is the absolute value of negative one minus two,
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or the absolute value of negative three which is equal to three. g of zero is the absolute value of
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zero minus two, which is the absolute value of negative two. The absolute value of negative two is two.
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g of one is the absolute value of one minus two, which is the absolute value of negative one,
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which is one. g of two is the absolute value of two minus two, which is zero, and the absolute value of zero is zero.
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g of three is the absolute value of three minus two, which is the absolute value of one,
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which is one. And finally, g of four is the absolute value of four minus two,
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which is the absolute value of two, which is equal to two. Now let's plot these points on our graph.
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We have negative two, four. Negative one, three. Zero, two. One, one. Two, zero. Three, one. And four, two.
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Connecting these three points with a line of slope one gives us the right side of this absolute value
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graph. And connecting this left side with a straight line and a slope of negative one
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gives us the left side of this graph. So here you can see the graph of f and the graph of g.
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Now let's see how they compare to each other. They have the same shape, but all of the ordered pairs
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on f have been shifted to the right. By how many units? two. Notice that our function was g of
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x equals absolute value of x minus two, and every ordered pair here has been shifted to the right
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two units from the green graph, which was f, to the red graph, which is g. So we say that g
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is a horizontal shift of two units to the right when compared to graph f.
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Reflecting on our last examples of horizontal shifts, we can now look at the rules for
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horizontal shifting. Let c be a positive number, the graph of g of x equal to f of x plus c
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is the graph of y equals f of x shifted c units to the left. Now this may seem a bit counterintuitive.
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Here we are adding a positive number to x, but yet we are shifting
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to the left. We can think about why that may be. We're trying to get the same output value
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of f of x. Well, we need a value that is c units less than the given input of f of x.
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The graph of g of x equal to f of x minus c is the graph of y equals f of x shifted c units to the
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right. Again notice, we have x minus a positive number, and we're shifting to the right. Again,
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we're looking for the same output value with a minus sign inside of our grouping symbols here,
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for x minus a positive number. And so in order to get that, we have to shift this graph to the right.
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Let's take a look at these two examples. We're going to start by graphing the function h of x
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equals the absolute value of x plus four. Which rule applies here? Well, we're adding a positive
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number to x inside of our grouping symbols, or the absolute value symbols. This represents
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f of x plus c. Therefore, we're going to shift c units to the left, where c is four.
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Here's our function f of x equal to the absolute value of x, and we're going to shift it c units or
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four units to the left. So starting with our vertex we will count four units to the left.
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And again, the right side of this absolute value function is going to look the same
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it's going to have a linear shape with a slope of one. And you can see from this graph
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that, or this portion of the graph, that these points are all shifted one, two, three,
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four units to the left. Let's complete the other side of the graph now. The left side of the graph
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has this linear shape with a slope of negative one. All of the points on this graph of f of x have been shifted to the left
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four units. This is the graph of h. Let's now take a look at k of x. k of x
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is the absolute value of x minus five plus three. So we have to apply two of our rules here. First,
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inside of our absolute value symbols, we see x minus a positive number, so
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our second rule here will apply for shifting. This is the graph of f of x minus c. So this
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minus five inside of our absolute value symbols will shift the graph to the right five units.
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Recall that when we add a number outside of the grouping symbols, or in this case
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the absolute value symbols, then we shift either up or down. When we have a positive value c,
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we shift up. So in addition to the absolute value of x graph being shifted five units
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to the right, we're also going to shift this graph three units up. Let's start with our absolute value function,
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and we will shift five units to the right and three units up. And now we'll complete both sides of our graph. To the right
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and then to the left. Every point on the original graph of f is now shifted five units to the right
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and three units up. So this is the graph of k. Consider the functions f of x equals the absolute value of x and g of x
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equals the negative absolute value of x. We're going to complete the table and graph f and g on the same axis.
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And then we're going to describe the relationship that we notice between the graph of f and the graph of g.
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The graph of f of x equals the absolute value of x is already graphed for us here,
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and the table is completed. Let's focus on g of x. First, when x is negative two,
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we're looking for g of negative two. This is the negative absolute value of negative two.
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The absolute value of negative two is positive two, and then we're multiplying
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that value by negative one, giving us a value of negative two. g of negative one is the negative
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absolute value of negative one. The absolute value of negative one is positive one.
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Then multiplied by negative one gives us g of negative one equal to negative one.
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g of zero is the negative absolute value of zero. The absolute value of zero is zero, and zero times anything is still zero. So
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g of zero is zero. g of one is the negative absolute value of one. That is negative one.
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And g of two is the negative absolute value of two, which is negative two. And now let's plot these ordered pairs on our
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graph. We'll begin with negative two, negative two. Next, we have negative one, negative one. Zero, zero.
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One, negative one. And two, negative two. Let's connect these points to form our graph. On the right side of our graph,
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we have this linear shape with a slope of negative one. And on the left side of our graph, we also have a linear
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shape, but this side has a slope of positive one. Now let's compare these two graphs of f and g.
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All of the ordered pairs on f have positive y-values, while all of the ordered pairs on g
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have negative y-values. This appears to be a mirror reflection of f across the x-axis. So we say that g is a
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reflection of f about the x-axis. Reflections about the x-axis. The graph of g of x equal to negative f of x is the graph of y
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equals f of x reflected about the x-axis. Let's look at this next example. We are to graph
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h of x equal to the negative absolute value of x plus four minus two.
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Let's look at the different changes to the graph of f of x that we see here. In h of x,
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I notice that there is a negative outside of the grouping symbols or outside of the absolute
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value symbols. There is a plus four inside of the absolute value symbols. And there is a minus two
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outside of the absolute value symbols. If you were to evaluate this function for a given value of x,
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where would you begin? You would begin by adding four to the value of x. That's
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what we'll begin with this transformation. The plus four inside of the absolute value symbol
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means that we're going to shift the graph of the absolute value of x to the left four units.
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The next thing we would do if we were evaluating this function is to multiply that value by negative one.
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This negative in front of our absolute value symbols is a reflection of the graph of f of x about the x-axis.
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And the last thing that we would do if we were evaluating this function for a given value of x is subtract two.
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And this is a vertical shift of negative two. So we're going to shift the graph of f of x down two. So we have
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a horizontal shift left four, a reflection about the x-axis, and a vertical shift of negative two.
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Let's put that on our graph. Beginning here with our vertex, we'll shift left four.
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A reflection about the x-axis does not change that because zero is neither positive nor negative.
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And we have minus two of a vertical shift. I could take any other ordered pair and make the same transformation,
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or knowing the shape of this graph, I could complete the graph at this point. Let's choose another ordered
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pair. Here's a point on our parent graph one, one. I would shift this point to the left four.
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It has a reflection about the x-axis, meaning that the y-value would now be
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negative one instead of positive one. And I have a vertical shift of negative two,
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which will shift us down two units from there. And now you can clearly see
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that this side of the graph will have this linear shape with a slope of negative one.
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And the other side of the graph will also have a linear shape. It will have a slope of positive one.
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And this is the graph of h, where every ordered pair from the graph of f has the same
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transformation shifted left four, reflected about the x-axis, and shifted down two units.