WEBVTT
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In previous sections, we have studied quadratic equations. In this video, we're going to discuss
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quadratic functions and their graphs. A quadratic function is a function that can be written in
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the form f of x equals a x squared plus b x plus c where a, b, and c are real numbers
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and a is not equal to zero. We're going to begin with the quadratic function
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y equals x squared. And if we want to graph this function, we can choose values of x,
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calculate the values of y, and then plot those points on our graph. So let's do that.
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If x is equal to negative three, then y is equal to negative three squared which is nine.
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If x is equal to negative two, then y is equal to negative two squared
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which is four. For an x-value of negative one, y will equal negative one squared
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which is one. For x equals zero, y is equal to zero squared or zero. For x equal to one,
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y is one squared which is one. For x equal to two, y is two squared which is four.
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And for x equal to three, y is three squared which is nine. Now let's plot
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these as ordered pairs on our graph. We have the ordered pair negative three,
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nine. I'm going to use a scaling here of one for each tick mark. So that would mean that this
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x-value here is one, two, three, etc. And in the y-values, we will go one, two, three, also. So
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negative three, nine would be negative three on the x and positive nine on the y, putting
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our ordered pair here. The ordered pair negative two, four will be negative two on the x direction
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and four from the y direction. Negative one, one is here. Zero, zero is at the origin.
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Then we have one, one so positive one on the x, positive one on the y goes here.
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Two, four. Positive two in the x, and up four in the y direction. And three, nine. Three,
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nine would be here. Let's connect these points with a smooth curve.
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This is the graph of the function y equals x squared. The shape of this function
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is called a parabola. You'll notice that this parabola opens in the upward direction.
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And it has what we call a turning point. The graph is decreasing here until it gets to x equals zero,
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and then it begins to increase. This turning point at zero, zero is called the vertex.
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So the vertex is located at zero, zero. In addition, you might notice the symmetry of
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this graph. This graph is symmetric about the y-axis. This line here, the line x equals zero,
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is called the axis of symmetry. So x equals zero is the axis of symmetry. Throughout this video,
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we're going to graph variations of this parent graph, y equals x squared, and we're going to
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identify characteristics of the graph such as the direction in which the parabola opens,
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where the vertex is located, and where the axis of symmetry is located.
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Let's continue graphing variations of the function y equals x squared. Consider the function
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f of x equals negative two x squared. We're going to complete this table and graph the
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function f of x equals negative two x squared. If x is equal to negative two,
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then f of x will be negative two times negative two squared. Negative two squared is four,
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and when I multiply four times negative two, I get negative eight. When x is negative one, we have
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f of x equal to negative two times negative one squared. Negative one squared is positive one,
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and when I multiply positive one times negative two, we get negative two.
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When x is zero, we have negative two times zero squared. Anything times zero is zero. When x
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is one, f of x is negative two times one squared. That will be negative two times one, or negative
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two. And when x is two, f of x is negative two times two squared which will be negative two
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times four which is equal to negative eight. Let's plot these as ordered pairs on our graph.
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Notice that I already have the graph y equals x squared written here in green
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so that we can compare the two graphs after we graph f of x equals negative two x squared. First,
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we have the ordered pair negative two, negative eight. Negative two on the x-axis,
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and negative eight on the y-axis will be here. Negative one on the x-axis,
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and negative two in the y-direction will be here. Zero, zero is at the origin. One, negative two is here.
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And two, negative eight is here. Let's connect these ordered pairs or these points with a smooth curve.
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This is a sketch of the graph f of x equals negative two x squared.
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Now the next part of this question asks us to state the vertex and the
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axis of symmetry of the graph of f. The vertex is here at the turning point.
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Notice it is the highest point of this graph which is zero, zero. The axis of symmetry is still located here on the y-axis
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which is the line x equals zero. Next, we're asked to describe the transformations required to graph f starting from the
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graph of y equals x squared. Well, first you may notice that this graph is reflected across the x-axis, so
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there is a reflection. Our new graph f of x opens down instead of up. So we say we have a reflection
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over the x-axis. You may also notice that the y-values are farther away from the x-axis in
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the graph of f of x than they were from the graph y equals x squared. For example, this
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ordered pair on y equals x squared was located at one, one. When x is one, the y-value was one.
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On our graph of f of x equals negative two x squared, we are now two units away from the x-axis
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when x is one. When x is two on the parent function, y equals x squared, our y-value is four.
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But on the graph of f of x equals negative two x squared, I am not four units away from the x-axis,
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but I am now eight units away from the x-axis in the negative direction. That is a vertical stretch
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of a factor of two. So we'll say that we have a vertical stretch by a factor of two. Meaning that each of the
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y-values are two times farther away from the x-axis than they were from the parent graph.
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In this next example, let's consider the two functions f of x equals x squared and
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g of x equals x squared minus two. We're going to begin by graphing f and g on the same axis.
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Notice, we've already discussed the graph of y equals x squared, and so I have used that
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information to complete the table here for f of x equals x squared, and its corresponding graph
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is here on our coordinate plane. Now let's use g of x equals x squared minus two to determine the
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values the y-values for these corresponding x-values. When x is equal to negative two,
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g will equal negative two squared minus two. Negative two squared is four, and four minus
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two is two. When x is negative one, g of x will be negative one squared minus two.
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Negative one squared is positive one, and one minus two is negative one.
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When x is zero, g of zero will be zero squared, which is zero, minus two,
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giving us a value of negative two. When x is one, g of one will give us one squared minus two
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which will be one minus two, or negative one. And when x is two, we'll have g of two
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equal to two squared minus two which is four minus two, or two. Let's plot these ordered pairs
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for the graph of g. We have the ordered pair negative two, two. Negative two, two.
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We have the ordered pair negative one, negative one. That's here. Zero, negative two. One, negative one.
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And two, two. And now we'll connect these points with a smooth curve. The next part
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of our problem asks us to state the vertex and the axis of symmetry of the graphs of f and g.
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I'll put f here and g here. Well, for the graph of f, that was our parent function f
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of x is equal to x squared, we notice that the vertex is at zero, zero.
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And for g notice that the vertex is here at zero, negative two. The axis of symmetry for both of these graphs
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is the y-axis, or the line x equals zero. Both graphs are symmetric about the y-axis.
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The last question asks us to describe the graph of g as a transformation of the graph of
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f. Well, first let me label g on this graph. This is g of x equal to x squared minus two.
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And our green graph was f of x equals x squared. Notice that each of these points of the red graph,
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g of x, are two units lower. The y-value is two units less than the y-value of the parent graph.
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This y-value is at one. This y-value is two units lower. The same here and here. In fact, for any
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point that's on the parabola the red graph is two units lower than the corresponding points on the
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green graph. So we say that the transformation of g is that it is a vertical shift down two units.
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All quadratic functions can be graphed using transformations of the graph
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y equals x squared. In order to precisely graph a quadratic function, you need a minimum of three
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points, one of which is the vertex. You can see here that I have graphed our equation,
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or our function, y equals x squared using three points, one of which is the vertex.
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Now let's take a look at this example. We're going to graph the function f of x equals one-fourth
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times x minus four squared plus three. And we're going to graph it, instead of using a table
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of values, we're going to use transformations of this function y equals x squared. So let's
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recall what we have learned previously about transformations. When we add or subtract inside
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of grouping symbols, the transformation is a horizontal shift left or right. So this minus four
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inside grouping symbols will require us to shift our graph to the right four units.
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The one-fourth that is multiplied outside of these parentheses is either a vertical stretch
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or a vertical compression. Because this number is a value between zero and one, less than one,
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then this is a vertical compression. So all of our y-values will be multiplied by one-fourth.
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They'll be a fourth of the distance from the x-axis as our parent graph originally has.
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And then our plus three at the end of the function, or outside of the grouping symbols,
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is a vertical shift. Adding three here will be a vertical shift up three. So we'll begin
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with the points that we have identified here, and we will apply these three transformations.
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Let's start with our vertex. We're going to shift our vertex to the right four units. Multiplying
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this y-value by one-fourth still gives us a value of zero, and then adding three puts us here.
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So this ordered pair is four, three. Now let's move this ordered pair from our original graph,
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which was at negative two, four. We'll begin with the horizontal shift to the right four units.
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Next, we'll multiply this y-value by one-fourth. The y-value here on the parent graph is four. When
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we multiply that by one-fourth, the new y-value is one. And then we have a vertical shift of positive
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three, so this will shift this point back up three units, putting us here at our new point two, four.
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And our third point, which was originally at two, four, will also follow these transformations.
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We'll move this point to the right four. We'll multiply the y-value by one-fourth,
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which brings us to one, and then we will add three for our vertical shift,
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bringing us to six, four. And now we'll connect these three points with a smooth curve.
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This is the graph of f of x equals one-fourth x minus four squared plus three. Now let's
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state the vertex and the axis of symmetry of the graph of f. The vertex is here at four, three.
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And the axis of symmetry runs through the vertex. It is the line x equals four.
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The graph of a function written in this form f of x equals a times x minus h squared plus
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k is a parabola that has the following features. The vertex will be h comma k.
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If a is greater than zero, a positive value, then the parabola will open upward. If a is less than
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zero, the parabola will open downward. The axis of symmetry in either case will be the vertical
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line x equals h. Here you can see I've graphed a parabola that opens downward on this side,
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where a is less than zero. Here's the vertex at h, k. And the axis of symmetry is x equals h.
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And here is a parabola that opens upwards, so a is greater than zero. The vertex is h comma k,
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and the axis of symmetry is x equals h. Let's continue with some more examples.
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Let's graph the function f of x equals negative x plus one squared plus five.
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Using what we just learned about the vertex and axis of symmetry,
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from this equation, we can identify that the vertex is located at negative one, five.
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And the axis of symmetry is the vertical line that runs through the x-value negative one. So we have an axis of symmetry
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of the line x equals negative one. To find another point that's on this parabola,
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let's find the y-intercept. To find the y-intercept, we set x equal to zero. So
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we're going to find f of zero. This gives us negative zero plus one squared plus five.
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This will be one squared multiplied by negative one, which is negative one,
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plus five, giving us four. So f of zero is equal to four. This is our y-intercept.
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Recognizing that our axis of symmetry is here at x equals negative one means that there will be
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a point on the other side of the axis of symmetry one unit from the axis of symmetry with the same
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y-value. Now we have three points, and we can sketch the graph of this function.
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The next part of the problem asks us to state the vertex and the axis of symmetry of the graph
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of f which we began with. We noticed that the vertex was located at negative one, five,
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and the axis of symmetry is the vertical line x equals negative one.