Graph each function. See Examples 6–8.h(x) = 2x² - 1

Question

Accepted Answer

Welcome back. I am so glad you're here. We're asked to draw the graph of the following function. Our function is F of X equals four, X squared minus three. And we're given a graph, we have a vertical Y axis, a horizontal X axis. They come together at the origin. The domain for what's shown for our X axis is drawn from negative 10 to positive 10. And then the range of what's drawn for our Y axis is from negative 10 to positive 15. All right. So we want to look at our given function and see what the base function is in there. And we see that the base function here is this X squared. And so we recall from previous lessons that when we have Y equals X squared, that's going to be a Parabola, it's going to open upward and it's going to have its vertex at the origin. So I am not drawing this one to scale but just for the general idea of what that Parabola looks like heading up toward positive infinity with its arms and then that vertex at the origin. Now what are the differences between our Y equals X squared. And our given function F of X equals four X squared minus three. There's two main differences. First, this X squared is being multiplied by four and then it's also having this three subtracted from it. So what are those transformations? Let's first look at this multiplication by four, that multiplication. Even though it's in front of the X, it's not happening directly to the function. If it were, it would be inside of parentheses with the X, we'd have four X and then that would be being squared because it's happening after the X is squared with the order of operations, it's happening, it's a vertical change. So it's vertical. And because that number, the four is greater than one, then it is going to be vertical stretching. So one of the transformations is that we have vertical stretching by a factor of four. The other thing that's happening here is we're subtracting three again, that's not happening directly to the X. It's not inside of parentheses. We don't have X minus three and then squaring that it happens after the X is squared. And so because it's not directly to the X, again, it's a vertical shift. And specifically specifically because we're subtracting three, it's a vertical shift down three units. So in general, we are stretching our Y equals X squared and we're bringing it down three units bring specifically, we can note that that vertex goes down three units. So it'll look something more like this, but we can be specific about it. So in order to graph this specifically what we can do is create an Xyt, choose some values for X, plug them into our function and then get the corresponding Y values then graph those because we know in general what it looks like and what's happening, we only need to pick a few and I'm going to choose for the X values negative 10 and positive one, we plug in negative one for X in our four X squared minus three. So it'll be four multiplied by negative one squared minus three. Negative one squared is 14, multiplied by one is four and four minus three is positive one. From the origin, we go to the left one unit and up one unit. Now for an X value of zero, that's four multiplied by zero squared minus 30 squared is 04, multiplied by zero is zero and zero minus three is negative three. So from the origin, we'll go down three units and the last X value of 14 multiplied by one squared minus 31 squared is one, multiplied by four, is 44 minus three is a positive one. Again, from the origin, we go to the right one and up one. So we can see we connect these, we extend this upward toward positive infinity for both arms and the vertex is at zero negative three. And we can see that there was vertical stretching and that it has shifted down three units and there's our graph well done. We'll catch you on the next one.

Graph each function. See Examples 6–8.h(x) = 2x² - 1

Key Concepts

Quadratic Functions

Vertex of a Parabola

Graphing Techniques

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