Graph each function. See Examples 6–8.g(x) = (x - 4)²

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 the quantity of X minus six squared. Then we're given a blank graph. We have a vertical Y axis, a horizontal X axis. They come together at the origin in the middle, the domain for what's drawn for our X axis is from negative 20 to positive 20. And the range for what's drawn for our Y axis is from negative 20 to positive 20 as well. All right, looking at our function, we first want to ask ourselves, what's the base function here? What's happening to our X and our X is being squared? So our base function is Y equals X squared. We recall from previous lessons. What that looks like? That's a parabola with its vertex at the origin and its opening upward. So we have one arm heading up toward positive infinity for the Y values and negative infinity for the X values. But it's closer to the Y axis than the X axis. And then it comes down and it curves through the origin. This is not exact and hits the origin then heads up toward positive infinity for both the X and Y values in the first quadrant. And it's closer to the Y axis than the X axis for most of its time. And there is our parabola, our Y equals X squared. Now, what's the difference between our function and the base function? The difference is that this one has a six being subtracted from the X before it gets squared. So what does that mean? Well, that transformation is happening directly to the X value before it gets squared. And when that happens, it indicates a horizontal shift and specifically the horizontal shift here because we are subtracting six is that it's going to move to the right six units. So instead of having a vertex at the origin, we'll have a vertex at 60. So it'll look something like this. I, I didn't draw that very well. It's floating above, but it's supposed to be with the vertex down at 60. And that gives us a really good idea of what's going on. But we can also be more specific by creating an XY table, choosing a few values for X plugging those in to our function X minus six squared coming up with the associated Y values and plotting those points. So for the X values here, I am going to choose and eight. Now we can plug those in, get the Y values and then we can draw basically our final answer of a graph which will be a little more accurate. But since I'm the one drawing, it maybe not too much more accurate. All right now plugging in four X A four. So we have instead of X minus six, it's four minus six squared, four minus six is a negative two and negative two squared is a positive four. So one of our points is at 44 from the origin. we'll go to the right four and up four. All right. Next plugging in six for X, we have, instead of X minus six, we have six minus six squared, six minus six is zero and zero squared is zero. So for our 0.60 from the origin, we go to the right six. All right, now we have eight for X. So instead of X minus six, we have eight minus six squared, eight minus six is two and two squared is four. So for the 0.84 from the origin, we go to the right eight and we go up four and there is more precisely some of our points. And we can draw this heading up toward positive infinity for both the X or for the Y values, but not the X values to the left of our vertex and then hits the vertex at 60 and then heads toward positive infinity for both the X and Y values to the right of the vertex. So opening upward and there is our F of X equals X minus six squared. Well done. We'll catch you on the next one.

Graph each function. See Examples 6–8.g(x) = (x - 4)²

Key Concepts

Quadratic Functions

Vertex of a Parabola

Transformations of Functions

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