In Exercises 53-66, begin by graphing the standard quadratic func...

Question

In Exercises 53-66, begin by graphing the standard quadratic function, f(x) = x². Then use transformations of this graph to graph the given function. g(x) = x² - 2

Accepted Answer

Welcome back. I am so glad you're here. We are given two functions. The first is F of X which equals two, X squared minus X plus one. We are asked to graph that and then apply transformations in order to graph the second function G of X which equals two, X squared minus x. But plus five. Alright, so let's start off by graphing that F of X. I'm going to draw a rough sketch of a coordinate plane will have a horizontal X axis of vertical Y axis. They come together at the origin in the middle. I am going to draw an X. Y. Table, choose a few values for X, plug them into the original function, find the corresponding Y values and then plot those points for the X values. I'm going to put in negative 10 and positive one, plugging in negative one for X. That will be two times not X but negative one squared minus not X but negative one plus one. Negative one squared is a positive one. So two times one minus negative one is plus one and then +12 times one is +22 plus one plus one equals four. Now plugging in zero, we've got two times zero squared minus zero plus one. Well two times zero squared is zero minus 00. And that plus one. That just leaves us with one. Now plugging one end we have two times one squared minus one plus 11 squared is one. So that's two times one minus one plus 12 times one is 22 minus one plus 12 minus one is one plus one brings us back up to two. Alright now let's plot these points from the origin in our graph at 00 to plot the point negative 14. We're going to go to the left one unit for our X. Value of negative one. And we're going to go up four units for our Y. Value for that's approximately here for the next point of 01 from the origin. We're going to go nowhere for our X values and we're going to go up one unit for our Y value, that's approximately here. And we can label that 01 for our Y intercept. And now for 12 from the origin we're going to go to the right one unit for our X. Value of one and up two units for our Y value of two. And that's approximately here. So we can graph this and it looks sort of like this, here's our affects. Alright. Now for our G. Of X to graph that we could go through the X. Y. Table but we are asked to apply transformations from the F. Of X graph. So we need to look and see what's the same and what's different between our F. Of X and our G. Of X. They both have this two X squared. They both have this minus X. And then very similarly one has a +11 has a plus five. So we could insert our F Of X inside of our G. Of X. Except the only difference is that this between this plus one and this plus five the difference would be a plus four. So we could say that our G. Of X is equal to F. Of X plus four. And then look at what transformations are happening. Well the only transformation is this plus four plus four is on the outside of the function. It's not happening directly to the F. Of X. It's happening afterward. And because it's not happening directly it indicates a vertical shift. And because it is a vertical shift and we're adding four it is a vertical shift up four units. And so looking specifically at our F. Of X graph that has a Y intercept at 01. If we shift that up four units are G. Of X has a Y intercept at 05. And if you went back in and plotted points, you would see that that makes sense, plugging in zero for x two times zero squared minus zero plus five. Yeah, all that's left is going to be that five for the Y value. Otherwise it will look exactly the same as the F. Of X. So here's our G. Of X shifted up four units. All right. Now let's look at our answer choices. We want to find an F. Of X with a Y intercept at 01 and we want to find a G of X with a Y intercept at 05 and looking at our answer choice A. That's exactly what they have. And they say that the graph ships shifts four units vertically upward, which is exactly what we determine. So answer choice A. Works well done. We'll catch you on the next one.

In Exercises 53-66, begin by graphing the standard quadratic function, f(x) = x². Then use transformations of this graph to graph the given function. g(x) = x² - 2

Key Concepts

Quadratic Functions

Graph Transformations

Vertex of a Parabola

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