Graph each function. See Examples 6 – 8.

h(x) = -(x + 1)³

Question

Graph each function. See Examples 6 – 8.

h(x) = -(x + 1)³

Accepted Answer

Welcome back. I am so glad you're here. We are asked to draw the graph of the following function. Our function is F FX equals negative quantity of X plus five cubed. Then 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 drawn for our X axis is from negative 15 to positive 20. And the range for what's drawn for our Y axis is from negative 10 to positive 15. All right. So looking at our function, we want to identify what is our base function here. And we can see that this X is being cubed. So our base function is Y equals X cubed. We recall from previous lessons that the graph of Y equals X cubed looks approximately and this is not exact, but it's comes down toward negative infinity for both X and Y in the third quadrant. And then it comes up, it's a lot closer to the Y axis than to the X axis. And then it kind of curves once it gets close to the origin passes through the origin and then uh mirrors it exactly going up into the first quadrant heading toward positive infinity for both the Y and the X axis, but closer to the Y axis than to the X axis. So there's what our base function Y equals X cubed looks like. And now we want to ask ourselves what's different between our base function and our given function. And we see two differences here. One is that A five is being added to the X before it's cubed. And then the other difference is this negative being multiplied in front of the X after it's been cubed. So what do each of those mean? Well, those two transformations will start with this plus five that's happening directly to the X before it gets cubed. So that indicates a horizontal change. And specifically because we are adding a number to our X, it's a horizontal shift and because we're adding five, it's going to go to the left five units. Now this negative being multiplied after we have added five and cubed our X that's not happening directly to the X before it gets cubed. And so that indicates a vertical change. And specifically the change here because it's being multiplied by a negative is that it's going to have reflection over the X axis. So those are our two shifts. So we know that we're gonna flip this thing over the X axis and go to the left five. So it'll look approximately like this when we do those two transformations but we can be a little bit more specific with this by creating an XY table, choosing some X values, plugging those into our function F of X equals a negative parentheses X plus five and parentheses cubed. And then we'll get the Y values and plug those in. No, because we know approximately what's going on. We can be strategic with our X values and we don't have to choose very many. So I'm going to choose for X values a negative six, a negative five and a negative four. And we'll plug those in to find the Y values. So we'll have a negative open parentheses instead of X, it'll be a negative six plus five closed parentheses. Cubed, negative six plus five is a negative one. So we have a negative negative one. Cubed, negative one. Cubed is a negative one. But if we have a negative negative one, that's going to be a positive one. So clearing out this graph now we can draw our final answer graph. The first point we have is negative 61 from the origin. We'll go to the left six and then we'll go up one. All right. Now plugging in negative five. For X, we have a negative open parentheses, negative five plus five, closed parentheses. Cubed, negative five plus five is 00. Cubed is zero, negative zero is zero. This whole thing is zero. So from the origin, we'll go to the left five and then neither up nor down. All right, the last X value negative four, we have a negative open parentheses, negative four plus five, closed parentheses, cubed, negative four plus five is a positive one. So now we have a one cubed which is one with the negative being multiplied by it, that is a negative one. So from the origin, we go to the left four and down one. All right. So now we know how this curve goes this time. It's heading up toward positive infinity for the Y values and toward negative infinity for the X values in the third quadrant heading faster toward our positive Y axis than the negative X axis. Then coming down, going through these points we've identified and then it's going to remain in the, excuse me, I was saying the third quadrant. So it was heading toward positive Y values and negative X values in the second quadrant. Then it goes down, passes through those points that we identified and heads toward negative infinity for the Y values as it's heading toward positive infinity for the X values. And as long as we can see it, it's still mostly in the third quadrant. Eventually it'll get over to the fourth quadrant. And there is our graph well done. We'll catch you on the next one.

Graph each function. See Examples 6 – 8. h(x) = -(x + 1)³

Key Concepts

Graphing Polynomial Functions

Transformation of Functions

Identifying Key Features of the Graph

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