Graph each function. See Examples 6–8.g(x) = ½ x³ - 4

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Accepted Answer

Hello, everyone. We are asked to draw the graph of the following function. We are given F of X equals 1/4 X cubed minus six. We are provided a Cartesian plane with X and Y axis with a scale of one unit. And the X axis ranges from negative 15 to 15. The Y axis ranges from negative 10 to 15. So again, our beginning function is F of X equals 1/ X cubed minus six. So the first thing I want to do is identify what is the parent function. So the parent function is what the function is. If we don't stretch it, flip it or shift it. And in this case, I'm gonna call it GVAX, but G of X will equal X cubed and our X cubed function. If I draw a little rough sketch of it here, I always thought of it as a squiggle. But that's probably not the most technical term. It is kind of a loop up in the first quadrant and a loop down into the third quadrant. So this is what we're starting with as our base. So we're gonna try to move that around a little bit and figure out what we are asked to stretch or shift or flip. So first, I'm looking at the minus six. So subtracting six from the function means I'm going to shift every point down six units. So I like to think about what used to be on the origin as the exact center. So I'm gonna shift that down six units. So I know I have a point at zero negative six from there 1/4 means we are compressing our grasp graph by a scale of 1/4. So it's gonna get thinner. So if I use some of the points that I know are on the parent function, for example, on the parent function, I know I have a point that is 28. So if I want to use two as a value on my new function F of two is 1/4 of two cubed minus six, two cubed is 8, 1/4 of eight is 22 minus six is negative four. So I know I have the 0.2 negative four on my function we're graphing. So I'm gonna put that on my graph to negative four. I'm also gonna pick probably two more points just to make sure I have the full graph on here. So I'm gonna do F of negative two. So we'll have 1/4 multiplied by negative two cubed minus six, negative two cubed is negative 8, 1/4 of negative eight is negative two, negative two minus six is negative eight. So we'd have the point negative two, negative eight, negative two, negative eight on my graph. And I want to pick one more just so I can see where it goes in the first quadrant. So I'm gonna pick F of four just because I know I can take 1/4 of four cubed and come out with an even number. So F of four would be 1/4 of four cubed minus six, four, cubed is 64 1/4 of four cubed. So 1/4 of 64 is 16, 16, minus six is 10. So we'd also have the 0.4 10. So 4 10 on my graph and we're gonna apply our little curve and so we shifted our graph down six units, compressed it by a scale of 1/4 and it might not be the prettiest drawing anyone has ever seen. But that is generally what the graph of this function looks like. Have a nice day.

Graph each function. See Examples 6–8.g(x) = ½ x³ - 4

Key Concepts

Graphing Polynomial Functions

Understanding the Behavior of Cubic Functions

Finding Key Features of the Graph

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