Video: Matching the Rule of a Cubic Function to Its Graph

Which graph represents the function 𝑦 = βˆ’2.5π‘₯Β³ + 3?

02:34

Video Transcript

Which graph represents the function 𝑦 equals negative 2.5 π‘₯ cubed plus three?

A cubic function is a function where the greatest power of a variable is three. If the leading coefficient of a cubic function, meaning the number in front of the first term which it would be an π‘₯ cubed, if it is positive, our graph will be increasing left or right, so going up. And if there would be a negative leading coefficient β€” the number in front of π‘₯ cubed, the first term π‘₯ cubed β€” then our graph would be decreasing left or right, it will be going down. Therefore, since we already know that it’s a cubic function and we have a negative leading coefficient, negative 2.5, our graph should be decreasing left or right. So we can go ahead and eliminate options a and b because these are increasing left or right.

Now notice, it is 𝑦 equals negative 2.5 π‘₯ cubed plus three. And our last two options c and d, we have our graph crossing the 𝑦-axis at positive three for c and our graph crossing the 𝑦 axis at negative three for option d. Since we want to be crossing the 𝑦-axis at positive three, our correct answer would be graph c.

So graph c represents the function 𝑦 equals negative 2.5 π‘₯ cubed plus three.

To further our thinking about this question, let’s talk about why some of the graphs look a little different than the others. When the coefficient of π‘₯ cubed is greater in magnitude, then as the π‘₯-coordinates increase, the 𝑦 coordinates increase faster and the curve accelerates away from the π‘₯-axis at a faster rate. Obviously, both curves extend infinitely in the positive and negative π‘₯-directions. But for these limited snapshots of the curve, that makes the curve look thinner. While this information doesn’t change our answer, it’s just something to keep in mind.

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