Which graph represents the function 𝑦 equals negative 2.5 𝑥 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 𝑥 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 — the number in front of 𝑥 cubed, the first term 𝑥 cubed — 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 𝑥 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 𝑥 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 𝑥 cubed is greater in magnitude, then as the 𝑥-coordinates increase, the 𝑦 coordinates increase faster and the curve accelerates away from the 𝑥-axis at a faster rate. Obviously, both curves extend infinitely in the positive and negative 𝑥-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.