Which of the following is the graph of a polynomial of even degree with a positive leading coefficient?
Here, we have four different polynomials that all exhibit different characteristics. To solve the question, we’ll need to remember what the characteristics of an even degree polynomial with a leading coefficient is. In an even degree polynomial, as 𝑥 approaches both positive and negative ∞, that’s the 𝑥-values in both the directions, you have two possible options. The output will either be both sets of 𝑦-values approaching negative ∞ or both sets of 𝑦-values approaching positive ∞. That means that the 𝑦-values are pointing in the same direction on either side.
We see that, here, the 𝑦-values approach negative ∞ and here as the 𝑦-values approach positive ∞. On the red graph, one side is approaching positive ∞ and the other side negative ∞. And that means that answer choice a, the red graph, is an odd degree polynomial. And the same is true for the purple. It is also an odd degree polynomial.
So how do we take b and c, the yellow and green graph, and determine which of them has a positive leading coefficient. They’re both even. But one is positive. And one is negative. As 𝑓 of 𝑥 approaches plus or minus ∞ with a positive leading coefficient, the 𝑦-values will be positive. Only the green graph have the 𝑦-values approaching positive ∞. The yellow graph would be an even degree polynomial with a negative leading coefficient.
The green graph, answer choice c, is the only graph of an even degree polynomial with a positive leading coefficient.