Question

Describe the characteristics of the graphs of odd-degree and even-degree polynomial functions whose leading coefficients are positive.

Short answer

Odd-degree polynomial functions whose leading coefficients are positive goes up on the right side and down on the left side.

Even-degree polynomial functions whose leading coefficients are positive goes up on both the right side and the left side.

Step-by-step solution

Step 1. Given

Odd-degree and even-degree polynomial functions whose leading coefficients are positive.

Step 2. To determine

We have to determine the characteristics of the graphs.

Step 3. Calculation

Let the leading term is $a{x}^n$, where a is positive andn = odd.

On the right end of the graph $x\to \infty$.

For $x\to \infty$, $a{x}^n\to \infty$.

So, the graph will go upwards on the right side.

On the left end of the graph $x\to -\infty$.

For $x\to -\infty$, $a{x}^n\to -\infty$.

So, the graph will go downwards on the left side.

Let the leading term is $a{x}^n$, where a is positive andn = even.

On the right end of the graph $x\to \infty$.

For $x\to \infty$, $a{x}^n\to \infty$.

So, the graph will go upwards on the right side.

On the left end of the graph $x\to -\infty$.

For $x\to -\infty$, $a{x}^n\to \infty$.

So, the graph will go upwards on the left side.

Hence, odd-degree polynomial functions whose leading coefficients are positive goes up on the right side and down on the left side.

Even-degree polynomial functions whose leading coefficients are positive goes up on both the right side and the left side.