# Algebra II : Graphing Polynomial Functions

## Example Questions

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### Example Question #1 : Graphing Polynomial Functions

Where does the graph of cross the axis?

Explanation:

To find where the graph crosses the horizontal axis, we need to set the function equal to 0, since the value at any point along the axis is always zero.

To find the possible rational zeroes of a polynomial, use the rational zeroes theorem:

Our constant is 10, and our leading coefficient is 1. So here are our possible roots:

Let's try all of them and see if they work! We're going to substitute each value in for using synthetic substitution. We'll try -1 first.

Looks like that worked! We got 0 as our final answer after synthetic substitution. What's left in the bottom row helps us factor down a little farther:

We keep doing this process until is completely factored:

Thus, crosses the axis at .

### Example Question #2 : Graphing Polynomial Functions

Where does cross the axis?

5

3

-7

7

-3

7

Explanation:

crosses the axis when equals 0. So, substitute in 0 for :

### Example Question #3 : Graphing Polynomial Functions

Which equation best represents the following graph?

None of these

Explanation:

We have the following answer choices.

The first equation is a cubic function, which produces a function similar to the graph. The second equation is quadratic and thus, a parabola. The graph does not look like a prabola, so the 2nd equation will be incorrect. The third equation describes a line, but the graph is not linear; the third equation is incorrect. The fourth equation is incorrect because it is an exponential, and the graph is not an exponential. So that leaves the first equation as the best possible choice.

### Example Question #1 : Understand Linear And Nonlinear Functions: Ccss.Math.Content.8.F.A.3

Which of the graphs best represents the following function?

None of these

Explanation:

The highest exponent of the variable term is two (). This tells that this function is quadratic, meaning that it is a parabola.

The graph below will be the answer, as it shows a parabolic curve.

### Example Question #4 : Graphing Polynomial Functions

Turns on a polynomial graph.

What is the maximum number of turns the graph of the below polynomial function could have?

3 turns

7 turns

4 turns

8 turns

7 turns

Explanation:

When determining the maximum number of turns a polynomial function might have, one must remember:

Max Number of Turns for Polynomial Function = degree - 1

First, we must find the degree, in order to determine the degree we must put the polynomial in standard form, which means organize the exponents in decreasing order:

Now that f(x) is in standard form, the degree is the largest exponent, which is 8.

We now plug this into the above:

Max Number of Turns for Polynomial Function = degree - 1

Max Number of Turns for Polynomial Function = 8 - 1

which is 7.

The correct answer is 7.

### Example Question #5 : Graphing Polynomial Functions

End Behavior

Determine the end behavior for  below:

Explanation:

In order to determine the end behavior of a polynomial function, it must first be rewritten in standard form. Standard form means that the function begins with the variable with the largest exponent and then ends with the constant or variable with the smallest exponent.

For f(x) in this case, it would be rewritten in this way:

When this is done, we can see that the function is an Even (degree, 4) Negative (leading coefficent, -3) which means that both sides of the graph go down infinitely.

In order to answer questions of this nature, one must remember the four ways that all polynomial graphs can look:

Even Positive:

Even Negative:

Odd Positive:

Odd Negative:

### Example Question #6 : Graphing Polynomial Functions

Which of the following is a graph for the following equation:

Cannot be determined

Explanation:

The way to figure out this problem is by understanding behavior of polynomials.

The sign that occurs before the  is positive and therefore it is understood that the function will open upwards. the "8" on the function is an even number which means that the function is going to be u-shaped. The only answer choice that fits both these criteria is:

### Example Question #4 : Understand Linear And Nonlinear Functions: Ccss.Math.Content.8.F.A.3

None of the above

Explanation:

Starting with

moves the parabola by  units to the right.

Similarly moves the parabola by  units to the left.

Hence the correct answer is option .

### Example Question #7 : Graphing Polynomial Functions

Explanation:

When we look at the function we see that the highest power of the function is a 3 which means it is an "odd degree" function. This means that the right and left side of the function will approach opposite directions. *Remember O for Odd and O for opposite.

In this case we also have a negative sign associated with the highest power portion of the function - this means that the function is flipped.

Both of these combine to make this an "odd negative" function.

Odd negative functions always have the right side of the function approaching down and the left side approaching up.

We represent this mathematically by saying that as x approaches negative infinity (left side), the function will approach positive infinity:

...and as x approaches positive infinity (right side) the function will approach negative infinity: