# Precalculus : Graphs of Polynomial Functions

## Example Questions

### Example Question #1 : Graphs Of Polynomial Functions

For what values of  will the given polynomial pass through the x-axis if plotted in Cartesian coordinates?

Explanation:

One can remember that if the multiplicity of a zero is odd then it passes through the x-axis and if it's even then it 'bounces' off the x-axis. You can think about this analytically as well. What happens when we plug a number into our function just slightly above or below a zero with an even multiplicity? You find that the sign is always positive. Whereas a zero with an odd multiplicity will yield a positive on one side and a negative on the other. For zeros with odd multiplicity this alters the sign of our output and the function passes through the x-axis. Whereas the zero with even multiplicity will output a number with the same sign just above and below its zero, thus it 'bounces' off the x-axis.

### Example Question #1 : Graphs Of Polynomial Functions

For this particular question we are restricting the domain of both  to nonnegative values, or the interval .

Let  and .

For what values of  is ?

Explanation:

The cubic function will increase more quickly than the quadratic, so the quadratic function must have a head start.  At , both functions evaluate to 8.  After than point, the cubic function will increase more quickly.

The domain was restricted to nonnegative values, so this interval is our only answer.

### Example Question #3 : Graphs Of Polynomial Functions

Graph .

Explanation:

is a parabola, because of the general  structure.  The parabola opens downward because .

Solving tells the x-value of the x-axis intercept;

The resulting x-axis intercept is: .

Setting  tells the y-value of the y-axis intercept;

The resulting y-axis intercept is:

### Example Question #1 : Write The Equation Of A Polynomial Function Based On Its Graph

Which could be the equation for this graph?

Explanation:

This graph has zeros at 3, -2, and -4.5. This means that , , and . That last root is easier to work with if we consider it as and simplify it to . Also, this is a negative polynomial, because it is decreasing, increasing, decreasing and not the other way around.

Our equation results from multiplying , which results in .

### Example Question #1 : Write The Equation Of A Polynomial Function Based On Its Graph

Write the quadratic function for the graph:

Explanation:

Because there are no x-intercepts, use the form , where vertex  is , so , , which gives

### Example Question #1 : Write The Equation Of A Polynomial Function Based On Its Graph

Write the quadratic function for the graph:

Explanation:

Method 1:

The x-intercepts are .  These values would be obtained if the original quadratic were factored, or reverse-FOILed and the factors were set equal to zero.

For , .  For , .  These equations determine the resulting factors and the resulting function; .

Multiplying the factors and simplifying,

.

Method 2:

Use the form , where is the vertex.

is , so , .

### Example Question #1 : Write The Equation Of A Polynomial Function Based On Its Graph

Write the equation for the polynomial in this graph:

Explanation:

The zeros for this polynomial are .

This means that the factors are equal to zero when these values are plugged in for x.

multiply both sides by 2

so one factor is

multiply both sides by 3

so one factor is

so one factor is

Multiply these three factors:

### Example Question #1 : Write The Equation Of A Polynomial Function Based On Its Graph

Write the equation for the polynomial shown in this graph:

Explanation:

The zeros of this polynomial are . This means that the factors equal zero when these values are plugged in.

One factor is

One factor is

The third factor is equivalent to . Set equal to 0 and multiply by 2:

Multiply these three factors:

The graph is negative since it goes down then up then down, so we have to switch all of the signs:

### Example Question #1 : Write The Equation Of A Polynomial Function Based On Its Graph

Write the equation for the polynomial in the graph:

Explanation:

The zeros of the polynomial are . That means that the factors equal zero when these values are plugged in.

The first factor is or equivalently multiply both sides by 5:

The second and third factors are and

Multiply:

Because the graph goes down-up-down instead of the standard up-down-up, the graph is negative, so change all of the signs:

### Example Question #7 : Write The Equation Of A Polynomial Function Based On Its Graph

Write the equation for the polynomial in this graph:

Explanation:

The zeros for this polynomial are . That means that the factors are equal to zero when these values are plugged in.

or equivalently multiply both sides by 4

the first factor is

multiply both sides by 3

the second factor is

the third factor is

Multiply the three factors: