Write the Equation of a Polynomial Function Based on Its Graph - Pre-Calculus

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

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 .

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:

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:

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:

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:

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 .

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,

.

Answer: .

Method 2:

Use the form , where is the vertex.

is , so , .

Answer:

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

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:

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:

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:

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:

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,

.

Answer: .

Method 2:

Use the form , where is the vertex.

is , so , .

Answer:

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

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 .

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:

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:

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: