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

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Question

Write the quadratic function for the graph:

Answer

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,

$f(x) = x\cdotx + 2\cdotx + 6\cdotx + 6\cdot2 = x^{2} + 8x + 12$ .

Answer: $f(x) = x^{2} + 8x + 12$ .

Method 2:

Use the form $(x - h)^{2} + k$ , where is the vertex.

is , so , .

$(x - [-4])^{2} + (-4) = (x + 4)^{2} - 4$

Answer: $f(x) = (x + 4)^{2} - 4$

$= x^{2} + 8x +16 - 4$

$= x^{2} + 8x + 12$

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Question

Write the quadratic function for the graph:

Answer

Because there are no x-intercepts, use the form $f(x) = (x - h)^{2} + k$ , where vertex is , so , , which gives

$f(x) = (x - 3)^{2} + 3$

$= x^{2} + 2(-3)(x) + (-3)^{2} + 3$

$= x^{2} - 6x + 9 + 3$

$= x^{2} - 6x + 12$

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Question

Which could be the equation for this graph?

Answer

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 $x+\frac{9}{2}=0$ 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 .

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Question

Write the equation for the polynomial in this graph:

Answer

The zeros for this polynomial are $-1.5, \frac{1}{3}, 4$ .

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

$x - \frac{1}{3} = 0$ multiply both sides by 3

so one factor is

so one factor is

Multiply these three factors:

$(6x^2 + 7 x - 3 ) ( x - 4 ) = 6x^3 + 7x^2 - 3 x \enspace -24x^2 - 28x + 12$

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Question

Write the equation for the polynomial shown in this graph:

Answer

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:

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Question

Write the equation for the polynomial in this graph:

Answer

The zeros for this polynomial are $-1.75, \frac{1}{3}, 4$ . That means that the factors are equal to zero when these values are plugged in.

or equivalently $x + \frac{7}{4} = 0$ multiply both sides by 4

the first factor is

$x-\frac{1}{3} = 0$ multiply both sides by 3

the second factor is

the third factor is

Multiply the three factors:

$(12x^2 +17x - 7 )(x-4) = 12x^3 +17x^2 -7x + \enspace -48x^2 -68x +28$

Compare your answer with the correct one above

Question

Write the equation for the polynomial in the graph:

Answer

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 $x - \frac{3}{5} = 0$ multiply both sides by 5:

The second and third factors are and

Multiply:

$(5x^2 -18x + 9 ) ( x + 2 ) = 5x^3 -18x^2 +9x \enspace + 10x^2 -36x +18$

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

Compare your answer with the correct one above

Question

Which could be the equation for this graph?

Answer

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 $x+\frac{9}{2}=0$ 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 .

Compare your answer with the correct one above

Question

Write the quadratic function for the graph:

Answer

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,

$f(x) = x\cdotx + 2\cdotx + 6\cdotx + 6\cdot2 = x^{2} + 8x + 12$ .

Answer: $f(x) = x^{2} + 8x + 12$ .

Method 2:

Use the form $(x - h)^{2} + k$ , where is the vertex.

is , so , .

$(x - [-4])^{2} + (-4) = (x + 4)^{2} - 4$

Answer: $f(x) = (x + 4)^{2} - 4$

$= x^{2} + 8x +16 - 4$

$= x^{2} + 8x + 12$

Compare your answer with the correct one above

Question

Write the quadratic function for the graph:

Answer

Because there are no x-intercepts, use the form $f(x) = (x - h)^{2} + k$ , where vertex is , so , , which gives

$f(x) = (x - 3)^{2} + 3$

$= x^{2} + 2(-3)(x) + (-3)^{2} + 3$

$= x^{2} - 6x + 9 + 3$

$= x^{2} - 6x + 12$

Compare your answer with the correct one above

Question

Write the equation for the polynomial in this graph:

Answer

The zeros for this polynomial are $-1.5, \frac{1}{3}, 4$ .

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

$x - \frac{1}{3} = 0$ multiply both sides by 3

so one factor is

so one factor is

Multiply these three factors:

$(6x^2 + 7 x - 3 ) ( x - 4 ) = 6x^3 + 7x^2 - 3 x \enspace -24x^2 - 28x + 12$

Compare your answer with the correct one above

Question

Write the equation for the polynomial shown in this graph:

Answer

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:

Compare your answer with the correct one above

Question

Write the equation for the polynomial in this graph:

Answer

The zeros for this polynomial are $-1.75, \frac{1}{3}, 4$ . That means that the factors are equal to zero when these values are plugged in.

or equivalently $x + \frac{7}{4} = 0$ multiply both sides by 4

the first factor is

$x-\frac{1}{3} = 0$ multiply both sides by 3

the second factor is

the third factor is

Multiply the three factors:

$(12x^2 +17x - 7 )(x-4) = 12x^3 +17x^2 -7x + \enspace -48x^2 -68x +28$

Compare your answer with the correct one above

Question

Write the equation for the polynomial in the graph:

Answer

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 $x - \frac{3}{5} = 0$ multiply both sides by 5:

The second and third factors are and

Multiply:

$(5x^2 -18x + 9 ) ( x + 2 ) = 5x^3 -18x^2 +9x \enspace + 10x^2 -36x +18$

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

Compare your answer with the correct one above

Question

Write the quadratic function for the graph:

Answer

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,

$f(x) = x\cdotx + 2\cdotx + 6\cdotx + 6\cdot2 = x^{2} + 8x + 12$ .

Answer: $f(x) = x^{2} + 8x + 12$ .

Method 2:

Use the form $(x - h)^{2} + k$ , where is the vertex.

is , so , .

$(x - [-4])^{2} + (-4) = (x + 4)^{2} - 4$

Answer: $f(x) = (x + 4)^{2} - 4$

$= x^{2} + 8x +16 - 4$

$= x^{2} + 8x + 12$

Compare your answer with the correct one above

Question

Write the quadratic function for the graph:

Answer

Because there are no x-intercepts, use the form $f(x) = (x - h)^{2} + k$ , where vertex is , so , , which gives

$f(x) = (x - 3)^{2} + 3$

$= x^{2} + 2(-3)(x) + (-3)^{2} + 3$

$= x^{2} - 6x + 9 + 3$

$= x^{2} - 6x + 12$

Compare your answer with the correct one above

Question

Which could be the equation for this graph?

Answer

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 $x+\frac{9}{2}=0$ 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 .

Compare your answer with the correct one above

Question

Write the equation for the polynomial in this graph:

Answer

The zeros for this polynomial are $-1.5, \frac{1}{3}, 4$ .

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

$x - \frac{1}{3} = 0$ multiply both sides by 3

so one factor is

so one factor is

Multiply these three factors:

$(6x^2 + 7 x - 3 ) ( x - 4 ) = 6x^3 + 7x^2 - 3 x \enspace -24x^2 - 28x + 12$

Compare your answer with the correct one above

Question

Write the equation for the polynomial shown in this graph:

Answer

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:

Compare your answer with the correct one above

Question

Write the equation for the polynomial in this graph:

Answer

The zeros for this polynomial are $-1.75, \frac{1}{3}, 4$ . That means that the factors are equal to zero when these values are plugged in.

or equivalently $x + \frac{7}{4} = 0$ multiply both sides by 4

the first factor is

$x-\frac{1}{3} = 0$ multiply both sides by 3

the second factor is

the third factor is

Multiply the three factors:

$(12x^2 +17x - 7 )(x-4) = 12x^3 +17x^2 -7x + \enspace -48x^2 -68x +28$

Compare your answer with the correct one above

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