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Precalculus : Write the Equation of a Polynomial Function Based on Its Graph

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Example Questions

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

Which could be the equation for this graph?

Polynomial

Possible Answers:

y=2x^3-x^4+2x-1

y=-2x^3 +7x^2 +21x -54

y=2x^3 -7x^2 -21x + 54

y=-2x^3 -7x^2 + 21x + 54

y=2x^3 +7x^2 -21x -54

Correct answer:

y=-2x^3 -7x^2 + 21x + 54

Explanation:

This graph has zeros at 3, -2, and -4.5. This means that x-3=0 , x+2 =0 , and x+4.5=0 . That last root is easier to work with if we consider it as $x+\frac{9}{2}=0$ and simplify it to 2x+9=0 . Also, this is a negative polynomial, because it is decreasing, increasing, decreasing and not the other way around.

Our equation results from multiplying -(x-3)(x+2)(2x+9) , which results in -2x^3 -7x^2 +21x +54 .

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Example Question #2 : Write The Equation Of A Polynomial Function Based On Its Graph

Write the quadratic function for the graph:

Varsity8

Possible Answers:

$f(x) = x^{2} + 6x - 12$

$f(x) = -x^{2} - 6x - 12$

$f(x) = -x^{2} - 12x + 3$

$f(x) = x^{2} + 12x + 3$

$f(x) = x^{2} - 6x + 12$

Correct answer:

$f(x) = x^{2} - 6x + 12$

Explanation:

Because there are no x-intercepts, use the form $f(x) = (x - h)^{2} + k$ , where vertex (h, k) is (3, 3) , so h = 3 , k = 3 , which gives

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

= (x - 3)(x - 3) + 3

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

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

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

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Example Question #3 : Write The Equation Of A Polynomial Function Based On Its Graph

Write the quadratic function for the graph:

Varsity9

Possible Answers:

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

$f(x) = x^{2} - 6x - 2$

$f(x) = -6x^{2} - 4x - 2$

$f(x) = x^{2} - 8x + 12$

$f(x) = x^{2} - 4x + 12$

Correct answer:

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

Explanation:

Method 1:

The x-intercepts are x = -6, 2 . These values would be obtained if the original quadratic were factored, or reverse-FOILed and the factors were set equal to zero.

For x = -6 , x + 6 = 0 . For x = -2 , x + 2 = 0 . These equations determine the resulting factors and the resulting function; f(x) = (x + 6)(x + 2) .

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 (h, k) is the vertex.

(h, k) is (-4, -4) , so h = -4 , k = -4 .

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

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

= (x + 4)(x + 4) - 4

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

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

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Example Question #4 : Write The Equation Of A Polynomial Function Based On Its Graph

Write the equation for the polynomial in this graph:

Graph 1 write funct

Possible Answers:

6x^3 -35x^2 + 47 x - 12

6x^3 + 31 x^2 + 25 x - 12

6x^3 + 13 x ^ 2 - 41 x + 12

6x^3 - 17x ^ 2 - 31 x + 12

6x^3 + 17x^2 - 31 x - 12

Correct answer:

6x^3 - 17x ^ 2 - 31 x + 12

Explanation:

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.

x + 1.5 = 0 multiply both sides by 2

2 x + 3 = 0 so one factor is 2x+3

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

3x - 1 = 0 so one factor is 3x-1

x - 4 = 0 so one factor is x- 4

Multiply these three factors:

(2x+3) ( 3x-1) = 6x^2 + 9 x - 2 x - 3 = 6x^2 + 7 x - 3

$(6x^2 + 7 x - 3 ) ( x - 4 ) = 6x^3 + 7x^2 - 3 x \enspace -24x^2 - 28x + 12$ = 6x^3 - 17x^2 -31x+12

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Example Question #5 : Write The Equation Of A Polynomial Function Based On Its Graph

Write the equation for the polynomial shown in this graph:

Graph 2 write funct

Possible Answers:

-2x^3 +3x^2 +32x+48

-2x^3 -19x^2 -56x-48

-2x^3 -3x^2 +32x+48

-2x^3 +19x ^2 -56x + 48

-2x^3 -13x^2 -8x+48

Correct answer:

-2x^3 -3x^2 +32x+48

Explanation:

The zeros of this polynomial are 4, -4 , -1.5 . This means that the factors equal zero when these values are plugged in.

One factor is x - 4

One factor is x + 4

The third factor is equivalent to x + 1.5 . Set equal to 0 and multiply by 2:

2(x + 1.5) = 2(0 )

2x+3 = 0

Multiply these three factors:

(x-4)(x+4) = x^2 - 16

(x^2 - 16) (2x+3 ) = 2x^3 +3x^2 - 32x - 48

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

y = -2x^3 -3x^2 + 32x + 48

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Example Question #6 : Write The Equation Of A Polynomial Function Based On Its Graph

Write the equation for the polynomial in the graph:

Graph 4 write funct

Possible Answers:

-5x^3 +8x^2 +27x-18

-5x^3 -2x^2 +33x -18

-5x^3 -22x^2 -15x + 18

-5x^3 -8x^2 +27x +18

-5x^3 +28x^2 -45x +18

Correct answer:

-5x^3 +8x^2 +27x-18

Explanation:

The zeros of the polynomial are 0.6, 3, -2 . That means that the factors equal zero when these values are plugged in.

The first factor is x - 0.6 = 0 or equivalently $x - \frac{3}{5} = 0$ multiply both sides by 5:

5x- 3 = 0

The second and third factors are x - 3 = 0 and x + 2 = 0

Multiply:

(5x - 3 ) ( x - 3 ) = 5x^2 -3x -15x +9 = 5x^2 -18x + 9

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

= 5x^3 - 8x^2 -27x + 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:

-5x^3 +8x^2 +27x - 18

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

Write the equation for the polynomial in this graph:

Graph 3 write funct

Possible Answers:

12x^3 + 31x^2 -75x -28

12x^3 +65x^2 -61x-28

12x^3 -73x^2 +107 x -28

12x^3 +23x^2 -93x + 28

12x^3 -31x^2 -75x+28

Correct answer:

12x^3 -31x^2 -75x+28

Explanation:

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.

x + 1.75 = 0 or equivalently $x + \frac{7}{4} = 0$ multiply both sides by 4

4x+7 = 0 the first factor is 4x+7

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

3x-1 = 0 the second factor is 3x - 1

x - 4 = 0 the third factor is x - 4

Multiply the three factors:

(4x+7)(3x-1) = 12x^2 +21x -4x - 7 = 12x^2+ 17x - 7

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

= 12x^3 -31x^2 -75x +28

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Precalculus : Write the Equation of a Polynomial Function Based on Its Graph

Study concepts, example questions & explanations for Precalculus

All Precalculus Resources

Example Questions

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

Example Question #2 : Write The Equation Of A Polynomial Function Based On Its Graph

Example Question #3 : Write The Equation Of A Polynomial Function Based On Its Graph

Example Question #4 : Write The Equation Of A Polynomial Function Based On Its Graph

Example Question #5 : Write The Equation Of A Polynomial Function Based On Its Graph

Example Question #6 : Write The Equation Of A Polynomial Function Based On Its Graph

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

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