Question

Which polynomial function could be represented by the graph below?

Answer 1

Note that

1.

$f(x)=x^3+x^2-6x=x(x^2+x-6)=x(x-2)(x+3)$

The x-intercepts are at points x=-3, x=0, x=2. The graph should be increasing - decreasing - increasing.

2.

$f(x)=x^3-x^2-6x=x(x^2-x-6)=x(x+2)(x-3)$

The x-intercepts are at points x=-2, x=0, x=3. The graph should be increasing - decreasing - increasing.

3.

$f(x)=-2x^3-2x^2+12x=-2x(x^2+x-6)=-2x(x-2)(x+3)$

The x-intercepts are at points x=-3, x=0, x=2. The graph should be decreasing - increasing - decreasing.

4.

$f(x)=-2x^3+x^2+12x=-2x(x^2-x-6)=-2x(x+2)(x-3)$

The x-intercepts are at points x=-2, x=0, x=3. The graph should be decreasing - increasing - decreasing.

From the diagram you can see that x-intercepts are at points x=-3, x=0, x=2 and the graph is   decreasing-increasing-decreasing.

Answer: correct choice is 3.

Answer 2

The polynomial function is $\boxed{f\left( x \right) = - 2{x^3} - 2{x^2} + 12x}$ that is represented by the graph. Option (3) is correct.

Further explanation:

Given:

The options of the equations are as follows.

1.$f\left( x \right) = {x^3} + {x^2} - 6x$

2. $f\left( x \right) = {x^3} - {x^2} - 6x$

3. $f\left( x \right) = - 2{x^3} - 2{x^2} + 12x$

4. $f\left( x \right) = - 2{x^3} + 2{x^2} + 12x$

Explanation:

The graph passes through the points $\left( {-3, 0}\right)$ and $\left( { 2,0} \right).$

Solve the polynomial $f\left( x \right) = {x^3} + {x^2} - 6x$ to obtain the zeros of x.

$\begin{aligned}f\left( x \right)&= {x^3} + {x^2} - 6x\\&= x\left({{x^2} + x - 6}\right)\\&= x\left({x - 2}\right)\left({x + 3}\right)\\\end{aligned}$

The zeros of the polynomial are -3, 0 and 2.

The graph of the polynomial $f\left( x \right) = {x^3} + {x^2} - 6x$ is increasing-decreasing-increasing.

Solve the polynomial $f\left( x \right) = {x^3} - {x^2} - 6x$ to obtain the zeros of x.

$\begin{aligned}f\left( x \right)&={x^3} - {x^2} - 6x\\&= x\left({{x^2} - x - 6}\right)\\&= x\left({x + 2} \right)\left({x - 3} \right)\\\end{aligned}$

The zeros of the polynomial are -2, 0 and 3.

The graph of the polynomial $f\left( x \right) = {x^3} - {x^2} - 6x$ is increasing-decreasing-increasing.

The graph doesn’t passes through the point  $\left({ - 3,0} \right).$Therefore, the polynomial doesn’t satisfy the graph.

Solve the polynomial $f\left( x \right)= - 2{x^3} - 2{x^2} + 12x$ to obtain the zeros of x.

$\begin{aligned}f\left( x \right)&= - 2{x^3} + {x^2} + 12x\\&= - 2x\left({{x^2} + x - 6} \right)\\&= - 2x\left( {x - 2}\right)\left({x + 3}\right)\\\end{aligned}$

The zeros of the polynomial are -2, 0 and 3.

The graph of the polynomial $f\left( x \right)=- 2{x^3} - 2{x^2} + 12x$ is decreasing-increasing-decreasing.

Solve the polynomial $f\left( x \right)= - 2{x^3} + 2{x^2} + 12x$ to obtain the zeros of x.

$\begin{aligned}f\left( x \right)&= - 2{x^3} + 2{x^2} + 12x\\&=- 2x\left( {{x^2} - x - 6} \right)\\&=- 2x\left({x + 2} \right)\left({x - 3} \right)\\\end{aligned}$

The zeros of the polynomial are -2, 0 and 3.

The graph of the polynomial $f\left( x \right) = - 2{x^3} + 2{x^2} + 12x$ is decreasing-increasing-decreasing.

The graph doesn’t passes through the point $\left({ - 3,0}\right).$ Therefore, the polynomial doesn’t satisfy the graph.

From the graph it has been observed that the graph is decreasing-increasing-decreasing.

The polynomial function is $\boxed{f\left( x \right)= - 2{x^3} - 2{x^2} +12x}$ that is represented by the graph. Option (3) is correct.

Learn more:

1. Learn more about inverse of the functionhttps://brainly.com/question/1632445.

2. Learn more about equation of circle brainly.com/question/1506955.

3. Learn more about range and domain of the function brainly.com/question/3412497

Answer details:

Grade: High School

Subject: Mathematics

Chapter: polynomials

Keywords: quadratic equation, equation factorization, polynomial, quadratic formula, zeroes, function.