Question

which polynomial function could be represented by the graph below?
$f(x)=x^{2}+x-2$
$f(x)=2x^{2}+2x-4$
$f(x)=x^{2}-x-2$
$f(x)=2x^{2}-2x-4$

Explanation:

Step1: Identify y-intercept from graph

The graph crosses the y-axis at $(0, -4)$, so the y-intercept is $-4$.

Step2: Check y-intercept of options

For a polynomial $f(x)=ax^2+bx+c$, the y-intercept is $c$.

• $f(x)=x^2+x-2$: $c=-2$ (does not match)
• $f(x)=2x^2+2x-4$: $c=-4$ (matches)
• $f(x)=x^2-x-2$: $c=-2$ (does not match)
• $f(x)=2x^2-2x-4$: $c=-4$ (matches)

Step3: Find x-intercepts from graph

The graph crosses the x-axis at $x=1$ and $x=-2$.

Step4: Test x-intercepts on remaining options

First test $f(x)=2x^2+2x-4$:
Substitute $x=1$: $2(1)^2+2(1)-4=2+2-4=0$ (matches)
Substitute $x=-2$: $2(-2)^2+2(-2)-4=8-4-4=0$ (matches)
Test $f(x)=2x^2-2x-4$:
Substitute $x=1$: $2(1)^2-2(1)-4=2-2-4=-4
eq0$ (does not match)

Answer:

B. $f(x)=2x^2+2x-4$