The quadratic function intersects the -axis at and and the -axis at . What is the value of ?

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SAT Nonlinear Functions Question 75 | Free SAT Prep | Aniko

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Question 75·Hard·Nonlinear Functions

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The quadratic function $h$ intersects the $x$ -axis at $(2,0)$ and $(10,0)$ and the $y$ -axis at $(0,12)$ . What is the value of $h(6)$ ?

For quadratic function questions where you are given the x-intercepts and one additional point, immediately write the function in factored form $h(x)=a(x-r_1)(x-r_2)$ using the roots, then plug the extra point in to solve for $a$ . After that, substitute the requested x-value and simplify carefully, watching your fraction arithmetic and signs. As a quick check, use the fact that values between the roots must all be above or all be below the x-axis (depending on the sign of $a$ ) to rule out answer choices with the wrong sign.

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Start from the x-intercepts

When you know the x-intercepts of a quadratic, you can write it in factored form using those intercepts as roots. How would you express a quadratic whose roots are $2$ and $10$ ?

Use the y-intercept to find the coefficient

Your factored form should look like $h(x)=a(x-2)(x-10)$ . Use the point $(0,12)$ and plug $x=0$ and $h(0)=12$ into this equation to solve for $a$ .

Substitute $x=6$

Once you know $a$ , write the full function $h(x)$ and substitute $x=6$ to get $h(6)$ . Be careful with the sign when you compute $(6-10)$ .

Check the sign

Because $x=6$ is between the two x-intercepts at $2$ and $10$ , the value of $h(6)$ must be negative. Use this to eliminate any impossible answer choices after you compute.

Desmos Guide

Define the general quadratic

In Desmos, type h(x) = a(x-2)(x-10) so that you have a slider for the parameter a.

Use the y-intercept to find the value of a

Either (a) add the point (0,12) and adjust the a slider until the curve passes through this point, or (b) type a = 12/((0-2)*(0-10)) so Desmos calculates the correct value of a for you.

Evaluate h(6) in Desmos

Once a is set, type h(6) into Desmos. The displayed value is the correct value of $h(6)$ ; compare it to the answer choices.

Step-by-step Explanation

Write the quadratic using its x-intercepts

A quadratic with x-intercepts at $(2,0)$ and $(10,0)$ can be written in factored form as

h(x) = a(x-2)(x-10)

where $a$ is a constant we still need to find.

Use the y-intercept to find the leading coefficient

We are told the graph intersects the y-axis at $(0,12)$ , so $h(0)=12$ .

Substitute $x=0$ and $h(0)=12$ into $h(x)=a(x-2)(x-10)$ and simplify:

\begin{aligned} 12 &= a(0-2)(0-10) \\ &= a(-2)(-10) \\ &= 20a \end{aligned}

Now solve for $a$ :

a = \frac{12}{20} = \frac{3}{5}

Write the full function and substitute $x=6$

Now we know

h(x) = \frac{3}{5}(x-2)(x-10).

We want $h(6)$ , so substitute $x=6$ :

h(6) = \frac{3}{5}(6-2)(6-10).

Compute inside the parentheses:

h(6) = \frac{3}{5}(4)(-4).

Simplify to get the final value

First multiply $4\cdot(-4) = -16$ :

h(6) = \frac{3}{5}(-16) = -\frac{48}{5}.

So the value of $h(6)$ is $-\dfrac{48}{5}$ , which corresponds to choice B.

Strategy

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Desmos Guide

Step-by-step Explanation

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Desmos Guide

Step-by-step Explanation