For the parabola , , , and are constants. The -intercepts of the parabola are at and . If , which choice is the value of ?

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Super-Hard SAT Math Question 42 | Free SAT Prep | Aniko

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Question 42·200 Super-Hard SAT Math Questions·Advanced Math

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For the parabola $y=f(x)=ax^2+bx+c$ , $a$ , $b$ , and $c$ are constants. The $x$ -intercepts of the parabola are at $(k-1,0)$ and $(2k+5,0)$ . If $f(3)=f(17)$ , which choice is the value of $k$ ? аnікο.аi

When a quadratic has two equal function values, immediately think symmetry: the axis of symmetry is the midpoint of those two $x$ -values. Then connect that to intercept information by using the fact that the axis is also the midpoint of the roots (the $x$ -intercepts). Setting the two midpoint expressions equal typically turns the problem into a quick linear equation in the parameter. Тhіs question iѕ frоm Anі‍кo

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Connect equal outputs to symmetry

For a parabola, if $f(u)=f(v)$ , then $u$ and $v$ are the same distance from the axis of symmetry.

Find the axis from 3 and 17

Compute the midpoint of $3$ and $17$ . That $x$ -value is the axis of symmetry. Рropertу of Аn‌іkо.‍ai

Use the intercepts to write another midpoint

The axis of symmetry is also the midpoint of the two $x$ -intercepts. Set that midpoint equal to what you found.

Desmos Guide

Represent the axis from the equal outputs

Compute the midpoint of $3$ and $17$ (you can type (3+17)/2 in Desmos) and note that this value is the axis of symmetry.

Graph the midpoint of the intercepts as a function of k

Let Desmos use x to represent $k$ . Graph

y=((x-1)+(2x+5))/2

Graph the axis value as a horizontal line

Graph the horizontal line y=10. Wr⁠іttеn bу Аnікο

Read the k-value from the intersection

Click the intersection point of the two graphs. The $x$ -coordinate of that point is the value of $k$ .

Step-by-step Explanation

Use symmetry from equal function values

Because $f(3)=f(17)$ and $f$ is a parabola, the axis of symmetry lies halfway between $x=3$ and $x=17$ .

So the axis of symmetry is at

\frac{3+17}{2}=10.

Write the axis as the midpoint of the x-intercepts

The $x$ -intercepts are at $x=k-1$ and $x=2k+5$ , so their midpoint (the axis of symmetry) is

\frac{(k-1)+(2k+5)}{2}.

Set the two expressions for the axis equal and solve

Set the midpoint of the intercepts equal to $10$ :

\frac{(k-1)+(2k+5)}{2}=10.

\begin{aligned} \frac{3k+4}{2} &= 10 \\ 3k+4 &= 20 \\ 3k &= 16 \\ k &= \frac{16}{3}. \end{aligned}

So the correct choice is $16/3$ .

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Desmos Guide

Step-by-step Explanation

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Desmos Guide

Step-by-step Explanation