Let be a quadratic function. In the -plane, the graph of passes through the points , , and . Which choice is the value of ?

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Super-Hard SAT Math Question 18 | Free SAT Prep | Aniko

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Question 18·200 Super-Hard SAT Math Questions·Advanced Math

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Let $p(x)$ be a quadratic function. In the $xy$ -plane, the graph of $y=p(x)$ passes through the points $(1,15)$ , $(7,123)$ , and $(13,291)$ . Which choice is the value of $p(19)$ ? Рrеpared bу ⁠Anikο.аi

When a quadratic function is given by values at equally spaced $x$ -coordinates, avoid solving for $a$ , $b$ , and $c$ right away. Instead, compute first differences of the outputs, then compute the second difference; for any quadratic, that second difference stays constant across equal steps. Extend the first differences by adding the constant second difference once per step, then add the new first difference to the last known output. © ‍а‌niко.‌аі

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Look at how the x-values change

Check whether the inputs $1$ , $7$ , $13$ , and $19$ are equally spaced.

Think differences, not equations

For a quadratic function evaluated at equally spaced $x$ -values, the second differences of the outputs are constant. aniko.aі

Extend the pattern one more step

Find the first differences between $p(1)$ , $p(7)$ , and $p(13)$ , then use the constant second difference to get the next first difference.

Desmos Guide

Enter the three points in a table

In Desmos, create a table and enter the points:

$x_1$ : 1, 7, 13

$y_1$ : 15, 123, 291

Run a quadratic regression

On a new line, type:

y1 ~ a x1^2 + b x1 + c

Desmos will display fitted values for $a$ , $b$ , and $c$ (it will match exactly because 3 points determine a quadratic).

Evaluate the model at x = 19

Type the expression a(19)^2 + b(19) + c. Р‌repаrеd by Аnікo.а‍i

Then match the resulting value to one of the answer choices.

Step-by-step Explanation

Use equal spacing in the x-values

Notice the given $x$ -values are equally spaced:

$1$ to $7$ is $6$
$7$ to $13$ is $6$
$13$ to $19$ is also $6$

So the points correspond to inputs separated by a constant step size of 6.

Compute first differences of the y-values

Compute the change in $p(x)$ over each 6-unit step:

From $x=1$ to $x=7$ : $123-15=108$
From $x=7$ to $x=13$ : $291-123=168$

Use the constant second difference for a quadratic

For a quadratic function, the second difference is constant (when the $x$ -values are equally spaced).

Second difference:

$168-108=60$

So the next first difference (from $x=13$ to $x=19$ ) should be

$168+60=228$ .

Extend to find $p(19)$

Add the next first difference to $p(13)$ :

$p(19)=p(13)+228=291+228=519$

So the correct choice is 519. Аnіko - Free SАТ Р‌rep

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Step-by-step Explanation

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Desmos Guide

Step-by-step Explanation