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 $p(19)$ ?

Three points on a quadratic function, with no root or vertex given, are a cue to use quadratic regression. Enter the points in a Desmos table, fit a quadratic, and evaluate it at the requested input. If the inputs are equally spaced, you can also track how the output increases change. Repeating one increase treats the function as linear.

Hints

Hint 1
A point $(u,v)$ on $y=p(x)$ means $p(u)=v$ : the first coordinate is the input, and the second is the output. Which coordinate belongs in each Desmos table column?
Hint 2
The function is quadratic, so fit the table with $y_1\sim ax_1^2+bx_1+c$ . The symbol $\sim$ tells Desmos to find the three coefficients from the paired values.
Hint 3
$p(19)$ means put 19 in for $x$ . After Desmos finds the coefficients, define $p(x)$ with those stored letters and evaluate it at the new input.

Step-by-step

Approach 1: Fit the quadratic in Desmos
Step 1
Pair each input with its output
Enter the points in a Desmos table. Put each input in the $x_1$ column and its output in the same row of the $y_1$ column, so Desmos fits $p(x)$ rather than reversing the coordinates.
Step 2
Find the quadratic coefficients
Type $y_1\sim ax_1^2+bx_1+c$ . This regression finds the quadratic that fits all three points. Under PARAMETERS, Desmos reports $a\approx0.8333$ , $b\approx11.3333$ , and $c\approx2.8333$ .
Step 3
Define the fitted function
Type $p(x)=ax^2+bx+c$ . Use Desmos's stored coefficients, not copies of the rounded decimals it displays. This gives you a function you can evaluate at a new input.
Step 4
Evaluate at the requested input
Type $p(19)$ . Desmos prints $519$ , the output at the new input, not a coefficient or one of the given outputs. So $p(19)=519$ . Choice C.

Approach 2: Use constant second differences
Step 1
Find the output rises
The inputs $1,7,13,19$ are each 6 apart, so compare outputs over equal input jumps. Type $d=[123-15,291-123]$ in Desmos. It stores the two rises, $[108,168]$ .
Step 2
Find how much the rises change
Type $t=d[2]-d[1]$ ; Desmos prints $60$ . This is the second difference, the change between consecutive rises. For equal input jumps, a quadratic has a constant second difference.
Step 3
Find the next rise
Add the same $60$ to the most recent rise: type $n=d[2]+t$ . Desmos prints $228$ . The next rise isn't another $168$ , because the rises of this quadratic are changing.
Step 4
Add the rise to the last output
The output at $13$ is $291$ , so type $291+n$ . Desmos prints $519$ , the output at $19$ . So $p(19)=519$ . Choice C.

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

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