Question 87·200 Super-Hard SAT Math Questions·Advanced Math

0 / 200 Questions completed

The function $f$ is quadratic. If $f(0)=1$ and $f(1)=f(5)=10$ , which choice is the value of $f(10)$ ?

For a quadratic function, two inputs with the same output become zeros after you subtract that output. Use those zeros to write a factored expression, then use the third function value to find its scale. A one-unknown regression in Desmos can find that scale before you evaluate the requested input. Equal outputs aren't zeros of the original function.

Hints

Hint 1
A zero is an input where a function's output is $0$ . The two matching outputs here are $10$ , not $0$ . What could you subtract from $f(x)$ to make both inputs zeros?
Hint 2
If a quadratic is $0$ at $x=1$ and $x=5$ , its expression has factors $(x-1)$ and $(x-5)$ . Include an unknown scale and restore the output you subtracted.
Hint 3
Use $f(0)=1$ to find the scale: plugging in $0$ gives a number for each factor. Then evaluate at the requested input rather than stopping when you've found the scale.

Step-by-step

Approach 1: Use the matching outputs
Step 1
Turn the matching outputs into zeros
Both $1$ and $5$ give an output of $10$ , so they are zeros, inputs that give $0$ , of $f(x)-10$ . Subtract the shared output before turning the inputs into factors.
Step 2
Write the formula from the zeros
A quadratic with those zeros has factors $(x-1)$ and $(x-5)$ . Restore that output to write $f(x)=a(x-1)(x-5)+10$ . Here $a$ is the unknown scale. Without $+10$ , the function would give $0$ , not $10$ , at both inputs.
Step 3
Use the remaining value to find the scale
The given $f(0)=1$ means put $0$ in for $x$ and set the output equal to $1$ : $1=a(0-1)(0-5)+10$ . This input is different from $1$ and $5$ , so the equation can tell you the scale instead of repeating a value already built into the formula.
Step 4
Fit the one unknown in Desmos
Type $1\sim x_1(0-1)(0-5)+10$ , using $x_1$ for the unknown $a$ . The tilde tells Desmos to fit that unknown. Under PARAMETERS, Desmos shows the scale $x_1=-1.8$ . That's a number for the formula, not the requested function value.
Step 5
Evaluate the requested input
For $f(10)$ , put $10$ in for $x$ in the same formula. Type $x_1(10-1)(10-5)+10$ on a new line. Desmos uses its stored scale and prints $-71$ , so $f(10)=-71$ . Choice A.

Approach 2: Fit the quadratic from three points
Step 1
Fit all three values at once
Each $f(u)=v$ gives a point $(u,v)$ , with the input first. Put $(0,1)$ , $(1,10)$ , and $(5,10)$ in a Desmos table, then type $y_1\sim ax_1^2+bx_1+c$ . This quadratic regression finds the coefficients without using the matching-output pattern. Under PARAMETERS, Desmos shows $a=-1.8$ , $b=10.8$ , and $c=1$ .
Step 2
Use the fitted coefficients
Type $a(10)^2+b(10)+c$ . Desmos uses the stored coefficients and prints $-71$ , so the quadratic gives $f(10)=-71$ . Choice A.

Question 87·200 Super-Hard SAT Math Questions·Advanced Math

Strategy

Hints

Step-by-step

Lessons that teach this

Study Hints

Strategy

Hints

Step-by-step

Lessons that teach this