Super-Hard SAT Advanced Math Question 15 | Free SAT Prep

00:00

Try Full-Screen!

Question 15·25 Super-Hard SAT Advanced Math Questions·Advanced Math

0 / 25 Questions completed

A quadratic function $f$ is defined by $f(x)=ax^2+bx+c$ , where $a$ , $b$ , and $c$ are constants and $a\ne0$ . Function $g$ is defined by $g(x)=f(x+2)-f(x-2)$ . In the $xy$ -plane, the graph of $y=g(x)$ passes through $(1,0)$ . The graph of $y=f(x)$ has two $x$ -intercepts whose $x$ -coordinates have a product of $-5$ , and the $y$ -intercept of $y=f(x)$ is $(0,35)$ . What is the value of $f(3)$ ?

Your answer

When a problem defines a new function by adding or subtracting shifted versions of a quadratic, expand only enough to see how the coefficients change. Then translate intercept facts into coefficient relationships: the $y$ -intercept is $c$ , and the product of the $x$ -intercepts is $\frac{c}{a}$ . Combine these relationships before substituting into the requested function value.

Hints

Click me!

Simplify $g(x)$

Write expressions for $f(x+2)$ and $f(x-2)$ , then subtract. Notice which terms cancel.

Use the point on $g$

The point $(1,0)$ means that $g(1)=0$ . Use this to relate $a$ and $b$ .

Use the product of the intercepts

For $ax^2+bx+c$ , the product of the two $x$ -intercepts is $\frac{c}{a}$ . The $y$ -intercept identifies $c$ .

Desmos Guide

Use algebra first

Algebra is faster because the conditions determine the coefficients exactly. After finding an equation for $f$ , Desmos can verify the result.

Graph the resulting quadratic

Enter $y=-7x^2+14x+35$ . Confirm that its $y$ -intercept is $35$ and click its two $x$ -intercepts; their coordinates have product $-5$ .

Verify the transformed condition

Enter $y=(-7(x+2)^2+14(x+2)+35)-(-7(x-2)^2+14(x-2)+35)$ . Confirm that the graph has an $x$ -intercept at $x=1$ .

Evaluate the requested value

Enter $-7(3)^2+14(3)+35$ in a Desmos expression line to verify the value of $f(3)$ .

Step-by-step Explanation

Use the definition of $g$

Expanding the difference gives $g(x)=f(x+2)-f(x-2)=8ax+4b$ . Since $g(1)=0$ , $8a+4b=0$ , so $b=-2a$ .

Use the intercept information

The $y$ -intercept of $f$ is $(0,35)$ , so $c=35$ . For a quadratic $ax^2+bx+c$ , the product of the $x$ -intercepts is $\frac{c}{a}$ . Therefore, $\frac{35}{a}=-5$ , which gives $a=-7$ .

Find the remaining coefficient and evaluate

Because $b=-2a$ , $b=14$ . Thus $f(x)=-7x^2+14x+35$ , and $f(3)=-7(3)^2+14(3)+35=14$ .

Strategy

Hints

Desmos Guide

Step-by-step Explanation

Study Hints

Strategy

Hints

Desmos Guide

Step-by-step Explanation