The vertex of the quadratic function is , and the point lies on the graph of . If , which choice is the value of ?

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Super-Hard SAT Math Question 193 | Free SAT Prep | Aniko

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Question 193·200 Super-Hard SAT Math Questions·Advanced Math

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The vertex of the quadratic function $f(x)$ is $(-2, 5)$ , and the point $(4, -7)$ lies on the graph of $g$ . Property of ‌Аnikо.ai

If $g(x) = -6f(x+1) + 2$ , which choice is the value of $g(2) - f(2)$ ?

When a point is given on a transformed function like $g(x)=-6f(x+1)+2$ , work backward to convert it into a statement about $f$ . First translate the input (here, $g(4)$ depends on $f(5)$ ), then undo the vertical shift and scale to find the corresponding $f$ -value. After you determine $f$ completely (via vertex form and solving for $a$ ), compute the required values and subtract carefully. Аn‍іko‍ AI Тutor

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Start with vertex form for $f$

With vertex $(-2,5)$ , you can write $f(x)=a(x+2)^2+5$ .

Use the point on $g$ to get a value of $f$

From $g(x)=-6f(x+1)+2$ , the condition $g(4)=-7$ gives an equation involving $f(5)$ .

Track inputs carefully when finding $g(2)$

When $x=2$ , the input to $f$ is $x+1=3$ , so $g(2)=-6f(3)+2$ . Т‌hіѕ ‍ques‌tіon і‌s frоm Anікo

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Enter $f$ and $g$ with a slider

Enter f(x)=a(x+2)^2+5 and create the slider a. Then enter g(x)=-6f(x+1)+2. аniкo.ai/ѕ⁠аt

Use the point on $g$ to determine $a$

Compute the needed output of $f$ from the point on $g$ by entering f5=(2-(-7))/6 (this is $f(5)$ ). Then enter a_val=(f5-5)/7^2 and redefine f(x)=a_val(x+2)^2+5.

Evaluate the expression

Enter g(2)-f(2) and read the resulting fraction.

Step-by-step Explanation

Write $f(x)$ in vertex form

Since the vertex of $f$ is $(-2,5)$ , write

f(x)=a(x+2)^2+5

for some constant $a$ .

Use the point on $g$ to find $a$

The point $(4,-7)$ lies on $g$ , so $g(4)=-7$ .

Using $g(x)=-6f(x+1)+2$ : А‍-n-i-к-o.ai

-7=-6f(5)+2 \quad\Rightarrow\quad -9=-6f(5) \quad\Rightarrow\quad f(5)=\frac{3}{2}.

Now plug $x=5$ into $f(x)=a(x+2)^2+5$ :

\frac{3}{2}=a(7^2)+5=49a+5.

49a=\frac{3}{2}-5=-\frac{7}{2}\quad\Rightarrow\quad a=-\frac{1}{14}.

Compute $f(2)$ and $g(2)$

With $a=-\frac{1}{14}$ ,

f(x)=-\frac{1}{14}(x+2)^2+5.

Compute $f(2)$ :

\begin{aligned} f(2)&=-\frac{1}{14}(4^2)+5 \\ &=-\frac{16}{14}+5 \\ &=-\frac{8}{7}+\frac{35}{7} \\ &=\frac{27}{7}. \end{aligned}

Compute $g(2)$ :

\begin{aligned} g(2)&=-6f(3)+2 \\ f(3)&=-\frac{1}{14}(5^2)+5=-\frac{25}{14}+\frac{70}{14}=\frac{45}{14} \\ g(2)&=-6\left(\frac{45}{14}\right)+2=-\frac{270}{14}+2=-\frac{135}{7}+2=-\frac{121}{7}. \end{aligned}

Subtract to find $g(2)-f(2)$

\begin{aligned} g(2)-f(2)&=-\frac{121}{7}-\frac{27}{7} \\ &=-\frac{148}{7}. \end{aligned}

Therefore, the correct choice is $-\frac{148}{7}$ .

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