The functions and are defined by the equations shown, where and are integer constants and . If the graph of each function is considered only for , which of the following equations displays, as a constant or a coefficient, the maximum value of the graph of the corresponding function? I. II.

Solution available with step-by-step explanation

Super-Hard SAT Math Question 47 | Free SAT Prep | Aniko

00:00

Question 47·200 Super-Hard SAT Math Questions·Advanced Math

0 / 200

The functions $f$ and $g$ are defined by the equations shown, where $a$ and $b$ are integer constants and $a>b>0$ . If the graph of each function is considered only for $x\ge 0$ , which of the following equations displays, as a constant or a coefficient, the maximum value of the graph of the corresponding function?

I. $f(x)=a-(x-b)^2$

II. $g(x)=a-b(0.9)^x$

For maximum-value questions, first decide whether a maximum exists (is it actually reached at some $x$ in the domain?). Then use the function form: a downward-opening parabola has a maximum at its vertex, while a shifted exponential like $a-b(0.9)^x$ often approaches an asymptote. Finally, check whether the maximum value is explicitly written as a single constant or coefficient in the equation, rather than requiring substitution or limit-like reasoning.

Hints

Click me!

Look for whether a maximum is actually attained

A function can approach a value without ever reaching it. Check whether the “top” value occurs at some $x\ge 0$ .

Use key shapes

For I, identify the vertex of the parabola. For II, think about what happens to $(0.9)^x$ as $x$ gets large.

Compare to the constant $a$ in II

Decide whether $g(x)$ can ever equal $a$ when $b>0$ and $(0.9)^x$ is positive.

Desmos Guide

Graph both functions with sliders

Enter

$y=a-(x-b)^2$
$y=a-b(0.9)^x$

Create sliders for $a$ and $b$ , and set them to integers with $a>b>0$ (for example, $a=5$ , $b=2$ ).

Find the maximum of function I visually

On the parabola, click the highest point (the vertex). Confirm the $y$ -value equals $a$ for your slider settings.

Check whether function II ever reaches $y=a$

Graph the horizontal line $y=a$ . Notice the exponential curve stays below that line and gets closer as $x$ increases but never intersects it, indicating there is no maximum value attained.

Decide which statement meets the prompt condition

Since only the parabola’s maximum is actually reached and shown as the constant $a$ , select the answer choice that includes I but not II.

Step-by-step Explanation

Analyze function I (parabola)

$f(x)=a-(x-b)^2$ is a parabola opening downward because the squared term is subtracted.

Its vertex is at $x=b$ , and the vertex value is

f(b)=a-(b-b)^2=a.

Since $b>0$ , the vertex occurs in the domain $x\ge 0$ , so the maximum value of $f$ on $x\ge 0$ is $a$ , which is displayed as a constant in the equation.

Analyze function II (shifted exponential)

Because $0<0.9<1$ , the term $(0.9)^x$ is positive for all $x$ and decreases as $x$ increases.

So $-b(0.9)^x$ is negative and increases toward $0$ , meaning $g(x)=a-b(0.9)^x$ increases toward $a$ .

But $(0.9)^x$ is never $0$ for any real $x$ , so $b(0.9)^x>0$ always, and therefore

g(x)=a-b(0.9)^x<a

for every $x\ge 0$ . The value $a$ is a horizontal asymptote, not a maximum value.

Choose the correct option

Only statement I corresponds to a function whose maximum value is shown directly as a constant (the $a$ in $f(x)$ ), while statement II has no maximum value on $x\ge 0$ .

Therefore, the correct choice is I only.

Strategy

Hints

Desmos Guide

Step-by-step Explanation

Study Hints

Strategy

Hints

Desmos Guide

Step-by-step Explanation