Function is defined by where and are constants with . In the -plane, the graph of passes through the points and . What is the value of ?

Solution available with step-by-step explanation

SAT Nonlinear Functions Question 108 | Free SAT Prep | Aniko

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Question 108·Hard·Nonlinear Functions

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Function $g$ is defined by

g(x)=k\,r^{x}-1,

where $k$ and $r$ are constants with $r>0$ . In the $xy$ -plane, the graph of $y=g(x)$ passes through the points $(1,7)$ and $(3,31)$ . What is the value of $r$ ?

Your answer

(Express the answer as an integer)

For exponential function questions where the function has unknown parameters and you’re given points, plug each point into the function to create equations, then use algebra to eliminate one variable. Often dividing one equation by the other is the fastest way to cancel a common factor (like $k$ here) and reduce the problem to a single equation in one variable. After solving, always check any domain or sign conditions (such as $r>0$ ) before choosing your final answer.

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Use the definition of g(x) with each point

Substitute $x=1$ , $y=7$ into $g(x)=k r^x -1$ to get one equation, and substitute $x=3$ , $y=31$ to get another equation.

Create a system and think about eliminating a variable

Your two equations will both include $k$ and $r$ . How can you combine them (for example, using division) so that $k$ disappears and only powers of $r$ remain?

Pay attention to the condition on r

After you get an equation involving $r$ only, you may end up with two possible values. Use the fact that $r>0$ to decide which value to keep.

Desmos Guide

Express k in terms of r from the first point

In Desmos, enter the expression k = 8/r. This comes from the equation $k r = 8$ based on the point $(1,7)$ .

Use the second point to form an equation in r only

Enter a second expression using the equation from $(3,31)$ : 8*r^2 - 1. This comes from substituting $k = 8/r$ into $k r^3 - 1$ .

Find the value of r that makes the expression equal to 31

Create a table for the expression 8*r^2 - 1 or use a slider for $r$ and adjust it until the expression equals 31. The positive $r$ value that makes 8*r^2 - 1 = 31 is the answer.

Step-by-step Explanation

Write equations from the two given points

The function is

g(x)=k r^x -1.

Use each point $(x,y)$ to get an equation.

From $(1,7)$ :

7 = k r^1 - 1

k r = 8.

From $(3,31)$ :

31 = k r^3 - 1

k r^3 = 32.

Eliminate k by dividing the equations

You now have a system:

k r = 8 \quad\text{and}\quad k r^3 = 32.

Divide the second equation by the first to cancel $k$ :

\frac{k r^3}{k r} = \frac{32}{8}.

This simplifies to

r^2 = 4.

Solve for r using r > 0

From

r^2 = 4,

you get two possible values:

r = 2 \quad\text{or}\quad r = -2.

But the problem says $r>0$ , so the only valid value is

\boxed{2}.

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Desmos Guide

Step-by-step Explanation

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Desmos Guide

Step-by-step Explanation