The function is defined for all real numbers by where , with , and and are constants. The graph of passes through the points and and has a horizontal asymptote at . What is the value of ?

Solution available with step-by-step explanation

SAT Nonlinear Functions Question 90 | Free SAT Prep | Aniko

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Question 90·Hard·Nonlinear Functions

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The function $h$ is defined for all real numbers $x$ by

h(x)=m\,k^{\,x+c}+n,

where $m>0$ , $k>0$ with $k\neq1$ , and $c$ and $n$ are constants. The graph of $y=h(x)$ passes through the points $(-1,9)$ and $(3,153)$ and has a horizontal asymptote at $y=3$ . What is the value of $k$ ?

For exponential functions written as $m k^{x+c} + n$ , immediately identify the horizontal asymptote as $y=n$ to fix the vertical shift. Then plug in any given points to form equations; when you have two equations with the same unknown parameters, divide them to cancel shared factors (like $m$ and the $k^c$ part), which usually leaves a simple power equation in the base $k$ . From there, either solve the resulting exponent equation directly (e.g., from $k^4=25$ ) or quickly test the answer choices by raising each to the required power and seeing which matches the constant on the other side.

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Start with the asymptote

In an exponential function of the form $h(x)=m k^{x+c}+n$ with $m>0$ and $k>0$ , what part of the equation represents the horizontal asymptote? Use this to find $n$ .

Use each point to form equations

After you find $n$ , plug in $x=-1$ and $h(x)=9$ to get one equation, and then plug in $x=3$ and $h(x)=153$ to get another equation. Keep the exponents in the form $c-1$ and $c+3$ to make the next step easier.

Eliminate unknowns by dividing

You will have two equations involving $m$ , $k$ , and $c$ . How can you combine them (for example, by dividing one by the other) so that $m$ and $c$ cancel, leaving an equation with only $k$ ?

Solve the exponent equation

After simplifying the division, you should get an equation of the form $k^4 = \text{(a number)}$ . Use properties of exponents or test the answer choices to find $k$ .

Desmos Guide

Reduce the problem to an equation in $k$

First, use algebra (as in the written solution) to get an equation involving only $k$ . From the two points and the asymptote, you should arrive at an equation of the form $k^4 = 25$ .

Graph to find the positive solution

In Desmos, enter the function y = x^4 - 25. Look for the positive $x$ -intercept (where the graph crosses the $x$ -axis). The $x$ -coordinate of that intercept is the value of $k$ .

Step-by-step Explanation

Use the horizontal asymptote to find $n$

The function is

h(x)=m k^{x+c}+n.

For an exponential function of this form with $m>0$ and $k>0$ , the horizontal asymptote is $y=n$ because $k^{x+c}$ goes to $0$ as $x$ goes to $-\infty$ (if $0<k<1$ ) or $+\infty$ (if $k>1$ ).

The problem says the horizontal asymptote is $y=3$ , so

n=3.

Now the function is $h(x)=m k^{x+c}+3$ . This reduces the number of unknowns to $m$ , $k$ , and $c$ .

Plug in the first point $(-1,9)$

Because $(-1,9)$ is on the graph of $y=h(x)$ , we have $h(-1)=9$ .

Substitute $x=-1$ and $n=3$ into $h(x)=m k^{x+c}+3$ :

9 = m k^{-1+c} + 3.

Subtract $3$ from both sides:

6 = m k^{-1+c}.

Write the exponent in a standard order:

6 = m k^{c-1}.

So one equation is

(1)\quad m k^{c-1} = 6.

Plug in the second point $(3,153)$

Because $(3,153)$ is on the graph, we have $h(3)=153$ .

Substitute $x=3$ and $n=3$ into $h(x)=m k^{x+c}+3$ :

153 = m k^{3+c} + 3.

Subtract $3$ from both sides:

150 = m k^{3+c}.

Write the exponent in a standard order:

150 = m k^{c+3}.

So a second equation is

(2)\quad m k^{c+3} = 150.

Eliminate $m$ and $c$ using division

We now have two equations involving $m$ , $k$ , and $c$ :

(1)\; m k^{c-1} = 6, \quad (2)\; m k^{c+3} = 150.

Divide equation (2) by equation (1):

\frac{m k^{c+3}}{m k^{c-1}} = \frac{150}{6}.

On the left, $m$ cancels and the exponents of $k$ subtract:

\frac{m k^{c+3}}{m k^{c-1}} = k^{(c+3)-(c-1)} = k^{4}.

On the right,

\frac{150}{6} = 25.

So we get

(3)\quad k^{4} = 25.

Solve for $k$ and match to the choices

From step 4 we have

k^4 = 25.

We need the positive value of $k$ (the problem states $k>0$ ). Taking the fourth root of both sides gives

k = 25^{1/4}.

Rewrite $25$ as $5^2$ :

25^{1/4} = (5^2)^{1/4} = 5^{2/4} = 5^{1/2} = \sqrt{5}.

Thus the value of $k$ is $\sqrt{5}$ , which corresponds to choice B.

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Step-by-step Explanation

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Strategy

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Desmos Guide

Step-by-step Explanation