The function is defined by , where and are constants. If and , what is the value of ?

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SAT Nonlinear Functions Question 170 | Free SAT Prep | Aniko

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Question 170·Medium·Nonlinear Functions

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The function $p$ is defined by $p(x)=a\,3^{x}+b$ , where $a$ and $b$ are constants. If $p(1)=10$ and $p(3)=100$ , what is the value of $p(2)$ ?

When a function on the SAT is given with unknown parameters (like $p(x)=a\,3^x+b$ ) and you are told its values at two points, immediately turn those into two equations and solve the resulting linear system for the parameters, usually by subtracting the equations to eliminate one variable. Then, instead of fully solving for every parameter if it’s not necessary, look for a way to express the desired quantity (here $p(2)$ ) in terms of what you already know (such as differences like $p(2)-p(1)$ ) so you can plug in quickly and avoid extra algebra.

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Use the definition of the function

Write out what $p(1)$ and $p(3)$ equal using the formula $p(x) = a\,3^x + b$ . This should give you two equations involving $a$ and $b$ .

Solve for the coefficient $a$ first

You should now have equations of the form $3a + b = 10$ and $27a + b = 100$ . How can you combine these to eliminate $b$ and find $a$ ?

Find how $p(2)$ compares to $p(1)$

Once you know $a$ , write expressions for $p(2)$ and $p(1)$ in terms of $a$ and $b$ . What is $p(2) - p(1)$ ? Use this difference and the fact that $p(1)=10$ to get $p(2)$ .

Avoid solving for both $a$ and $b$ if you can

You don't actually need the exact value of $b$ to find $p(2)$ . Focus on the difference $p(2) - p(1)$ , which cancels out $b$ .

Desmos Guide

Solve for $a$ and $b$ using a graph of the system

In two expression lines, type the equations 3a + b = 10 and 27a + b = 100. Make sure Desmos is in the $(a,b)$ -plane (it will treat a and b as variables like x and y). Find the intersection point of these two lines; its coordinates give you the values of $a$ and $b$ .

Evaluate $p(2)$ with the found parameters

In a new expression line, type p(x) = a*3^x + b using the $a$ and $b$ values from the intersection (or just keep a and b as the parameters Desmos already knows). Then type p(2) in another line. The numeric output that Desmos shows for p(2) is the value you should choose from the answer options.

Step-by-step Explanation

Write equations using the given function values

The function is defined by $p(x) = a\,3^x + b$ .

Use the given information:

For $x=1$ : $p(1) = a\,3^1 + b = 10$ , so

3a + b = 10

For $x=3$ : $p(3) = a\,3^3 + b = 100$ , so

27a + b = 100

Now you have a system of two equations in $a$ and $b$ . (We will use this to find $a$ first.)

Eliminate $b$ to solve for $a$

Subtract the first equation from the second to eliminate $b$ :

(27a + b) - (3a + b) = 100 - 10

This simplifies to

24a = 90

a = 90/24 = 15/4.

Now you know $a$ .

Relate $p(2)$ to $p(1)$ using the form of the function

Notice that

p(2) = a\,3^2 + b = a\cdot 9 + b,

and

p(1) = a\,3^1 + b = a\cdot 3 + b.

Subtract these to see how much $p(2)$ is above $p(1)$ :

p(2) - p(1) = (9a + b) - (3a + b) = 6a.

So once we know $a$ , we can find $p(2)$ from $p(1)$ by adding $6a$ .

Compute the value of $p(2)$

We know $a = 15/4$ and $p(1) = 10$ .

From the relation in the previous step:

p(2) - p(1) = 6a = 6\cdot 15/4 = 90/4 = 22.5.

p(2) = p(1) + 22.5 = 10 + 22.5 = 32.5.

Therefore, the value of $p(2)$ is 32.5, which corresponds to choice B.

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