The function is defined for all real numbers by where and are positive constants. The graph of in the -plane passes through the points and . What is the value of ?

Solution available with step-by-step explanation

SAT Nonlinear Functions Question 54 | Free SAT Prep | Aniko

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Question 54·Hard·Nonlinear Functions

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

h(x)=p^{x}+q^{x},

where $p$ and $q$ are positive constants. The graph of $y=h(x)$ in the $xy$ -plane passes through the points $(1,9)$ and $(2,41)$ . What is the value of $pq$ ?

For exponential-function questions with unknown bases but known function values at a few points, first convert each point into an equation by substituting its $x$ -coordinate into the function. Then, instead of solving for each unknown separately, focus on what the question actually asks for (such as a sum or product) and look for algebra identities like $(p+q)^2=p^2+2pq+q^2$ that connect your equations to that target quantity. This approach saves time, reduces algebra, and avoids unnecessary work solving the full system.

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Turn the graph information into equations

Use the fact that the point $(x,y)$ lies on the graph of $y=h(x)$ to write equations for $x=1$ and $x=2$ . What do you get for $h(1)$ and $h(2)$ ?

Write out what you know about $p$ and $q$

From the two equations you just formed, you should have values for $p+q$ and for $p^2+q^2$ . Write them clearly; you will use both together.

Use an algebra identity instead of solving for $p$ and $q$

You want $pq$ , not $p$ and $q$ separately. Recall the expansion of $(p+q)^2$ . How can you use $p+q$ and $p^2+q^2$ in that identity to isolate $pq$ ?

Substitute and solve for $pq$

After writing $(p+q)^2=p^2+2pq+q^2$ , plug in your known values for $p+q$ and $p^2+q^2$ , then solve the resulting equation for $pq$ .

Desmos Guide

Graph the two equations in the $(p,q)$ -plane

In Desmos, treat $p$ as the horizontal axis and $q$ as the vertical axis. Type the equations

p+q=9
p^2+q^2=41 These will appear as a straight line and a circle; their intersection points correspond to the possible $(p,q)$ pairs.

Find a valid intersection point

Zoom or pan until you see where the line and circle intersect in the first quadrant (both coordinates positive). Click on one of those intersection points and note its coordinates; these are values of $p$ and $q$ that satisfy both equations.

Compute the product using Desmos

Using the coordinates you just read (call them $p$ and $q$ ), type a new expression like p_value * q_value, replacing p_value and q_value with the actual numbers from the intersection. The resulting output is the value of $pq$ you should choose from the answer options.

Step-by-step Explanation

Translate the points into equations

We are given

h(x)=p^x+q^x

and that the graph passes through $(1,9)$ and $(2,41)$ .

Plug in $x=1$ :

h(1)=p^1+q^1=p+q=9

Plug in $x=2$ :

h(2)=p^2+q^2=41

So we know:

$p+q=9$
$p^2+q^2=41$

Focus on what you are asked for: $pq$

We are not asked to find $p$ and $q$ individually; we only need the product $pq$ .

There is a standard algebra identity that connects $p+q$ , $p^2+q^2$ , and $pq$ :

(p+q)^2 = p^2 + 2pq + q^2

We already know $p+q$ and $p^2+q^2$ , so this identity is the key to finding $pq$ directly.

Apply the identity with the known values

Substitute $p+q=9$ and $p^2+q^2=41$ into

(p+q)^2 = p^2 + 2pq + q^2.

This gives

9^2 = (p^2+q^2) + 2pq.

Now replace $p^2+q^2$ with $41$ :

9^2 = 41 + 2pq.

Compute $9^2$ :

81 = 41 + 2pq.

Now isolate $2pq$ :

2pq = 81 - 41 = 40.

Solve for $pq$ and select the answer

From the previous step we have

2pq = 40.

Divide both sides by $2$ :

pq = \frac{40}{2} = 20.

So the value of $pq$ is $20$ , which corresponds to answer choice B) 20.

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Step-by-step Explanation

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Step-by-step Explanation