Question 713·Hard·Math

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The linear function $f$ satisfies $f(3p-2)=4p+5$ and $f(5p+1)=-2p+11$ , where $p$ is a constant. Which expression gives the slope of $f$ in terms of $p$ ?

Two function values give two input-output points on a line. Its slope is the change in output divided by the change in input, so subtract the two points in the same order on top and bottom. Simplify the result in $p$ to match a choice. Desmos can check the arithmetic at one sample value, but that check cannot prove the expression for every $p$ .

Hints

Hint 1
The notation $f(a)=b$ means input $a$ gives output $b$ , so it gives the point $(a,b)$ . Turn each function statement into a point, with the expression inside the parentheses first.
Hint 2
Slope is output change divided by input change. If you start with the second point's output on top, which input must you start with on the bottom? Keep the subtraction order the same.
Hint 3
Subtracting an expression in parentheses changes the sign of every term inside it. Distribute the minus signs in both differences before combining terms with $p$ .

Step-by-step

Find slope from two function values
Step 1
Read the two points
In $f(a)=b$ , $a$ is the input and $b$ is the output. So the two function statements give the points $(3p-2,4p+5)\quad\text{and}\quad(5p+1,-2p+11)$ . Keep each output with its own input.
Step 2
Write the slope
The slope is output change divided by input change. Take the second point minus the first in both places: $m=\frac{(-2p+11)-(4p+5)}{(5p+1)-(3p-2)}$ . Subtracting in opposite orders would flip the slope's sign.
Step 3
Check the subtraction in Desmos
As a zero check, type $p=1$ , then the two input expressions and the slope fraction into Desmos. The inputs show $1$ and $6$ , so they're different; the fraction shows $0$ . This checks one value of $p$ . Simplifying the fraction will show which expression works generally.
Step 4
Distribute the subtraction on top
Subtract both terms of the first output: $(-2p+11)-(4p+5)=-2p+11-4p-5$ . The $+5$ becomes $-5$ because the whole expression is being subtracted.
Step 5
Combine the output change
The terms containing $p$ are like terms: combine them, then combine the plain numbers: $-2p+11-4p-5=-6p+6$ . Keep the negative sign on $-6p$ .
Step 6
Distribute the subtraction on the bottom
Subtract both terms of the first input: $(5p+1)-(3p-2)=5p+1-3p+2$ . Subtracting $-2$ gives $+2$ .
Step 7
Combine the input change
Combine the $p$ terms and the plain numbers in the denominator, the bottom of the fraction: $5p+1-3p+2=2p+3$ . This is the input change, not the output change.
Step 8
Factor to match the slope
Factor $-6$ from the output change, leaving $p-1$ inside parentheses: $m=\frac{-6p+6}{2p+3}=\frac{-6(p-1)}{2p+3}$ . So the slope of $f$ is $\dfrac{-6(p-1)}{2p+3}$ . Choice D.

Question 713·Hard·Math

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