Question 18·SAT Desmos Regression Practice·Algebra

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The function $f$ is defined by

f(x)=3x+m(2x+1),

where $m$ is a constant and $f(2)=25$ . What is the value of $f(-3)$ ?

Your answer

A function with an unknown parameter, a fixed number in its rule, and a known input-output pair gives you an equation for that parameter. Find the constant from the known output before evaluating the new input. A Desmos regression can find the constant; then you can use it in the function. At a negative input, watch the sign of the entire parenthesized factor.

Hints

Hint 1
In function notation, $f(2)=25$ means that input $2$ gives output $25$ . Replace both occurrences of $x$ with $2$ , then set the resulting expression equal to $25$ . What equation does that give you for $m$ ?
Hint 2
A regression asks Desmos to find an unknown value that makes an equation work. Enter your equation with $\sim$ instead of $=$ , and read $m$ under PARAMETERS.
Hint 3
Once you know $m$ , evaluate the function at $-3$ . Keep the negative input inside parentheses when you consider $2x+1$ : is that factor positive or negative?

Step-by-step

Find the constant, then evaluate
Step 1
Turn the known output into an equation
In function notation, $f(2)=25$ means input $2$ gives output $25$ . Put $2$ in for both occurrences of $x$ and set the result equal to $25$ :
$3(2)+m(2(2)+1)=25.$
Simplify the known numbers:
$6+5m=25.$
Step 2
Find the constant
Type $6+5m\sim25$ . The regression symbol $\sim$ tells Desmos to find the unknown constant $m$ rather than graph an equation. Under PARAMETERS, Desmos shows $m=3.8$ .
Step 3
Evaluate at the requested input
At $x=-3$ , the factor $2(-3)+1=-5$ , so the $m$ term is negative, not positive. Type $f(x)=3x+m(2x+1)$ , then $f(-3)$ . Desmos uses the value it found for $m$ and shows $-28$ . The output at input $-3$ is $-28$ . Grid in -28.

Question 18·SAT Desmos Regression Practice·Algebra

Strategy

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