Question 473·Medium·Algebra

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A linear function $f$ satisfies $f(-2) = 7$ and $f(4) = -5$ . Which equation could represent $f$ ?

For a linear function, outputs at two different inputs determine one line. Write $f(x)=ax+b$ , then use a Desmos regression to find the coefficients $a$ and $b$ that fit both conditions. Don’t use a given output as $b$ unless its input is $0$ : $b$ is the output where the line crosses the $y$ -axis.

Hints

Hint 1
Write a linear function as $f(x)=ax+b$ . Here $a$ is the change in output for each increase of $1$ in the input, and $b$ is the output at input $0$ . How can you use each given input and output?
Hint 2
Each function value gives an input and its output. Substitute both pairs into $f(x)=ax+b$ , then put the two resulting expressions in one Desmos regression so they share the same $a$ and $b$ .

Step-by-step

Fit both function values in Desmos
Step 1
Turn the function values into equations
Write $f(x)=ax+b$ , where $a$ is the slope, the change in output per $1$ increase in $x$ , and $b$ is the $y$ -intercept, the output at $x=0$ . Substitute each given input and output:
$\begin{aligned}7&=a(-2)+b\\[1.4em]-5&=a(4)+b.\end{aligned}$
Don’t treat $7$ as the intercept: its input is $-2$ , not $0$ .
Step 2
Fit one pair of coefficients to both equations
Type $\left[a(-2)+b,a(4)+b\right]\sim\left[7,-5\right]$ . The $\sim$ tells Desmos to fit $a$ and $b$ ; the entries in matching positions form the two equations above. Under PARAMETERS, Desmos reports $a=-2$ and $b=3$ . It uses one shared pair of values: both given outputs must fit the same rule.
Step 3
Match the rule to an equation
Replace $a$ and $b$ with the fitted values:
$f(x)=-2x+3.$
The choices use $y$ for the output $f(x)$ , so the matching equation is $y=-2x+3$ . Choice D.

Question 473·Medium·Algebra

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