Question 831·Medium·Math

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The linear function $f$ satisfies $f(-2)=7$ and $f(6)=-1$ . Which equation defines $f$ ?

Two input-output values for a linear function give two points on the same line. Write $f(x)=mx+b$ , then use a Desmos list regression to find the slope $m$ and intercept $b$ that fit both points. Don't choose a rule because it matches only one value: several lines can pass through one point.

Hints

Hint 1
In $f(a)=c$ , $a$ is the input and $c$ is the output. A line in slope-intercept form is $f(x)=mx+b$ . What equations do you get by putting each given input into that rule?
Hint 2
A regression lets Desmos find $m$ and $b$ together. Match the list $[f(-2),f(6)]$ to the given outputs, keeping them in the same order.

Step-by-step

Approach 1: Fit both function values in Desmos

Step 1
Turn the function values into equations
A line has slope-intercept form $f(x)=mx+b$ : $m$ is the output change for each $1$ added to $x$ , and $b$ is the output when $x=0$ . Put the two given inputs and outputs into that form to make two equations:
$f(-2)=-2m+b=7$ $f(6)=6m+b=-1.$
Step 2
Find one slope and intercept that fit both
Type $f(x)=mx+b$ , then $[f(-2),f(6)]\sim[7,-1]$ in Desmos. The tilde tells Desmos to fit the first list to the second. Under PARAMETERS, it reports $m=-1$ and $b=5$ . Both conditions must use the same $m$ and $b$ .
Step 3
Write the function
Replace $m$ and $b$ with the values Desmos found:
$f(x)=(-1)x+5.$
Simplify the multiplication:
$f(x)=-x+5.$
That's the rule with both required function values. Choice C.

Approach 2: Find the slope and intercept by hand

Step 1
Find the change per input
The slope is the change in output divided by the change in input. Subtract the values from $f(-2)=7$ and $f(6)=-1$ in the same order on top and bottom:
$m=\frac{-1-7}{6-(-2)}.$
Subtract:
$m=\frac{-8}{8}.$
Divide:
$m=-1.$
Step 2
Find the output at zero
Use $f(-2)=7$ in $f(x)=mx+b$ to find the y-intercept $b$ , the output when $x=0$ . Substitute $m=-1$ :
$7=(-1)(-2)+b.$
Multiply:
$7=2+b.$
Subtract $2$ from both sides:
$5=b.$
The given output $7$ isn't the intercept because its input is $-2$ , not $0$ .
Step 3
Combine the two parts of the line
Put the slope $-1$ and intercept $5$ into $f(x)=mx+b$ :
$f(x)=-x+5.$
The slope and one point determine the given line, so this equation defines $f$ . Choice C.

Question 831·Medium·Math

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