Question 272·Easy·Algebra

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The values of the linear function $g$ are shown in the table.

$x$	$g(x)$
-3	11
-1	7
1	3

Which equation defines $g(x)$ ?

A linear function has a constant change in output for equal changes in input. With a table, first check whether the outputs rise or fall, then enter the pairs in Desmos and fit $y_1\sim mx_1+b$ . Desmos finds the slope $m$ and the output $b$ when $x=0$ . A falling table needs a negative slope.

Hints

Hint 1
Each table row pairs an input $x$ with an output $g(x)$ . The slope tells you which way the output moves as the input increases. In the two rightmost rows, does the output rise or fall?
Hint 2
Use slope-intercept form $g(x)=mx+b$ , where $m$ is the output change per $1$ input and $b$ is the output at input $0$ . A Desmos table regression can find both numbers.

Step-by-step

Fit a line to the table
Step 1
Check which way the outputs move
From $x=-1$ to $x=1$ , the input rises by $2$ , while $g(x)$ falls from $7$ to $3$ . The slope, the output change for each increase of $1$ in the input, must be negative. An equation with $+2x$ can't match that direction.
Step 2
Find the slope and intercept in Desmos
Enter the three table rows in Desmos, with inputs under $x_1$ and outputs under $y_1$ . Then type $y_1\sim mx_1+b$ . The regression symbol $\sim$ asks Desmos to fit a line to those pairs. Under PARAMETERS, Desmos shows $m=-2$ and $b=5$ . Here $m$ is the slope, and $b$ is the output when $x=0$ .
Step 3
Write the function
Put those values into slope-intercept form $g(x)=mx+b$ :
$g(x)=-2x+5.$
The negative coefficient matches the falling outputs in the table, so this equation defines $g$ . Choice C.

Question 272·Easy·Algebra

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