Question 632·Easy·Algebra

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Some values of the linear function $f$ are shown in the table below.

$x$	$f(x)$
1	5
3	13
5	21

Which of the following defines $f$ ?

A table of values for a linear function gives points on one line. Enter the rows in Desmos, then fit $y_1\sim mx_1+b$ to find the slope $m$ and the output $b$ when $x=0$ . Use those values to write the rule. The tempting trap is treating the first output as $b$ , even when its input isn't $0$ .

Hints

Hint 1
Each row pairs an input with an output, making a point $(x,f(x))$ . Enter all three rows in a Desmos table. What points does it plot?
Hint 2
A line has slope-intercept form $f(x)=mx+b$ . Fit $y_1\sim mx_1+b$ to the table to find both numbers. The intercept $b$ is the output at $x=0$ , not necessarily the first output shown.

Step-by-step

Fit a line to the table
Step 1
Plot the table rows
Enter the rows in a Desmos table, using $x_1$ for the input and $y_1$ for the output $f(x)$ . Desmos plots $(1,5)$ , $(3,13)$ , and $(5,21)$ . Each row is a point the same line must pass through.
Step 2
Find the slope and intercept
Because $f$ is linear, write its rule in slope-intercept form $f(x)=mx+b$ . Here $m$ is the output change for each increase of $1$ in $x$ , and $b$ is the output at $x=0$ . Under the table, type $y_1\sim mx_1+b$ ; the $\sim$ tells Desmos to fit both numbers to the rows. Under PARAMETERS, it shows $m=4$ and $b=1$ . The first output, $5$ , is not the intercept: its input is $1$ , not $0$ .
Step 3
Write the function
Substitute the fitted values into $f(x)=mx+b$ . The slope and the output at $x=0$ fix the whole line.
$f(x)=4x+1$
So this rule defines the function in the table. Choice C.

Question 632·Easy·Algebra

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Strategy

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