Question 617·Hard·Math

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x	-11	-10	-9	-8
$f(x)$	21	18	15	12

The table above shows some values of $x$ and their corresponding values $f(x)$ for the linear function $f$ . What is the $x$ -intercept of the graph of $y = f(x)$ in the $xy$ -plane?

A table of a linear function might not include the axis crossing you need. Its constant change lets you fit a line in Desmos, then find where the output reaches zero. At an x-intercept, the output is zero, not the input. Keep the line’s slope and its other intercept separate from the point you’re seeking.

Hints

Hint 1
An x-intercept is where a graph meets the horizontal axis, so the output is zero. None of the table’s outputs is zero. How could you find an input that gives zero?
Hint 2
A linear function changes by the same amount each time the input rises by $1$ . Enter the pairs in a Desmos table and fit $y_1\sim mx_1+b$ to find the rule for the line.
Hint 3
Use the fitted rule to set $f(x)=0$ . In Desmos, $\sim$ asks it to find the unknown input; the $=$ sign would graph the equation instead.

Step-by-step

Approach 1: Fit the line, then find its zero
Step 1
Identify which value must be zero
The graph is $y=f(x)$ , so a table column gives a point $(x,f(x))$ . An x-intercept lies on the horizontal axis, where $f(x)=0$ . The table doesn’t show that output, so you need the line’s rule.
Step 2
Fit a rule to the table
Enter the $x$ values and their matching $f(x)$ values as a Desmos table. Type $y_1\sim mx_1+b$ ; the $\sim$ tells Desmos to fit a line. Under PARAMETERS, it shows $m=-3$ and $b=-12$ , so $f(x)=-3x-12$ . The slope $-3$ is the output change per step right. $b$ is the output at $x=0$ , not the x-intercept.
Step 3
Find the input that gives zero
Type $-3t-12\sim0$ , using $t$ for the unknown input because $x_1$ already labels the table column. Desmos shows $t=-4$ under PARAMETERS. So $f(-4)=0$ , and the x-intercept is $(-4,0)$ . Choice B.

Approach 2: Continue the table’s pattern
Step 1
Find the change per step
From $x=-11$ to $x=-10$ , the input rises by $1$ while the output falls from $21$ to $18$ , a change of $-3$ . Because the function is linear, the same change repeats for every step of $1$ .
Step 2
Continue until the output reaches zero
Start at the table’s last point, $(-8,12)$ . Add $1$ to the input and subtract $3$ from the output each time:
$\begin{aligned} f(-7)&=12-3=9\\[1.4em] f(-6)&=9-3=6\\[1.4em] f(-5)&=6-3=3\\[1.4em] f(-4)&=3-3=0 \end{aligned}$
The zero output gives the x-intercept $(-4,0)$ . Choice B.

Question 617·Hard·Math

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