Question 557·Easy·Algebra

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For the function $f$ , the graph of $y = f(x)$ in the $xy$ -plane passes through the point $(4,2)$ and has a $y$ -intercept of $5$ . Which equation defines $f$ ?

A point and a $y$ -intercept give you two points on a line. Use slope-intercept form, $f(x)=mx+b$ : the intercept gives $b$ , and the change between the points gives the slope $m$ . Desmos can calculate that change as a fraction. Watch the direction of the line: if the output falls as the input rises, the slope is negative.

Hints

Hint 1
A $y$ -intercept is where the graph meets the vertical axis, so its input is $0$ . Write the intercept as a point before finding the slope.
Hint 2
The slope is the change in output divided by the change in input. Go from the intercept to the given point, and subtract in the same order on top and bottom.

Step-by-step

Find the slope from two points
Step 1
Turn the intercept into a point
The $y$ -intercept is where $x=0$ , so an intercept of $5$ gives the point $(0,5)$ . In slope-intercept form, $f(x)=mx+b$ , $b$ is that output at $x=0$ . So $b$ is $5$ , and $f(x)=mx+5$ .
Step 2
Set up the slope
Now use $(0,5)$ and the given point $(4,2)$ . The slope $m$ compares the change in output with the change in input. Divide the output change by the input change, subtracting in the same order: $m=\frac{2-5}{4-0}$ .
Step 3
Calculate the slope in Desmos
Type $\frac{2-5}{4-0}$ in Desmos. It shows $-0.75$ ; tap its fraction button to see $-\frac34$ . The negative sign fits: the output falls from $5$ to $2$ as the input rises from $0$ to $4$ .
Step 4
Write the function
Put $m=-\frac34$ and $b=5$ into $f(x)=mx+b$ : $f(x)=-\frac34x+5$ . It has the given intercept and passes through $(4,2)$ , so it matches both facts. Choice B.

Question 557·Easy·Algebra

Strategy

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Step-by-step

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Strategy

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