Key Features of Linear Functions

4.6.4 Key Features of Linear Functions

4.6.4 • Key Features of Linear Functions

Activity

Use the following information to answer questions 1 – 4.

The graph of $f (x) = 2 x + 6$ is graphed below. Answer the questions about the linear function and its key features.

Graph of a line with an x-intercept of negative 3 and y-intercept of 6.

1. What is the value of $f (- 3)$ ?

Compare your answer:

2. What key feature of the linear function is the point $(- 3, 0)$ ?

Compare your answer:

The $x$ -intercept, or zero

3. What is the slope of the linear function?

Compare your answer:

4. What part of the linear function is the point $(0, 6)$ ?

Compare your answer:

The $y$ -intercepts

Use the following information to answer questions 5 – 8.

The amount of money Adam saves in dollars is shown in the graph below.

Graph that shows the total amount of money in dollars as a function of number of days. The function is a line with a positive slope that passes through the points (0, 3) and (1, 6).

5. What is the $y$ -intercept of the graph?

Compare your answer:

The point is $(0, 3)$ .

6. What does the $y$ -intercepts mean in the context of the problem

Compare your answer: Your answer may vary.

When Adam started saving, he already had 3 dollars on day 0.

7. What is the slope?

Compare your answer:

The slope is 3.

8. What does the slope mean in the context of the problem

Compare your answer: Your answer may vary.

Every 1 day, Adam saves 3 dollars.

Use the following information to answer questions 9 – 12.

The graph of $f (x) = 13 x + 2$ is graphed below. Identify the appropriate label for each key feature of the graph.

A graph shows a diagonal blue line crossing points B (negative 6, 1), A (0, 2), and D (3, 3.5), with point C (0, 1) on the y-axis. Dashed lines highlight the rise and run for point D.

9. What is the key feature labeled $A$ on the graph?

Compare your answer:

$A$ , (0, 2), is the $y$ -intercept.

10. What is the key feature labeled $B$ on the graph?

Compare your answer:

$B$ , (-6, 0), is the $x$ -intercept or zero.

11. What is the key feature labeled $C$ on the graph?

Compare your answer:

$C$ , (0, 0), is the origin.

12. What is the key feature labeled $D$ on the graph?

Compare your answer:

$D$ , $\frac{1}{3}$ , is the slope.

Self Check

What is the zero of the linear function graphed below?

$(0, 0)$
$(0,6)$
$(0, -4)$
$(6,0)$

Additional Resources

Features of Linear Functions

A graph with an orange downward-sloping line crossing the y-axis near 4 and the x-axis near 2. A black dot marks the point (2, 0). The grid ranges from -8 to 8 on both axes.

The graph of a linear function forms a line that goes on forever in both directions. Since the ends go to infinity and negative infinity, linear functions do not have maximums or minimums, unless their domain is restricted to a specific part of the graph.

You can still pick out key features of linear functions.

The $x$ -intercept, or zero, is where the line crosses the $x$ -axis. This is where the $y$ -coordinate is 0.

In this linear function, the $x$ -intercept, or zero, is $x = 2$ and written as the point $(2, 0)$ .

The $y$ -intercept is where the line crosses the $y$ -axis and the $x$ -coordinate is 0.

In this linear function, the $y$ -intercept is $y = 4$ and written as the point $(0, 4)$ .

The slope of the line can be found as in previous units as the change in $y$ over the change in $x$ .

$m = \frac{r i s e}{r u n} = \frac{c h a n g e i n y}{c h a n g e i n x}$

The slope of this linear function is $m = \frac{4}{- 2} = \frac{- 4}{2} = - 2$ .

Try it

Features of Linear Functions

Find the $x$ -intercept and $y$ -intercept of the linear function graphed below. Write each answer as a point.

An orange diagonal line passes through the origin, extending from the bottom left to the top right. A black dot is marked at the point (4,5) on the line. The background is a grid with labeled axes.

Here is how to find the intercepts of a linear function:

Since the line crosses the $x$ -axis at about $(- 2.75, 0)$ , that is the $x$ -intercept.

Since the line crosses the $y$ -axis at $(0, 2)$ , that is the $y$ -intercept.