MA.8.A.1.2 Interpret the slope and the x- and y-intercepts when graphing a linear equation for a real-world
problem
Constant Rate of Change/Slope
In a Table
Relationships that have straight-lined graphs are called linear relationships. The rate
of change between any two points in a linear relationship is the same, or constant. A
linear relationship has a constant rate of change.
Example
The height of a hot air balloon after a few
seconds is shown. Determine whether the
relationship between the two quantities is
linear. If so, find the constant rate of
change. If not, explain your reasoning.
As the number of seconds increase by 1, the height of the balloon increases by 9 feet.
Since the rate of change is constant, this is a linear relationship. The constant rate of change is
second. This means that the balloon is rising 9 feet per second.
In a Graph
Slope formula:
or
y
x
or
rise
run
The slope m of a line passing through points (x1, y1) and (x2, y2) is the ratio of the
difference in the y-coordinates to the corresponding difference in the x-coordinates.
Example 1
Find the slope of the line that passes through A(-1, -1) and B(2,
3).
Slope formula:
(x1, y1) = (-1, -1),
(x2, y2) = (2, 3)
Simplify.
Check
When going from left to right, the graph of the line slants upward.
This is correct for a positive slope.
or 9 feet per
Example 2
Find the slope of the line that passes through C(1, 4) and
D(3, -2).
Slope formula:
(x1, y1) = (1, 4),
(x2, y2) = (3, -2)
Simplify.
Check When going from left to right, the graph of the line slants downward.
This is correct for a negative slope.
Slope-Intercept Form
Linear equations are often written in the form y = mx + b. This is called the slope-intercept form. When an
equation is written in this form, m is the slope and b is the y-intercept.
Example 1 State the slope and the y-intercept of the graph of y = x − 3.
Write the original equation.
Write the equation in the form y = mx + b.
m = 1, b = -3
The slope of the graph is 1, and the y-intercept is -3.
You can use the slope-intercept form of an equation to graph the equation.
Example 2 Graph y = 2x + 1 using the slope and y-intercept.
Find the slope and y-intercept.
y = 2x + 1
slope = 2, y-intercept = 1
Graph the y-intercept 1.
Write the slope 2 as
. Use it to locate a second point on the line.
Draw a line through the two points.
Graph Functions Using Intercepts
Standard form is when an equation is written in the form Ax + By = C.
Example
State the x- and y-intercepts of 3x + 2y = 6. Then graph the function.
STEP 1
Find the x-intercept.
STEP 2
To find the x-intercept, let y = 0.
3x + 2y = 6
Write the equation.
3x + 2(0) = 6
Replace y with 0.
3x + 0 = 6
Multiply.
3x = 6
Simplify.
x=2
Divide each side by 3.
The x-intercept is 2.
Find the y-intercept.
STEP 3
To find the y-intercept, let x = 0.
3x + 2y = 6
Write the equation.
3(0) + 2y = 6
Replace x with 0.
0 + 2y = 6
Multiply.
2y = 6
Simplify.
y=3
Divide each side by 2.
The y-intercept is 3.
Graph the points (2, 0) and (0, 3) on a coordinate plane. Then connect the points.
MA.8.A.1.2 Practice Problems
5.
6.
7.
8.
9.
10.
11. The table shows the cost for a clothing store to buy jeans
and khakis. The total cost for Saturday’s shipment, $1800, is
represented by the function 15x + 20y = 1800. Find and
interpret the x- and y-intercepts.
12. A taxi fare y can be determined by the equation y =
0.50x + 3.50, where x is the number of miles traveled.
Graph the equation.
Find the cost of traveling 8 miles.
What is the slope and y-intercept?
Interpret the slope and y-intercept.
13.
14. For working 3 hours, Carmen earns $18.00. For working 5 hours, she earns $30. For working 6 hours, she earns
$36.00. Make a table of the information and then graph it with hours on the horizontal axis and money earned on the
vertical axis. Draw a line through the points.
What is the slope of the line?
What does the slope of the line represent?
15. South Seminole Middle School wants to make $4,740 from yearbooks. Print yearbooks x cost $60 and digital
yearbooks y cost $15. This can be represented by the function 60x + 15y = 4,740.
Find the x- and y-intercepts. Interpret the x- and y-intercepts.
16.
17. Lenny charted the growth rate of his new puppy for several weeks. He measured the weight of the puppy in pounds
and plotted the values on the graph below. He drew a trend line passing through points (2,4) and (10,20) on the graph to
estimate the puppy’s weight during any week. What is the slope of the line and what does the it represent?
18.
19. Student Council is selling spirit T-shirts during spirit week. It costs $20 for the design and $5 to print each shirt. The
cost y to print x shirts is given by y = 5x + 20.
How many shirts can be printed for $50?
What is the slope?
Interpret the slope.
What is the y-intercept?
Interpret the y-intercept.
20. The total number of legs, 1500, on four-legged and two-legged animals in a zoo can be represented by the function 4x
+ 2y = 1,500. Find and interpret the x- and y-intercepts.