Functions, Linear and Notation

Section 6.3 Linear Functions NOTES and Practice

Goal

Today’s lesson is writing a linear function from a graph or a table of values.

Remember:

 a function is a relation that pairs each input with exactly one output.

Linear Function

-is a function whose graph is a nonvertical line.

-can be written in the form y = mx + b, where m is the slope and b is the y-intercept.

Use the graph to write a linear function that relates y to x

• The points lie on a line. Find the slope by using the points (2, 0) and (4, 3).

•  $m=\frac{change\ in\ y}{\ change\ in\ x}=\frac{3−0}{4-2}=\frac{3}{2}$  

• Because the line crosses the y-axis at (0, −3), the y-intercept is −3.

• So, the linear function is  $y=\frac{3}{2}x-3$   

Use the table to write a linear function that relates y to x.

• Draw a table

• Plot the points in the table.

• The points lie on a line. Find the slope by using the points (− 2, 7) and (− 3, 9).

•  $m=\ \frac{change\ in\ y}{change\ in\ x}=\frac{9-7}{−3−(−2)}=\frac{2}{-1}=-2$  

• Because the line crosses the y-axis at (0, 3), the y-intercept is 3.

• So, the linear function is y = −2x + 3.

Multiple Choice

Manager A earns $15 per hour and receives a $50 bonus. The graph shows the earnings of Manager B.

Which manager has a higher hourly wage?

• Manager A

• Manager B

Multiple Choice

Manager A earns $15 per hour and receives a $50 bonus. The graph shows the earnings of Manager B.

After how many hours does Manager B earn more money than Manager A?

• After 3 hours

• After 4 hours

• After 5 hours

• After 6 hours

Comparing Linear Functions

The earnings y (in dollars) of a nighttime employee working x hours are represented by the linear function y = 7.5x + 30. The table shows the earnings of a daytime employee.

Which employee has a higher hourly wage?

• Nighttime Employee

•  $y=7.5x+30$  

• Daytime Employee

•  $\frac{change\ in\ earnings}{change\ in\ time}=\frac{\$12.50}{1hour}$  

• The nighttime employee earns  $7.50 per hour. 

• The daytime employee earns $12.50 per hour.

• So, the daytime employee has a higher hourly wage.

Which employee has a higher hourly wage?

• The nighttime employee earns  $7.50 per hour. 

• The daytime employee earns $12.50 per hour.

• So, the daytime employee has a higher hourly wage.

Write a linear function that relates the daytime employee’s earnings to the number of hours worked.

• Draw a coordinate plane.

• In the same coordinate plane, graph the linear functions that represent the earnings of the two employees.

• Interpret the graphs.

• Use a verbal model to write a linear function that represents the earnings of the daytime employee.

Employee Earnings

• Earnings = Hourly wage ⋅ Hours worked
  

•  $y=12.5x$  
  

• The graph shows that the daytime employee has a higher hourly wage but does not earn more money than the nighttime employee until each person has worked more than 6 hours.