1. Section 8.2 Linear Functions

2. 8.2 Lecture Guide: Linear Functions

3. Key Characteristics of a Linear Function 1. The graph of a linear function is a ____________ line. 2. The equation is first degree and can be written in the slope-intercept form with slope ______ and y-intercept ______. 3. Linear functions have a constant rate of change. The slope is the same between any two ____________ on the line. 4. The domain of all linear functions is ______. The range of all linear functions is also unless the function is a constant function, whose graph is a horizontal line.

4. AlgebraicallyNumerical Example Graphical Example A function of the form is called a linear function. Linear Function

5. Objective 1: Determine the slope of a line. From Section 3.1, we define slope of a line through two points as the ratio of the change in y to the change in x. This rise over the run can be expressed algebraically by or A line defined by the slope-intercept Form has a slope m.

6. Determine the slope of each line. 5.6.7.

7. 8. Calculate the slope of the line in the graph. 9. Calculate the slope of the line in the graph.

8. 10. Calculate the slope of the line in the graph. 11. Calculate the slope of the line in the graph.

9. 12. Calculate the slope of the line containing the points in the table.

10. 13. Calculate the slope of the line containing the points in the table.

11. Interpreting Slopes Positive SlopeZero SlopeNegative Slope Algebraically In m will be positive. Example: In m will be 0 and Example: In m will be negative. Example:

12. Interpreting Slopes Positive SlopeZero SlopeNegative Slope Graphically The line will slope ____________ to the right. Example: This is a ________ line that does not slope upward or downward. Example: The line slopes ____________ to the right. Example:

13. Interpreting Slopes Positive SlopeZero SlopeNegative Slope Numerically The y-values will ____________ as the x-values increase. The y-values will _______________ as the x-values change. The y-values will ____________ as the x-values increase.

14. The equation of a vertical line is of the form This equation does not represent a function --- it fails the vertical line test. The slope of a vertical line is ____________.

15. Objective 2: Sketch the graph of a linear function.

16. Use the slope and y-intercept to graph each line. 14. Slope: ______ y-intercept: ______ Graph:

17. Use the slope and y-intercept to graph each line. 15. Slope: ______ y-intercept: ______ Graph:

18. Objective 3: Write the equation of the line through given points. Determine the equation of the line in the form that passes through each pair of points. 16. and

19. Determine the equation of the line in the form that passes through each pair of points. 17. and

20. All the points listed in the table lie on the same line. Use the table to determine the equation of the line in the form 18.

21. All the points listed in the table lie on the same line. Use the table to determine the equation of the line in the form 19.

22. Use the information displayed in the graph to determine the equation of the line in the form 20.

23. Use the information displayed in the graph to determine the equation of the line in the form 21.

24. Objective 4. Determine the intercepts of a line. 22. Use the graph to determine the ­intercepts of the linear function.

25. Objective 4. Determine the intercepts of a line. 23. Use the table to determine the intercepts of the linear function.

26. 24. Determine the intercepts of the linear function. Then use the intercepts to sketch a graph of the function.

27. Objective 5: Determine the x-values for which a linear function is positive and the x-values for which a linear function is negative. Classifying a Function as Positive or Negative VerballyGraphicallyNumericallyAlgebraically The function is positive at is __________ the x-axis. has a positive y-value. The function is negative at is __________ the x-axis. has a negative y-value.

28. Determine the x-values for which each linear function is positive and the x-values for which each function is negative. 25.

29. Determine the x-values for which each linear function is positive and the x-values for which each function is negative. 26.

30. 27. Determine the x-values for which each linear function is positive and the x-values for which each function is negative.

31. 28. Determine the profit and loss intervals for the profit function graphed below. The x-variable represents the number of units of production and y-variable represents the profit generated by the sale of this production. Profit interval: Loss interval: Units Profit

32. Write in slope-intercept form the equation of a line passing through and perpendicular to 29.

33. 30. Use the function to determine the missing input and output values. (a) (b)

34. 31.Temperature Beginning at 6:00 am, when the temperature was Fahrenheit, the temperature increased by per hour over the next 6 hours. (a) Write a linear function T so that T(x) gives the temperature in degrees (b) Determine the temperature at noon.

35. (c) Use this function to determine how many hours until the temperature reaches. 31.Temperature Beginning at 6:00 am, when the temperature was Fahrenheit, the temperature increased by per hour over the next 6 hours.

36. d) Use this function to complete the table. 31.Temperature Beginning at 6:00 am, when the temperature was Fahrenheit, the temperature increased by per hour over the next 6 hours.