1. Chapter 1: Prerequisites for Calculus Section 1.1 - Lines
AP CALCULUS AB
Chapter 1:
Prerequisites for Calculus
Section Lines

2. What you’ll learn about…
…and why.
Linear equations are used extensively in business and economic applications.

3. Increments

4. Example Increments

5. Slope of a Line
A line that goes uphill as x increases has a positive slope. A line that goes downhill as x increases has a negative slope.

6. Slope of a Line

7. Parallel and Perpendicular Lines

8. Section 1.1 – Lines Parallel and Perpendicular Lines:
Two distinct non-vertical lines are parallel if and only if their slopes are equal. So

9. Section 1.1 – Lines Parallel and Perpendicular Lines:
2. Two non-vertical lines are perpendicular if and only if their slopes are negative reciprocals of each other. That is,

10. Section 1.1 – Lines
You try: Write an equation of the line passing through the point (1, 3) that is (a) parallel, and (b) perpendicular to each given line L.

11. Equations of Lines

12. Example Equations of Lines

13. Section 1.1 Forms of Equations of Lines
Point-slope Equation: The equation
is the point-slope equation of the line through the point with slope m.
Example: Find the equation of the line that contains the points (1, 2) and (3, 5).

14. Point Slope Equation

15. Example: Point Slope Equation

16. Section 1.1 – Lines
You try: Write the point-slope equation for the line that passes through the given point with the given slope.

17. Equations of Lines

18. Section 1.1 Forms of Equations of Lines
Slope-intercept Equation: The equation
y = mx + b
is the slope-intercept equation of the line with slope m and y-intercept b.
Example: Find the equation of the line with slope 2 and y-intercept (0, 5).
y = 2x + 5

19. Slope-Intercept Equation

20. Section 1.1 – Lines
You try: Write the slope-intercept equation of the line passing through each pair of points.

21. General Linear Equation
Although the general linear form helps in the quick identification of lines, the slope-intercept form is the one to enter into a calculator for graphing.

22. Section 1.1 Forms of Equation of Lines
Standard Linear Equation: The equation
Ax + By + C = 0
is a standard linear equation in x and y.

23. Section 1.1 Forms of Equations of Lines
Equation of a Vertical line: The equation of a vertical line going through the point (a, b) is
x = a.
Equation of an Horizontal line: The equation of an horizontal line going through the point (a, b) is
y = b.
Intercept Equation: The equation of the line with x-intercept (a, 0) and y-intercept (0, b) is

24. Example Analyzing and Graphing a General Linear Equation
[-10, 10] by [-10, 10]

25. Example Determining a Function
f(x) -1 1 5 3 11

26. Section 1.1 – Lines
You try: The table below gives the values for a linear function. Determine m and b.

27. Section 1.1 – Lines
You try: The table below gives the values for a linear function. Determine m and b.

28. Example Reimbursed Expenses

29. Section 1.1 Regression Analysis
Regression analysis is the process of finding an equation to fit a set of data. This allows us to:
Summarize the data with a simple expression, and
Predict values of y for other values of x.

30. Section 1.1 Regression Analysis
On the TI-83/TI-83 Plus/TI-84 Calculator
STAT, EDIT enter x’s in List1 and y’s in List2.
STAT PLOT turn Plot1 ON, choose scatter plot.
Choose an appropriate window to match your data.
GRAPH and look at the data points.
Decide what type of equation best fits your data (linear, quadratic, exponential, trigonometric, power, etc.).
STAT, CALC, type of regression equation, L1, L2, VARS, Y-VARS, FUNCTION, Y1 (this pastes the regression equation into y1).
GRAPH and see how close the regression curve is to your data points. If it appears too far off, choose a different type of equation and repeat steps 5-7.

31. Section 1.1 Regression Analysis
On the TI-89 Calculator:
{ x values separated by commas } STO ALPHA L 1.
{ y values separated by commas } STO ALPHA L 2.
Y= arrow up to Plot1, ENTER
Choose: Scatter, Box, x … L1, y … L2 ENTER
Choose appropriate window for your data.
Graph and look at data points.
Decide what type of equation best fits your data. QUIT
CATALOG type of regression equation, L1, L2 ENTER
CATALOG SHOW STAT
Enter equation shown in Y1
GRAPH and see how close the regression curve is to your data points. If it appears too far off, choose a different type of equation and repeat steps 7-11.