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

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

Increments

Example Increments

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.

Slope of a Line

Parallel and Perpendicular Lines

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

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,

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.

Equations of Lines

Example Equations of Lines

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).

Point Slope Equation

Example: Point Slope Equation

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

Equations of Lines

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

Slope-Intercept Equation

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

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.

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.

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

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

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

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

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

Example Reimbursed Expenses

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.

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.

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.