College Algebra Chapter 2 Functions and Graphs Section 2.5 Applications of Linear Equations and Modeling

1. Apply the Point-Slope Formula 2. Determine the Slopes of Parallel and Perpendicular Lines 3. Create Linear Functions to Model Data 4. Create Models Using Linear Regression

Apply the Point-Slope Formula Point-slope formula for a line: y – y1 = m(x – x1) m is the slope (x1, y1) is a point on the line

Example 1: Use the point-slope formula to write an equation of the line having the given conditions. Write the answer in slope-intercept form (if possible). Passes through (–4, 2) and m = 2

Example 2: Use the point-slope formula to write an equation of the line having the given conditions. Write the answer in slope-intercept form (if possible). Passes through (5, 0) and m =

Example 3: Use the point-slope formula to write an equation of the line having the given conditions. Write the answer in slope-intercept form (if possible). Passes through (–1, –4) and (–2, 1)

Example 4: Use the point-slope formula to write an equation of the line having the given conditions. Write the answer in slope-intercept form (if possible). Passes through (2,3) and the slope is undefined.

1. Apply the Point-Slope Formula 2. Determine the Slopes of Parallel and Perpendicular Lines 3. Create Linear Functions to Model Data 4. Create Models Using Linear Regression

Determine the Slopes of Parallel and Perpendicular Lines Parallel lines have matching slopes. If m1 and m2 represent the slopes of two nonvertical parallel lines, then m1 = m2. Perpendicular lines have slopes that are negative reciprocals. If m1 and m2 represent the slopes of two nonvertical perpendicular lines, then .

Examples 5 – 7: The slope of a line is given. Determine the slope of a line parallel and perpendicular to the given line, if possible. Parallel: ____ Perpendicular: ____ Parallel: ____ Perpendicular: ____ Parallel: ____ Perpendicular: ____

Example 8: Give the equation of a line that passes through (–1,2) and is parallel to the line defined by .

Example 9: Give the equation of a line that passes through (6,8) and is perpendicular to the line defined by .

1. Apply the Point-Slope Formula 2. Determine the Slopes of Parallel and Perpendicular Lines 3. Create Linear Functions to Model Data 4. Create Models Using Linear Regression

Example 10: The local hardware store charges $28 to rent a carpet cleaning machine for 24 hours and $10.98 for each medium-sized bottle of rug shampoo. a. Write a linear function S that represents the cost of renting the machine for x days along with 2 bottles of rug shampoo.

Example 10 continued: b. Evaluate S(2) and interpret the meaning in the context of this problem.

Create Linear Functions to Model Data A linear cost function models the cost C(x) to produce x items. m is the variable cost per item b is the fixed cost A linear revenue function models revenue R(x) for selling x items. p is the price per item A linear profit function models the profit for producing and selling x items.

Example 11: Alina is starting a summer business power-washing home driveways and sidewalks. She will charge $35 to pressure-clean a driveway and the sidewalk in front of a house. Her start-up costs include her initial purchase of a power washer for $330 and a fee of $2 per house she must pay to the homeowners association for the use of the water for each house.

Example 11 continued: a. Write a linear cost function for power-washing at x homes. b. Write a linear revenue function for power- washing at x homes.

Example 11 continued: c. Write a linear profit function for power-washing at x homes. d. How much profit will Alina make if she power- washes at 15 homes?

Example 11 continued: e. How many homes must Alina power-wash to make $330?

1. Apply the Point-Slope Formula 2. Determine the Slopes of Parallel and Perpendicular Lines 3. Create Linear Functions to Model Data 4. Create Models Using Linear Regression

Create Models Using Linear Regression Creating a Linear Regression Model 1. Graph the data in a scatter plot. 2. Inspect the data visually to determine if the data suggest a linear trend. 3. Invoke the linear regression feature on a calculator, graphing utility, or spreadsheet. 4. Check the result by graphing the line with the data points to verify that the line passes through or near the data points.

Example 12: Determine the equation for the least-squares regression line for the given data. x y 0.5 1 1.3 2 2.9 3 2.4 4 5 6 5.4 7 7.7 8 8.3

Example 12 continued: 1. Use the STAT button, then EDIT to enter the x and y data in two lists. Exit this screen.

Example 12 continued: 2. Use the STAT button, then CALC, choose 4:LinReg(ax + b).

Example 12 continued: Hit CALCULATE. The equation is

Example 12 continued: 4. To see the data and the line graphed: Above the y = key, Turn Plot1 ON and select STATPLOT. select the type of graph.

Example 12 continued: Graph

Example 12 continued: 5. Enter into the equation editor and see the line graphed.