Mrs. Rivas International Studies Charter School. Bell Ringer A line contains the points (0, 0) and (1, 4). Select all the equations that represent this line.

Mrs. Rivas International Studies Charter School.Objectives:  Calculate a line’s slope.  Write the point-slope form of the equation of a line. slope-intercept form  Write and graph the slope-intercept form of the equation of a line. horizontal or vertical lines  Graph horizontal or vertical lines.  Recognize and use the general form of a line’s equation.  Use intercepts to graph the general form of a line’s equation.  Model data with linear functions and make predictions.

Mrs. Rivas International Studies Charter School.

Mrs. Rivas International Studies Charter School.

Mrs. Rivas International Studies Charter School. Example: Example: Using the Definition of Slope Find the slope of the line passing through the points (4, –2) and (–1, 5)

Mrs. Rivas International Studies Charter School.

Mrs. Rivas International Studies Charter School. Point-Slope Form of the Equation of a Line

Mrs. Rivas International Studies Charter School. Example: Example: Writing an Equation in Point-Slope Form for a Line Write an equation in point-slope form for the line with slope 6 that passes through the point (2, –5). Then solve the equation for y. The equation in point-slope form is Solve for y.

Mrs. Rivas International Studies Charter School. Slope-Intercept Form of the Equation of a Line It is where the line intersects the y-axis. POINT Starting POINT

Mrs. Rivas International Studies Charter School. Graphing y = mx + b Using the Slope and y-Intercept y-intercept 1. Plot the point containing the y-intercept on the (0, b) y-axis. This is the point (0, b). slopemm y-intercept 2. Obtain a second point using the slope, m. Write m as a fraction, and use rise over run, starting at the point containing the y-intercept, to plot this point. 3. Use a straightedge to draw a line through the two points. Draw arrowheads at the ends of the line to show that the line continues indefinitely in both directions.

Mrs. Rivas International Studies Charter School. Example: Graphing Using the Slope and y-Intercept Graph the linear function: Step 1 Plot the point containing the y-intercept on the y-axis. The y-intercept is 1. We plot the point (0, 1).

Mrs. Rivas International Studies Charter School. Example: Graphing Using the Slope and y-Intercept Step 2 Step 2 Obtain a second point using the slope, m. Write m as a fraction, and use rise over run, starting at the point containing the y-intercept, to plot this point. rise = 3 y-intercept=1 We plot the point (5,4)

Mrs. Rivas International Studies Charter School. Example: Graphing Using the Slope and y-Intercept Step 3 Step 3 Use a straightedge to draw a line through the two points. Draw arrowheads at the ends of the line to show the line continues indefinitely in both directions.

Mrs. Rivas International Studies Charter School. Equation of a Horizontal Line

Mrs. Rivas International Studies Charter School. Equation of a Vertical Line

Mrs. Rivas International Studies Charter School. Example: Graphing a Horizontal Line

Mrs. Rivas International Studies Charter School. General Form of the Equation of a Line

Mrs. Rivas International Studies Charter School. Example: Finding the Slope and the y-Intercept Find the slope and the y-intercept of the line whose equation is The slope is The y-intercept is 2.

Mrs. Rivas International Studies Charter School. x-intercepty = 0 1. Find the x-intercept. Let y = 0 and solve for x. Plot the point containing the x-intercept on the x-axis. y-interceptx = 0 2. Find the y-intercept. Let x = 0 and solve for y. Plot the point containing the y-intercept on the y-axis. 3. Use a straightedge to draw a line through the points containing the intercepts. Draw arrowheads at the ends of the line to show that the line continues indefinitely in both directions.

Mrs. Rivas International Studies Charter School. Example: Using Intercepts to Graph a Linear Equation Graph using intercepts: Step 1 Step 1 Find the x-intercept. Let y = 0 and solve for x. x-intercept2 The x-intercept is 2, so the line passes through (2, 0).

Mrs. Rivas International Studies Charter School. Example: Using Intercepts to Graph a Linear Equation Step 2 Step 2. Find the y-intercept. Let x = 0 and solve for y. y-intercept–3 The y-intercept is –3, so the line passes through (0, –3).

Mrs. Rivas International Studies Charter School. Example: Using Intercepts to Graph a Linear Equation Step 3 Step 3 Graph the equation by drawing a line through the two points containing the intercepts.

Mrs. Rivas International Studies Charter School. A Summary of the Various Forms of Equations of Lines

Mrs. Rivas International Studies Charter School. Example: Application Use the data points (317, 57.04) and (354, 57.64) to obtain a linear function that models average global temperature, f(x), for an atmospheric carbon dioxide concentration of x parts per million. Round m to three decimal places and b to one decimal place.

Mrs. Rivas International Studies Charter School. Example: Application

Mrs. Rivas International Studies Charter School. Example: Application Use the function to project average global temperature at a concentration of 600 parts per million. The temperature at a concentration of 600 parts per million would be 61.6°F.

Mrs. Rivas International Studies Charter School.Objectives: slopes equations of parallel and perpendicular  Find slopes and equations of parallel and perpendicular lines. rate of change  Interpret slope as rate of change. average rate of change  Find a function’s average rate of change.

Mrs. Rivas International Studies Charter School. Slope and Parallel Lines Have the same slope

Mrs. Rivas International Studies Charter School. Writing an equation of a line Parallel to a given line Example 1 “Same Slope” Point-Slope Form Solve for y: Slope Intercept Form

Mrs. Rivas International Studies Charter School. Slope and Perpendicular Lines Opposite Reciprocal slope.

Mrs. Rivas International Studies Charter School. Writing an equation of a line Perpendicular to a given line. Example 2 “Opposite Reciprocal”

Mrs. Rivas International Studies Charter School. Slope as Rate of Change

Mrs. Rivas International Studies Charter School. Slope as a Rate Change Example 3 The line graphs for the number of women and men living alone are shown again in Figure Find the slope of the line segment for the women. Describe what this slope represents. We let x represent a year and y the number of women living alone in that year. The two points shown on the line segment for women have the following coordinates: The rate of change is 0.21 million women per year. The slope indicates that the number of American women living alone increased at a rate of approximately 0.21 million each year for the period from 1990 through The rate of change is 0.21 million women per year.

Mrs. Rivas International Studies Charter School. The Average Rate of Change of a Function

Mrs. Rivas International Studies Charter School. The man’s average growth rate between ages 13 and 18 is the slope of the secant line containing (13, 57) and (18, 76):

Mrs. Rivas International Studies Charter School. The Average Rate of Change of a Function

Mrs. Rivas International Studies Charter School. Finding the Average of Change Example 4 A B C average rate positive increasing (0,1) The average rate of change is positive and the function is increasing on the interval (0,1)

Mrs. Rivas International Studies Charter School. Finding the Average of Change Example 4 B average rate positive increasing(1,2) The average rate of change is positive and the function is increasing on the interval (1,2)

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Mrs. Rivas International Studies Charter School. Finding the Average of Change Example 4 C average rate negative decreasing(-2,0) The average rate of change is negative and the function is decreasing on the interval (-2,0)

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Mrs. Rivas International Studies Charter School. Finding the Average Rate of Change Example 5 Solution Solution At 3 hours, the drug’s concentration is 0.05 and at 7 hours, the concentration is The average rate of change in its concentration between 3 and 7 hours is

Mrs. Rivas International Studies Charter School. average velocity The average velocity of an object is its change in position divided by the change in time between the starting and ending positions. If a function expresses an object’s position in terms of time, the function’s average rate of change describes the object’s average velocity

Mrs. Rivas International Studies Charter School. Finding Average Velocity Example 6 Solution

Mrs. Rivas International Studies Charter School 1-4 Pages: 1-4 Pages: # 3, 5, 7, 9, 11, 15, 19, 21, 23, 25, 29, 33, 43, 45, 49, 51, 57, 59, 61, 67, 69, 82, Pages: 1-5 Pages: # 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 30, 32, 33-38