NAME ______________________________________________ DATE______________________________ PERIOD ______________

Unit 2 Homework Packet3-3 Rate of Change and Slope

Rate of Change The rate of change tells, on average, how a quantity is changing over time.

Exercises

1. LONGEVITY The graph shows the predicted life expectancy for men and women born in a given year.

a. Find the rates of change for women from 2000-2025 and 2025-2050.

b. What pattern do you see in the increase with each 25-year period?

c. Make a prediction for the life expectancy for 2050-2075. Explain how you arrived at your prediction.

Find Slope The slope of a line is the ratio of change in the y-coordinates (rise) to the change in the x-coordinates (run) as you move in the positive direction.

Slope of a Line m = riserun or m =

y2− y1

x2−x1 , where (x1, y1) and (x2, y2) are the coordinates of any two points

on a nonvertical line

ExercisesFind the slope of the line that passes through each pair of points.

1. (4, 9), (1, 6) 2. (–4, –1), (–2, –5) 3. (–4, –1), (–4, –5)

Find the value of r so the line that passes through each pair of points has the given slope.

4. (6, 8), (r, –2), m = 1 5. (–1, –3), (7, r), m = 34

NAME ______________________________________________ DATE______________________________ PERIOD ______________

4-1 Graphing Equations in Slope-Intercept Form

Slope-Intercept Form

Slope-Intercept Form y = mx + b, where m is the slope and b is the y-intercept

ExercisesWrite an equation of a line in slope-intercept form with the given slope and y-intercept.

1. slope: 8, y-intercept –3 2. slope: –2, y-intercept –1

Write an equation in slope-intercept form for each graph shown.

3. 4.

Graph each equation.

5. y = 2x + 1 6. y = –3x + 2

Exercises

2. POPULATION The population of the United States is projected to be 300 million by the year 2010. Between 2010 and 2050, the population is expected to increase by about 2.5 million per year.

a. Write an equation to find the population P in any year x between 2010 and 2050.

Projected UnitedStates Population

b. Graph the equation on the grid at the right.

c. Find the population in 2050.

NAME ______________________________________________ DATE______________________________ PERIOD ______________

4-2 Writing Equations in Slope-Intercept Form

ExercisesWrite an equation of the line that passes through the given point and has the given slope.

1. 2. (8, 2); slope −34 3. (–1, –3); slope 5

ExercisesWrite an equation of the line that passes through each pair of points.

4. 5. (–1, 6), (7, –10) 6. (0, 2), (1, 7)

NAME ______________________________________________ DATE______________________________ PERIOD ______________

4-3 Writing Equations in Point-Slope Form

Point-Slope Form

Point-Slope Formy− y1 = m(x−x1), where (x1 , y1) is a given point on a nonvertical line and m is the slope of the line

ExercisesWrite an equation in point-slope form for the line that passes through each point with the given slope.

1. 2. 3. (2, 1), m = 4

4. Write an equation in point-slope form for a horizontal line that passes through (4, –2).

Forms of Linear Equations

Slope-Intercept Form y = mx + b m = slope; b = y-intercept

Point-Slope Form y− y1 = m(x−x1) m = slope; (x1 , y1) is a given point

Standard Form Ax + By = C A and B are not both zero. Usually A is nonnegative and A, B, and C are integers whose greatest common factor is 1.

ExercisesWrite each equation in standard form.

5. y + 2 = –3(x – 1) 6. y – 1 = −13 (x – 6)

Write each equation in slope-intercept form.

7. y + 4 = 4(x – 2) 8. y – 5 = 13 (x – 6)

NAME ______________________________________________ DATE______________________________ PERIOD ______________

4-4 Parallel and Perpendicular Lines

Parallel Lines Two non-vertical lines are parallel if they have the same slope. All vertical lines are parallel.

ExercisesWrite an equation in slope-intercept form for the line that passes through the given point and is parallel to the graph of each equation.

1. 2. (–2, 2), y = 4x – 2

3. Find an equation of the line that has a y-intercept of 2 that is parallel to the graph of the line 4x + 2y = 8.

Perpendicular Lines Two non-vertical lines are perpendicular if their slopes are negative reciprocals of each other. Vertical and horizontal lines are perpendicular.

Determine whether the graphs of the following equations are parallel or perpendicular.

4. 2x + y = –7, x – 2y = –4, 4x – y = 5

Write an equation in slope-intercept form for the line that passes through the given point and is perpendicular to the graph of each equation.

5. (4, 2), y = 12x + 1 6. (2, –3), y =

−23 x + 4

NAME ______________________________________________ DATE______________________________ PERIOD ______________

4-7 Inverse Linear Functions

Inverse Relations An inverse relation is the set of ordered pairs obtained by exchanging the x-coordinates with the y-coordinates of each ordered pair. The domain of a relation becomes the range of its inverse, and the range of the relation becomes the domain of its inverse

ExercisesFind the inverse of each relation.

1. {(4, 7), (6, 2), (9, –1), (11, 3)} 2.

Graph the inverse of each relation.

3. 4.

Inverse Functions A linear relation that is described by a function has an inverse function that can generate ordered pairs of the inverse relation. The inverse of the linear function f (x) can be written as f−1 (x) and is read f of x inverse or the inverse of f of x.

ExercisesFind the inverse of each function.

5. f (x) = 4x – 3

6. TOOLS Jimmy rents a chainsaw from the department store to work on his yard. The total cost C(x) in dollars is given by C(x) = 9.99 + 3.00x, where x is the number of days he rents the chainsaw.

a. Find the inverse function C−1 (x).

b. What do x and C−1 (x) represent in the context of the inverse function?

x y–8 –15–2 –111 –85 1

11 8

NAME ______________________________________________ DATE______________________________ PERIOD ______________c. How many days did Jimmy rent the chainsaw if the total cost was $27.99?

5-3 Solving Multi-Step Inequalities

Solve Multi-Step Inequalities To solve linear inequalities involving more than one operation, undo the operations in reverse of the order of operations, just as you would solve an equation with more than one operation.

ExercisesSolve each inequality. Check your solution.

1. 11y + 13 ≥ –1 2. 8n – 10 < 6 – 2n 3. q7 + 1 > –5

Define a variable, write an inequality, and solve each problem. Check your solution.

4. Negative three times a number plus four is no more than the number minus eight.

Solve Inequalities Involving the Distributive Property When solving inequalities that contain grouping symbols, first use the Distributive Property to remove the grouping symbols. Then undo the operations in reverse of the order of operations, just as you would solve an equation with more than one operation.

ExercisesSolve each inequality. Check your solution.

5. 2(t + 3) ≥ 16 6. 4h – 8 < 2(h – 1)

Define a variable, write an inequality, and solve each problem. Check your solution.

7. Twice the sum of a number and 4 is less than 12.

NAME ______________________________________________ DATE______________________________ PERIOD ______________

5-5 Inequalities Involving Absolute Value

Inequalities Involving Absolute Value (<) When solving inequalities that involve absolute value, there are two cases to consider for inequalities involving < (or ≤). Remember that inequalities with and are related to intersections.

ExercisesSolve each inequality. Then graph the solution set.

1. |y| < 3 2. |x−4| < 4 3. |y+3| ≤ 2

Solve Absolute Value Inequalities (>) When solving inequalities that involve absolute value, there are two cases to consider for inequalities involving > (or ≥). Remember that inequalities with or are related to unions.

ExercisesSolve each inequality. Then graph the solution set.

1. |c−2| > 6 2. |x−3| > 0 4. |x| ≥ 2

NAME ______________________________________________ DATE______________________________ PERIOD ______________

5-6 Graphing Inequalities in Two Variables

Graph Linear Inequalities The solution set of an inequality that involves two variables is graphed by graphing a related linear equation that forms a boundary of a half-plane. The graph of the ordered pairs that make up the solution set of the inequality fill a region of the coordinate plane on one side of the half–plane.

ExercisesGraph each inequality.

1. y < 4 2. x ≥ 1 3. y < – 12 x – 3

4. x – y ≥ 1 5. 2x – 3y ≤ 6