Unit #3 Linear Functions, Equations and Their Equations Algebra
Name: ____________________________________
Date: ___________________
UNIT #3 – LINEAR FUNCTIONS, EQUATIONS, AND THEIR ALGEBRA COMMON CORE ALGEBRA II Part I Questions 1. The distance that a person drives at a constant speed varies directly with the amount of time they have been driving. If, at a particular speed, a person drives 107 miles in two hours, then how far will they drive, at the same speed, in 1 14 hours? (1) 75 miles
(3) 91 miles
(2) 44 miles
(4) 67 miles
2. Given the function f x x 2 2 x 7 , what is its average rate of change over the interval 3 x 11 ? (1) 8
(3) 5
(2) 12
(4) 7
3. Which of the following an equation for the line that is parallel to the line y 2 x 9 and passes through the point 1, 5 ? (1) y 14 x 9
(3) y 2 x 7
(2) y 2 x 5
(4) y 14 x 4
4. Given the line pictured below, which of the following could be its equation?
3 (1) y x 8 4 (2) y 2 x 7
1 (3) y x 4 2 (4) y
3 x 1 2
COMMON CORE ALGEBRA II, UNIT REVIEWS – UNIT #3 eMATHINSTRUCTION, RED HOOK, NY 12571, © 2015
5. At what x-coordinate would a line whose equation is y 2 x 3 intersect a perpendicular line whose yintercept is 17? (1) x 12
(3) x 5
(2) x 11
(4) x 8
6. The speed of a car, in miles per hour, is a linear function of time, in minutes. Which of the following would be the units of the slope of this linear function? (1) miles (2)
miles/hour minute
(3) hours (4)
miles minute
7. Which of the following is the equation of the inverse of the linear function y 4 x 2 ? (1) y
1 1 x 4 2
(3) y 4 x 2
(2) y
1 x2 4
1 (4) y x 8 4
8. Which of the following is the equation of the piecewise linear function shown below?
x4 x2 (1) f x 3x 5 x 2 1 x4 x2 (2) f x 2 3 x 1 x 2 1 x 5 x 2 (3) f x 4 3 x 3 x 2 2 x 4 x 2 (4) f x 4x 1 x 2 9. The graph of a function and the graph of its inverse always have symmetry across (1) the x-axis
(3) the line y x
(2) the y-axis
(4) the line y x COMMON CORE ALGEBRA II, UNIT REVIEWS – UNIT #3 eMATHINSTRUCTION, RED HOOK, NY 12571, © 2015
10. Given the linear graph shown below answer the following questions. (a) Write the equation of the line in y mx b form.
(b) Create a graph of this linear function's inverse on the same set of graph paper.
(c) Determine the equation of the inverse.
11. Selected values of a linear function f x are given in the table below. Find the value of k. Explain how you found your answer. x
8
2
4
12
14
18
f x
33
12
9
k
44
58
12. Write an equation for the line passing through the points 5, 15 and 20, 25 . Show how you arrived at your answer.
COMMON CORE ALGEBRA II, UNIT REVIEWS – UNIT #3 eMATHINSTRUCTION, RED HOOK, NY 12571, © 2015
13. After a recent Arlington High School basketball game, traffic was exiting the parking lot at a constant rate of 28 cars per minute. The parking lot started with 922 cars. (a) How many cars are still in the parking lot after (b) Determine a formula for the number of cars, n, 10 minutes? in the parking lot after m-minutes.
(c) After 25 minutes, the rate at which the cars leave rises to 34 cars per minute. How many total minutes does it take for the parking lot to completely clear? Round to the nearest minute. Show your analysis.
14. A single pump is filling a storage container with water at a rate of 60 gallons per minute. After 30 minutes, an additional pump turns on and the container begins to fill at a total rate of 130 gallons per minute for an additional 30 minutes. The container already had 1,500 gallons of water when it began to be filled. (b) Write a piecewise defined function for the volume, V, as a function of time, t, measured in minutes.
Volume (gal)
(a) On the grid below, graph the amount of water the tank contains for the first 60 minutes.
Time (min)
COMMON CORE ALGEBRA II, UNIT REVIEWS – UNIT #3 eMATHINSTRUCTION, RED HOOK, NY 12571, © 2015
15. Determine a piecewise equation for the function shown graphed below.
x 2 x 1 16. For the piecewise function g x 3 answer the following questions by using algebraic 4 x 2 x 1 techniques. Explain your thinking and show your work. (a) What is the y-intercept of this function?
(b) Determine the x-intercept(s) of this function.
(c) Which has the greater average rate of change over the interval 12 x 8 , the function g x or the function f x 2 x 7
(d) Provide evidence that this function is not one-to-one. Explain how your evidence supports that g x is not one-to-one.
COMMON CORE ALGEBRA II, UNIT REVIEWS – UNIT #3 eMATHINSTRUCTION, RED HOOK, NY 12571, © 2015
17. Solve the following system of equations algebraically. 3x 5 y 2 z 5 5 x y 6 z 33 2 x 10 y 3 z 40
18. One application of solving a system of three linear equations is, somewhat ironically, in algebraically finding the equation of a parabola in standard form. Given a parabola that passes through the points 4, 5 , 1, 10 , and 2, 11 : (a) Substitute each point into the general form y ax 2 bx c , to produce three equations with the three unknowns a, b, and c. The first is done for you.
5 a 4 b 4 c 5 16a 4b c 2
(b) Solve this system for a, b, and c and state the equation of the parabola.
COMMON CORE ALGEBRA II, UNIT REVIEWS – UNIT #3 eMATHINSTRUCTION, RED HOOK, NY 12571, © 2015
Date: ___________________
UNIT #3 – LINEAR FUNCTIONS, EQUATIONS, AND THEIR ALGEBRA COMMON CORE ALGEBRA II Part I Questions 1. The distance that a person drives at a constant speed varies directly with the amount of time they have been driving. If, at a particular speed, a person drives 107 miles in two hours, then how far will they drive, at the same speed, in 1 14 hours? (1) 75 miles
(3) 91 miles
(2) 44 miles
(4) 67 miles
2. Given the function f x x 2 2 x 7 , what is its average rate of change over the interval 3 x 11 ? (1) 8
(3) 5
(2) 12
(4) 7
3. Which of the following an equation for the line that is parallel to the line y 2 x 9 and passes through the point 1, 5 ? (1) y 14 x 9
(3) y 2 x 7
(2) y 2 x 5
(4) y 14 x 4
4. Given the line pictured below, which of the following could be its equation?
3 (1) y x 8 4 (2) y 2 x 7
1 (3) y x 4 2 (4) y
3 x 1 2
COMMON CORE ALGEBRA II, UNIT REVIEWS – UNIT #3 eMATHINSTRUCTION, RED HOOK, NY 12571, © 2015
5. At what x-coordinate would a line whose equation is y 2 x 3 intersect a perpendicular line whose yintercept is 17? (1) x 12
(3) x 5
(2) x 11
(4) x 8
6. The speed of a car, in miles per hour, is a linear function of time, in minutes. Which of the following would be the units of the slope of this linear function? (1) miles (2)
miles/hour minute
(3) hours (4)
miles minute
7. Which of the following is the equation of the inverse of the linear function y 4 x 2 ? (1) y
1 1 x 4 2
(3) y 4 x 2
(2) y
1 x2 4
1 (4) y x 8 4
8. Which of the following is the equation of the piecewise linear function shown below?
x4 x2 (1) f x 3x 5 x 2 1 x4 x2 (2) f x 2 3 x 1 x 2 1 x 5 x 2 (3) f x 4 3 x 3 x 2 2 x 4 x 2 (4) f x 4x 1 x 2 9. The graph of a function and the graph of its inverse always have symmetry across (1) the x-axis
(3) the line y x
(2) the y-axis
(4) the line y x COMMON CORE ALGEBRA II, UNIT REVIEWS – UNIT #3 eMATHINSTRUCTION, RED HOOK, NY 12571, © 2015
10. Given the linear graph shown below answer the following questions. (a) Write the equation of the line in y mx b form.
(b) Create a graph of this linear function's inverse on the same set of graph paper.
(c) Determine the equation of the inverse.
11. Selected values of a linear function f x are given in the table below. Find the value of k. Explain how you found your answer. x
8
2
4
12
14
18
f x
33
12
9
k
44
58
12. Write an equation for the line passing through the points 5, 15 and 20, 25 . Show how you arrived at your answer.
COMMON CORE ALGEBRA II, UNIT REVIEWS – UNIT #3 eMATHINSTRUCTION, RED HOOK, NY 12571, © 2015
13. After a recent Arlington High School basketball game, traffic was exiting the parking lot at a constant rate of 28 cars per minute. The parking lot started with 922 cars. (a) How many cars are still in the parking lot after (b) Determine a formula for the number of cars, n, 10 minutes? in the parking lot after m-minutes.
(c) After 25 minutes, the rate at which the cars leave rises to 34 cars per minute. How many total minutes does it take for the parking lot to completely clear? Round to the nearest minute. Show your analysis.
14. A single pump is filling a storage container with water at a rate of 60 gallons per minute. After 30 minutes, an additional pump turns on and the container begins to fill at a total rate of 130 gallons per minute for an additional 30 minutes. The container already had 1,500 gallons of water when it began to be filled. (b) Write a piecewise defined function for the volume, V, as a function of time, t, measured in minutes.
Volume (gal)
(a) On the grid below, graph the amount of water the tank contains for the first 60 minutes.
Time (min)
COMMON CORE ALGEBRA II, UNIT REVIEWS – UNIT #3 eMATHINSTRUCTION, RED HOOK, NY 12571, © 2015
15. Determine a piecewise equation for the function shown graphed below.
x 2 x 1 16. For the piecewise function g x 3 answer the following questions by using algebraic 4 x 2 x 1 techniques. Explain your thinking and show your work. (a) What is the y-intercept of this function?
(b) Determine the x-intercept(s) of this function.
(c) Which has the greater average rate of change over the interval 12 x 8 , the function g x or the function f x 2 x 7
(d) Provide evidence that this function is not one-to-one. Explain how your evidence supports that g x is not one-to-one.
COMMON CORE ALGEBRA II, UNIT REVIEWS – UNIT #3 eMATHINSTRUCTION, RED HOOK, NY 12571, © 2015
17. Solve the following system of equations algebraically. 3x 5 y 2 z 5 5 x y 6 z 33 2 x 10 y 3 z 40
18. One application of solving a system of three linear equations is, somewhat ironically, in algebraically finding the equation of a parabola in standard form. Given a parabola that passes through the points 4, 5 , 1, 10 , and 2, 11 : (a) Substitute each point into the general form y ax 2 bx c , to produce three equations with the three unknowns a, b, and c. The first is done for you.
5 a 4 b 4 c 5 16a 4b c 2
(b) Solve this system for a, b, and c and state the equation of the parabola.
COMMON CORE ALGEBRA II, UNIT REVIEWS – UNIT #3 eMATHINSTRUCTION, RED HOOK, NY 12571, © 2015
