Mth 100 - Quadratic functions and quadratic equations
Name___________________________________
Find the product.
1) 8ax 3 (2ax 3 + 5x2 + 12a)
2) (x + 4y)(x + 6y)
3) (3p - 1)(9p2 + 3p + 1)
4) (3x2 + 4x - 4)(x 2 - 3x + 3)
5) (4a - 7)2
Factor completely.
6) 9x2 - 4
7) 50a 4 b - 32b3
8) x2 - x - 6
9) u2 - 4uv - 32v2
10) 9y2 + 18y + 8
11) 20x2 + 27x + 9
12) 2x3 + 2x2 y - 24xy2
Solve the equation.
13) 24d2 + 38d + 15 = 0
14) 10m2 - 9m = 0
15) 2x 2 + 56 = x 2 + 15x
16) (x - 6)(x - 8) = 288
Use the square root property to solve the equation. All solutions are real numbers.
17) (7t + 2)2 = 13
18) (4s + 9)2 = 4
19) 5z 2 + 3 = 128
20) (4x + 5)2 = -8
1
The two graphs shown represent the graphic solution to what equation?
What are the approximate solutions to the equation?
What are the exact solutions to the equation?
21)
Solve the equation.
22) 3n 2 = -12n - 7
23)
z2 z 5
= +
3
2 6
24) 3x2 + 7x = -6
For the next three problems use the discriminant to determine whether the following equation has solutions that are: two
different real solutions; exactly one real solution; or two different imaginary solutions.
25) s 2 - 7s - 8 = 0
26) w2 + 3w + 5 = 0
27) 4x2 + 8x + 4 = 0
Answer the question.
28) Describe how the graph of y = x + 9 2 is shifted compared to the graph of y = x2 .
For the next two problems, find the coordinates of the vertex of the parabola.
29) y = -2x2 + 16x - 31
30) x = y2 + 16y + 60
Identify the vertex of the quadratic equation.
31) f(x) = (x + 5)2 + 7
32) f(x) = x2 - 4
33) f(x) = (x - 8)2
Tell whether the graph opens upward or downward and whether the graph is wider, narrower, or the same as f(x) = x 2 .
34) f(x) = -.7x2 + 9
2
For the next two problems identify which graph matches the equation. Do not use the calculator, explain how you decide
which one is the graph.
35) Do not use the calculator, explain how you decide which one is the graph.
f(x) = 5(x + 4)2 + 3
A
B
y
10
y
10
10 x
-10
10 x
-10
-10
-10
36) f(x) = (x - 2)2 - 4
A
10
9
8
7
6
5
4
3
2
1
B
y
10
9
8
7
6
5
4
3
2
1
-1 1 2 3 4 5 6 7 8 9
-10-9-8-7-6-5-4-3-2-1
-2
-3
-4
-5
-6
-7
-8
-9
-10
x
-1 1 2 3 4 5 6 7 8 9
-10-9-8-7-6-5-4-3-2-1
-2
-3
-4
-5
-6
-7
-8
-9
-10
Sketch the graph of the parabola.
1
37) f(x) = x2
2
10
y
5
-10
-5
5
y
10 x
-5
-10
3
x
38) f(x) = -4(x - 2)2 - 1
10
y
5
-10
-5
5
10 x
-5
-10
The calculator screen shows the x-intercepts of the graph. Use the graphs to solve each equation.
39) x2 + 3x - 18 = 0
40) -2x2 -3x +5 = 0
Answer the question.
41) The graph of a quadratic function y = f(x) is shown in the standard viewing window, without x-axis tick
marks. Which one of the following choices would be the only possible solution set for the equation f(x) = 0?
A) -2, 4
B) -4, 2
C) -2, -4
4
D) 2, 4
Graph the parabola. Use the coordinates of the vertex and two more points.
42) y = -x2 + 2x - 9
10
y
5
-10
-5
5
10 x
5
10 x
-5
-10
43) y = 4x2 + 2x - 2
10
y
5
-10
-5
-5
-10
Choose the equation that matches the graph.
44)
10
y
5
-10
-5
5
10 x
-5
-10
f(x) = x2 + 2x + 6, f(x) = x2 + 2x - 6
f(x) = x2 - 2x - 6, f(x) = -x2 + 2x - 6
5
45)
10
y
5
-10
-5
5
10 x
-5
-10
y = -x2 + 3x - 9, y = x2 + 3x - 9
y = -x2 - 3x - 9, y = x2 - 3x + 9
Solve the problem.
46) John owns a hotdog stand. He has found that his profit is represented by the equation P(x) = -x2 + 58x + 70,
with P being profits and x the number of hotdogs.
a) How many hotdogs must he sell to earn the most profit?
b) Sketch the graph of the function. Don't use the calculator.
Now check with the calculator and indicate the window values used.
c) Find P(10) and explain the meaning within the context of the problem.
d) How many hotdogs must he sell to earn more than $80? Solve graphically. Explain how you use your
calculator to solve. Label the points that help you answer this problem.
47) An object is thrown upward with an initial velocity of 14 ft per second. Its height is given by h = -14t2 + 56t at
time t seconds.
a) After how many seconds does it hit the ground?
b) Sketch the graph using your knowledge. Then use the calculator to check. Indicate window values used.
Label axes.
c) Make up three different questions related to this problem. Answer them.
48) Bob owns a watch repair shop. He has found that the cost of operating his shop is given by c = 4x2 - 288x + 47,
where c is cost and x is the number of watches repaired. How many watches must he repair to have the lowest
cost? Make up 3 different questions related to this problem and answer them.
A boy is standing on a flat field and tosses his ball toward a second boy standing at the other end of the field. The path of
the ball is a parabola, and the equation of the path is y = -4x 2 + 8x.
49) What is the name of the place on the path where the ball is highest from the ground?
50) What is the highest the ball will be above the flat field and for what value of x?
51) How many x-intercepts are there on the graph of the equation? Find them
6
52) What are the points on the parabola called where the ball is at ground level?
Decide whether the ordered pair is a solution of the given system.
53) (6, 1)
x+y=7
x-y=5
Solve the system using the graphing method.
54) 5x + y = -14
5x + 5y = 10
10
y
10 x
-10
-10
Solve by the elimination method.
55) x + 2y = -19
-2x + 2y = -4
56) 9x + 7y = 7
-3x + 4y = 4
57) 3x - 5y = 4
15x - 25y = 20
58) -4x - 6y = -2
12x + 18y = -6
Use the substitution method to solve the system of linear equations.
59) x + 2y = 2
8x - 5y = -5
60) 3x + y = 13
2x + 9y = -8
61) x + y = 0
2x + 3y = -7
7
Solve the problem.
62) The table shown was generated by a graphics calculator. The functions defined by y1 and y2 are linear.
a) Based on the table, find the coordinates of the point of intersection of the graphs.
b) Find the equations for Y1 and Y2. (Are they linear? Why?)
c) Solve by any method to check your answer to part (a).
63) The solution set of the system y1 = -x + 5 and y2 = -2x + 7 is {(2,3)}. Which of the two calculator generated
screens, left or right, is the appropriate one for this system?
64) Which of the ordered pairs listed could be possible solutions for the system whose graphs are shown in the
viewing window of a graphics calculator?
A) (-16, -7)
B) (-20, 12) or (16, 7)
C) (16, -7)
D) None of the listed pairs is a possible solution.
Answer the question.
65) What is indicated by the occurrence of a false statement such as "0 = 1" when you solve a system of two linear
equation (in two variables) using elimination?
66) What is indicated by the occurrence of a true statement such as "0 = 0" when you solve a system of two linear
equation (in two variables) using substitution?
67) Graphs of two linear functions f and g that are neither parallel nor coincident intersect in how many points?
8
Answer Key
Testname: REV-PARABOLAS.TST
1) 16a 2 x6 + 40ax 5 + 96a 2 x3
2) x2 + 10xy + 24y2
3) 27p3 - 1
4) 3x4 - 5x3 - 7x2 + 24x - 12
5) 16a 2 - 56a + 49
6) (3x + 2)(3x - 2)
7) 2b(5a 2 + 4b)(5a 2 - 4b)
8)
9)
10)
11)
(x + 2)(x - 3)
(u + 4v)(u - 8v)
(3y + 2)(3y + 4)
(4x + 3)(5x + 3)
12) 2x(x - 3y)(x + 4y)
3
5
13) - , 4
6
14)
9
,0
10
15) {8, 7}
16) {-10, 24}
13 - 2
17)
,7
18) -
13 + 2
7
7
11
,4
4
19) {5, -5}
-5 + i 8 -5 - i 8
20)
,
4
4
21) - 19 , 19
-6 + 15 -6 - 15
22)
,
3
3
23) -1,
24) x =
25)
26)
27)
28)
29)
30)
31)
32)
33)
34)
35)
36)
5
2
-7 ± i 23
6
Two different rational solutions
Two different imaginary solutions
Exactly one rational solution
The parabola is shifted 9 units to the left.
(4, 1)
(-4, -8)
(-5, 7)
(0, -4)
8, 0
Downward, wider
A
B
1
Answer Key
Testname: REV-PARABOLAS.TST
37)
10
y
5
-10
-5
5
10 x
5
10 x
5
10 x
-5
-10
38)
10
y
5
-10
-5
-5
-10
39) -6, 3
40) -2.5, 1
41) A
42)
10
y
5
-10
-5
-5
-10
2
Answer Key
Testname: REV-PARABOLAS.TST
43)
10
y
5
-10
-5
5
10 x
-5
-10
44) f(x) = x2 + 2x - 6
45) y = -x2 - 3x - 9
46)
47)
48)
49)
50)
51)
52)
53)
54)
29 hotdogs
4 sec
36 watches
The vertex
Highest height = 4 when x = 1
2
The x-intercepts
Yes
10
y
10 x
-10
-10
{(-4, 6)}
55)
56)
57)
58)
59)
60)
61)
62)
63)
{(-5, -7)}
{(0, 1)}
Infinitely many solutions
No solution
{(0, 1)}
{(5, -2)}
{(7, -7)}
(2, 7)
Right
64)
65)
66)
67)
C
The system is inconsistent.
The system h as an infinite number of solutions.
One
3