Math 2: Algebra 2, Geometry and Statistics
Ms. Sheppard-Brick 617.596.4133
http://lps.lexingtonma.org/Page/2434
Name:
Date:
Chapter 3 Test Review
Students Will Be Able To:
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Use the quadratic formula to solve (find the roots of) a quadratic equation (TB 3.02)
Use the discriminant (𝑏 ! − 4𝑎𝑐) to determine how many real solutions a quadratic equation
has and whether the real solutions are rational or irrational (TB 3.02)
Find (multiple) quadratic equations that have a given pair of roots (solutions) (TB 3.03)
Use factoring (including GCF, Difference of squares, monic quadratic factoring and non-monic
quadratic factoring) to find solutions to polynomials (TB 3.04)
Graph quadratic functions efficiently from vertex form, standard/normal form, and factored
form of a quadratic equation (TB 3.07 and TB 3.08)
Graph quadratic functions when given some of the critical values (TB 3.07 and TB 3.08)
Model projectile motion with quadratic function and find values such as launch point, vertex,
landing point within the context of the problem (TB 3.07)
Convert between different forms of a quadratic equation (TB 3.08)
Vocabulary: Find and write definitions for each of the vocabulary words below. You should know all
of these words and be able to use them in context.
Factor an Expression
Constant Term
Monomial
Quadratic Equation
Difference of Squares
Parabola
Vertex
Expand an Expression
Linear Term
Binomial
Monic Equation
Perfect Square Trinomial
Roots, Zeros, x-intercepts
Maximum of a Parabola
Solve a Quadratic Equation
Quadratic Term
Trinomial
Non-Monic Equation
Greatest Common Factor
y-Intercept
Minimum of a Parabola
Completing the Square
Coefficient
Standard (Normal) Form
Vertex Form
Factored Form
Mirror Point
Line of Symmetry
All of this can be done without a graphing calculator. A scientific calculator is acceptable and will be
allowed on the test. You will need separate paper to have room to do the problems.
1. Describe how to find the vertex of a parabola from each of the following forms of the equation.
a. Vertex form
b. Standard form
c. Factored form
2. A parabola has a vertex at (3, -8) and passes through the point (5, 20). Write an equation of the
parabola in vertex form.
3. A parabola has zeros at 4 and -3. Write an equation in intercept form (factored form) so that the yintercept is (0, 6).
Ch 3 Quadratic Test Review
page 2
4. Here is a quadratic in vertex form: y = 4(x − 2) 2 −100
Calculate your points and think about your scaling before graphing anything.
€
Sketch a graph clearly labeling:
The vertex
The zeros
The axis of symmetry
The y-intercept
A mirror point
5. Here is a quadratic in intercept form: y = −4(x − 2.5)(x + 8.5) .
Calculate your points and think about your scaling before graphing anything.
Sketch a graph clearly Free
labeling:
€Multi-Width Graph Paper from http://incompetech.com/graphpaper/
The vertex
The zeros
The axis of symmetry
The y-intercept
A mirror/sister point
Free Multi-Width Graph Paper from http://incompetech.com/graphpaper/m
Ch 3 Quadratic Test Review
page 3
6. Solve by any method you choose.
a. 3x 2 − 48 = 0
€
€
b. x 2 + 4 x = −4
c. x 6 − 9x 4 = 0
€
e. 8x2 + 14x – 15 = 0
€
f.
3(2x + 1)2 = (2x + 1)
g. 5x2 – 20x = 62
h. 3(x + 5) 2 = 0
i. x2 +1.5x + 3.5 = 0
j.
€
7. Given equation 2x 2 − 20x +15 = 0
a. Solve by completing the square:
€
8. Put in vertex form:
a. y = 5x 2 +12x − 8
€
d. x 3 = 16x
x2
14x
+3=
2
4
b. solve using the quadratic formula
b. y = 3x 2 +18x −11
9. A ball is thrown upward from a flat surface. Its height in meters as a function of time since it
€
was thrown (in seconds) is given by the equation
y = −5t 2 + 34t + 25 .
a.
b.
c.
d.
What was the highest the ball got, and when did it reach that height?
When does the ball land?
€
What does the 25 in the equation signify?
Graph the situation using an appropriate domain.
Free Multi-Width Graph Paper from http://incompetech.com/graphpa
Ch 3 Quadratic Test Review
page 4
Extra Practice: If you can do all the problems above correctly, you are in good shape. Here are some
extra practice problems if you need more work on a particular topic.
I.
Factoring Non-Monic Quadratics (TB 3.04)
1. Factor:
2. Factor:
3. Factor:
II.
Solving Quadratic Equations (TB 2.10, 2.11, 3.02)
III.
Finding the Equation of a Quadratic Function (TB 3.03)
4.
5.
6.
7.
Ch 3 Quadratic Test Review
IV.
Graphing Quadratic Functions (TB 3.07, 3.08)
V.
Modeling Projectile Motion
page 5
8.
9.
10.
11.
12.
13. A ball is thrown from some unknown height. After 3 seconds, its height is 50 meters and after 5
seconds its height is 10 meters.
a. Write the function h(t ) = −5t 2 + bt + c . Use the two points given to find b and c.
b. Exactly when does the ball land?
c. What is the highest the ball got and when did it reach that height?
14. A small rocket is launched from the top of a building. Its height (in meters) as a function of time (in
seconds) is given by the equation h(t ) = −5t 2 + 40t + 100 .
a. How high is it after 3 seconds?
b. How tall is the building?
c. What two times is its height 100 meters?
d. When is its height 160 meters?
e. When does it land?
f. What is its maximum height and when is it attained?
Ch 3 Quadratic Test Review
page 6
ANSWERS:
1. a. The vertex form of the equation of a parabola is 𝑦 = 𝑎(𝑥 − ℎ)! + 𝑘 where (h, k) are the
coordinates of the vertex.
!!
b. The standard form of the equation of a parabola is 𝑦 = 𝑎𝑥 ! + 𝑏𝑥 + 𝑐. To find the vertex, use !!
as the x-value of the vertex. Then plug that value into the original equation to find the y-value.
c. The factored form of the equation of a parabola is 𝑦 = 𝑎(𝑥 − 𝑚)(𝑥 − 𝑛) where m and n are the
roots of the parabola. Average the roots to find the x-value of the vertex. Then plug that value into
the original equation to find the y-value.
2. y = 7(x − 3) 2 − 8
3. y = −0.5(x − 4)(x + 3)
60
40
20
4. Vertex: (2, –100)
Zeros: –3 and 7
Axis of symmetry: x=2
y-intercept: (0, –84)
other point: (4, –84)
€
€
–5
5
– 20
– 40
– 60
120
– 80
100
– 100
80
5. Zeros: 2.5 and -8.5
Vertex is halfway between at x = –3.
Plug in –3 to get the y-value (–3, 121)
Axis of symmetry: x = –3
y-intercept: (0, 85)
– 120
€
6. a.
b.
c.
d.
e.
f.
g.
h.
i.
j.
4, –4
–2
0, –3, 3
0, 4, –4
–2.5, 0.75
–1/2, –1/3
2 ± 825
–5
no solutions
1, 6
60
40
20
– 10
–5
5
– 20
Factor out the GCF of 3 and then difference of squares
Factor/Complete the square
Factor out the GCF of x4 and then difference of squares
Factor out the GCF of x and then difference of squares
“Sasha’s Method” of factoring or quadratic formula
Factoring
Completing the square or quadratic formula
Solve directly by taking the square root of both sides
Multiply through by 2 to get rid of the non-monic fraction
7. Completing the square gives: 5 ±
same: 9.18 and 0.817
35
2
, Quadratic Formula gives: 5 ±
70
2
, Both answers are the
100
8. a. y = 5(x +1.2) −15.2
€
2
9. a.
b.
€
c.
d.
b. y = 3(x + 3) − 38
3.4 seconds, 82.8 feet
7.47 seconds
€ is 0)
starting height (when time
(graph at right à)
2
€
80
60
40
20
5
– 20
Ch 3 Quadratic Test Review
page 7
Extra Practice
1.
2.
3.
4.
5.
6.
a & b) Any equation with the form
𝑦 = 𝑎(𝑥 + 2)(𝑥 − 7) where a is any
real number.
c & d) Any equation with the form
𝑦 = 𝑎 𝑥 − 2 𝑥 + 7 where a is any
real number.
e. No Real solutions, the
discriminant is negative.
7. Answers will vary, but must be
8.
these equations multiplied by some
real number.
8.
8.
8.
8.
Ch 3 Quadratic Test Review
page 8
9.
10.
11.
12.
13.
a) b = 20 and c = 35 (plug in values
and solve the system of equations)
so the equation is:
ℎ 𝑡 = −5𝑡 ! + 20𝑡 + 35
14.
a) After 3 seconds, the height is 67
meters.
b) The building is 100 meters tall.
c) The rocket will be 100 meters at
0 seconds and 8 seconds.
d) The rocket will be 160 meters at
2 seconds and 6 seconds.
e) The rocket lands after 10
seconds.
f) The vertex of the parabola is (4,
180) so the rocket reaches its
highest point of 180 meters 4
seconds after launch
!!! !!
b) The ball lands at ! seconds
(remember that an exact answer
means DO NOT estimate a decimal
equivalent)
c) The vertex is at (2, 55) which
means that the ball reached its
highest point of 55 meters 2 seconds
after launch.