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: • Use the quadratic formula to solve (find the roots of) a quadratic equation (TB 3.02) • Use the discriminant (! − 4) 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 Expand an Expression Solve a Quadratic Equation Completing the Square Constant Term Linear Term Quadratic Term Coefficient Monomial Binomial Trinomial Standard (Normal) Form Quadratic Equation Monic Equation Non-Monic Equation Vertex Form Difference of Squares Perfect Square Trinomial Greatest Common Factor Factored Form Parabola Roots, Zeros, x-intercepts y-Intercept Mirror Point Vertex Maximum of a Parabola Minimum of a Parabola 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 y- intercept is (0, 6).
• 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 Expand an Expression Solve a Quadratic Equation Completing the Square Constant Term Linear Term Quadratic Term Coefficient Monomial Binomial Trinomial Standard (Normal) Form Quadratic Equation Monic Equation Non-Monic Equation Vertex Form Difference of Squares Perfect Square Trinomial Greatest Common Factor Factored Form Parabola Roots, Zeros, x-intercepts y-Intercept Mirror Point Vertex Maximum of a Parabola Minimum of a Parabola 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 y-
intercept is (0, 6).
Ch 3 Quadratic Test Review page 2
Free Multi-Width Graph Paper from http://incompetech.com/graphpaper/multiwidth/
Free Multi-Width Graph Paper from http://incompetech.com/graphpaper/multiwidth/
4. Here is a quadratic in vertex form:
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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 labeling:
The vertex
The zeros
The axis of symmetry
The y-intercept
A mirror/sister point
Ch 3 Quadratic Test Review page 3
Free Multi-Width Graph Paper from http://incompetech.com/graphpaper/multiwidth/
6. Solve by any method you choose.
a.
€
3x 2 − 48 = 0 b.
€
x 2 + 4x = −4
c.
€
x 6 − 9x 4 = 0 d.
€
x3 =16x
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. x2
2+3= 14x
4
7. Given equation
€
2x 2 − 20x +15 = 0 a. Solve by completing the square: b. solve using the quadratic formula
8. Put in vertex form:
a.
€
y = 5x 2 +12x − 8
b.
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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. What was the highest the ball got, and when did it reach that height? b. When does the ball land? c. What does the 25 in the equation signify? d. Graph the situation using an appropriate domain.
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)
4.
5.
III. Finding the Equation of a Quadratic Function (TB 3.03) 6.
7.
Ch 3 Quadratic Test Review page 5
IV. Graphing Quadratic Functions (TB 3.07, 3.08) 8.
9.
10.
11.
12.
V. Modeling Projectile Motion 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 cbttth ++−= 25)( . 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 100405)( 2 ++−= ttth . 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) 4. Vertex: (2, –100)
Zeros: –3 and 7 Axis of symmetry: x=2 y-intercept: (0, –84) other point: (4, –84)
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)
6. a. 4, –4 Factor out the GCF of 3 and then difference of squares
b. –2 Factor/Complete the square c. 0, –3, 3 Factor out the GCF of x4 and then difference of squares d. 0, 4, –4 Factor out the GCF of x and then difference of squares e. –2.5, 0.75 “Sasha’s Method” of factoring or quadratic formula f. –1/2, –1/3 Factoring g.
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2 ± 825 Completing the square or quadratic formula
h. –5 Solve directly by taking the square root of both sides i. no solutions j. 1, 6 Multiply through by 2 to get rid of the non-monic fraction
7. Completing the square gives:
€
5 ± 352 , Quadratic Formula gives:
€
5 ± 702 , Both answers are the
same: 9.18 and 0.817 8. a.
€
y = 5(x +1.2)2 −15.2 b.
€
y = 3(x + 3)2 − 38 9. a. 3.4 seconds, 82.8 feet
b. 7.47 seconds c. starting height (when time is 0) d. (graph at right à)
60
40
20
–20
–40
–60
–80
–100
–120
–5 5
120
100
80
60
40
20
–20
–10 –5 5
100
80
60
40
20
–20
5
Ch 3 Quadratic Test Review page 7
Extra Practice 1.
2.
3.
4.
5.
e. No Real solutions, the discriminant is negative.
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.
7. Answers will vary, but must be these equations multiplied by some real number.
8.
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 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.
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
