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Unit 3: QUADRATIC EQUATIONS Name: ________________________

Be prepared for daily quizzes. Every student is expected to do every assignment for the entire unit. Students who complete every assignment for this semester are eligible for a 2% semester

grade bonus and a pizza lunch paid by the math department. Try www.khanacademy.org or www.mathguy.us (Earl’s website) if you need help.

Check out our class webpage: DRHSAlgebra2.weebly.com

Day Date Assignment (Due the next class meeting)

Friday Monday

9/20/13 (A) 9/21/13(B)

3.1 Worksheet

Tuesday Wednesday

9/24/13 (A) 9/25/13 (B)

3.2 Worksheet

Thursday Friday

9/26/13 (A) 9/27/13 (B)

3.3 Worksheet

Monday Tuesday

9/30/13 (A) 9/31/13 (B)

3.4 Worksheet

Wednesday Thursday

10/1/13 (A) 10/2/13 (B)

3.5 Worksheet

Friday Monday

10/3/13 (A) 10/7/13 (B)

Unit 3 Practice Test STUDY for your test!

Next class is the LAST DAY you can turn in any late assignments from this unit.

Tuesday Wednesday

10/8/13 (A) 10/9/13 (B)

Unit 3 Test

3.1: Solving Quadratic Equations

WARM-UP: Graph: 𝒚 = − 𝒙 + 𝟏 𝟐 + 𝟒 1. Name the following: a. Vertex: ___________

b. Axis of Symmetry: ___________

c. y-intercept: __________

d. x-intercepts __________; __________

e. Max/Min Value: __________

f. Domain: _________ g. Range: _________

2. Describe the transformation of the graph

compared to the parent function 𝒚 = 𝒙𝟐.

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A. WHAT IS A QUADRATIC EQUATION?

A Quadratic Equation is an equation of the 2nd degree in the form

𝒂𝒙𝟐 + 𝒃𝒙 + 𝒄 = 𝟎, where 𝒂 ≠ 𝟎. 1. Decide whether or not each equation is a quadratic equation. a. 2𝑥2 + 3𝑥 − 2 = 0 b. 3𝑥 + 2 = 0 c. x2 = 9

d. 5𝑥 + 6 = 𝑥2 e. 1

𝑥2 + 2𝑥 − 3 = 0 f. 1

2𝑥2 = 4𝑥

B. SOLVING QUADRATIC EQUATIONS

A Quadratic Equation can have 2 Real solutions, 1 Real solution, or 2 Imaginary Solutions

SOLVING QUADRATIC EQUATIONS USING SQUARE ROOTS 1. Solve the following quadratics by using the Square Root method. If needed, write

answers in terms of i, and simplify radical answers. a. 𝑥2 = 9 b. 2𝑥2 − 100 = 0 c. −5𝑥2 + 10 = 70

d. 𝑥 − 2 2 − 9 = 0 e. 1

4 𝑦 − 6 2 = 8

QUESTION What is a Quadratic Equation & how do you solve it?

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SOLVING QUADRATIC EQUATIONS BY FACTORING 2. Solve each quadratic equation by using the factoring method. a. 𝑥2 + 8𝑥 + 12 = 0 b. 𝑥2 − 36 = 0 c. 2𝑥2 − 3𝑥 + 1 = 0 d. 3𝑥2 − 3𝑥 − 6 = 0

e. 0 = −𝑥2 + 25 f. 8 = 3x2 + 10𝑥

3. Solve each quadratic equation by factoring. Then include a quick sketch including

the x-intercepts of the graph.

a. 𝑦 = 𝑥2 + 7𝑥 + 10 b. 𝑦 = 𝑥2– 16

c. 𝑦 = − 3𝑥2 + 6𝑥 d. 𝑦 = 2𝑥2 – 7𝑥 − 4

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SOLVING QUADRATIC EQUATIONS USING THE QUADRATIC FORMULA

A quadratic equation in general form: 𝒂𝒙𝟐 + 𝒃𝒙 + 𝒄 = 𝟎. Let a, b, and c be real numbers such that 0a .

The Quadratic Formula: 2 4

2b b acx

a

Math Version:

x equals negative b plus or minus the square root

of b squared minus 4ac all over 2a

Actual Song:

All around the mulberry bush, The monkey chased the weasel.

The monkey thought 'twas all in fun. Pop! goes the weasel.

Or the story version… 4. Solve each quadratic equation using the quadratic formula.

a. 𝑥2 + 7𝑥 = −6 b. 2𝑥2 − 8𝑥 + 8 = 0

c. −𝑥2 + 2𝑥 = 5 d. 3𝑥2 + 7𝑥 + 11 = 5𝑥 + 7 e. 𝑦2 + 3𝑦 = 8 + 4𝑦 f. −4𝑥2 + 2𝑥 = 5

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Vertex Form

𝒇 𝒙 = 𝒂 𝒙 − 𝒉 𝟐 − 𝒌

Vertex

𝒉, 𝒌

𝑨𝒙𝒊𝒔 𝒐𝒇 𝒔𝒚𝒎𝒎𝒆𝒕𝒓𝒚

𝒙 = 𝒉

QUESTION How do you rewrite a quadratic equation into vertex form & solve the

equation by graphing?

3.2: Writing Quadratic Equation in Vertex Form

WARM-UP 1. Multiply 𝒙 + 𝟑 𝟐 2. Factor 𝐱𝟐 + 𝟏𝟎𝒙 + 𝟐𝟓

2. Write the equation 𝒚 = 𝒙 + 𝟐 𝟐 + 𝟑 in Standard Form ( 𝒚 = 𝒂𝒙𝟐 + 𝒃𝒙 + 𝒄)

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A. COMPLETING THE SQUARE

We can change a quadratic in standard form into vertex form by completing the square.

1. Relationship Between Coefficient of the Middle Term and Constant Term

Look at the following perfect square trinomials. How is the constant related to the middle term? a. 𝑥2 + 12𝑥 + 36 b. 𝑥2 − 20𝑥 + 100

To complete the square for the expression

𝒙𝟐 + 𝒃𝒙, add the square of half the coefficient of the term 𝒃𝒙.

𝒙𝟐 + 𝒃𝒙 + (𝒃

𝟐)𝟐

= (𝒙 +𝒃

𝟐)𝟐

2. For each of the following, find the value that completes the square. Then factor:

a. 𝑥 2 + 14𝑥 + _____ b. 𝑎 2 − 8𝑎 + _____

c. 𝑚 2 + 10𝑚 + _____ d. 𝑥 2 − 5𝑥 + _____

B. WRITING QUADRATIC EQUATIONS IN VERTEX FORM (WHEN 𝒂 = 𝟏

1. Write each quadratic equation in the form 𝒚 = 𝒙 − 𝒉 𝟐 + 𝒌

a. Rewrite 𝒚 = 𝒙𝟐 + 𝟔𝒙 + 𝟏𝟎 in Vertex Form.

Step 1: Separate the first two terms from the constant term.

𝒚 = 𝒙𝟐 + 𝟔𝒙 + 𝟏𝟎

Step 2a: Complete the square with the first two terms.

𝒚 = 𝒙𝟐 + 𝟔𝒙 + ________ + 𝟏𝟎

Step 2b: Since you added a number, you need to subtract the same number to the constant term.

𝒚 = 𝒙𝟐 + 𝟔𝒙 + ________ + 𝟏𝟎 − _______

Step 3: Rewrite it in vertex Form

𝒚 = 𝒙 + _________ 𝟐 + ________

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b. 𝑦 = 𝑥2 − 4𝑥 − 5 (Write your steps)

2. Examples

a. 𝑦 = 𝑥2 + 12𝑥 + 4 b. 𝑓 𝑥 = 𝑥2 + 2𝑥 − 8

c. 𝑔 𝑥 = 𝑥2 − 16𝑥 − 4 d. 𝑦 = −𝑥2 + 10𝑥 + 10 e. 𝑦 = −𝑥2 + 2𝑥 − 3 f. 𝑦 = 𝑥2 − 5𝑥 − 3

3. Which of the following is the vertex form for 𝑦 = 𝑥2 + 4𝑥 + 7 ?

A. 𝑦 = 𝑥 + 2 2 + 3 B. 𝑦 = 𝑥 + 2 2 + 7

C. 𝑦 = 𝑥 − 2 2 + 4 D. 𝑦 = 𝑥 − 2 2 + 3

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4. Write each equation in vertex from. Then solve the quadratic by graphing. In addition, identify the vertex, axis of symmetry, x-intercepts, and the maximum or minimum value. a. y = x2 – 4x + 3

b. 𝑦 = −𝑥2 + 2𝑥 − 4

A. WRITING QUADRATIC EQUATIONS IN VERTEX FORM (WHEN 𝒂 ≠ 𝟏 1. Write each quadratic equation in the form 𝒚 = 𝒂 𝒙 − 𝒉 𝟐 + 𝒌

a. Rewrite 𝒚 = 𝟐𝒙𝟐 − 𝟏𝟐𝒙 + 𝟏𝟕 into Vertex Form

Step 1: Separate the first two terms from the constant term.

𝒚 = 𝟐𝒙𝟐 − 𝟏𝟐𝒙 + 𝟏𝟕

Step 2: Factor out 𝑎 from the first two terms

𝒚 = 𝟐 𝒙𝟐 − 𝟔𝒙 + 𝟏𝟕

Step 3a: Complete the square with the first two terms inside the parentheses.

𝒚 = 𝟐 𝒙𝟐 − 𝟔𝒙 + ________ + 𝟏𝟕 Step 3b: Since you added a number, you need to subtract the same number to the

constant term (don’t forget to multiply it by 𝑎 .

𝒚 = 𝟐 𝒙𝟐 + 𝟔𝒙 + ________ + 𝟏𝟕 − _______

Step 4: Simplify it into Vertex Form

𝒚 = 𝟐 𝒙 + _________ 𝟐 + ________

3.3: Writing Quadratic Equation in Vertex Form (when 𝒂 ≠ 𝟏)

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b. 𝑦 = 3𝑥2 − 6𝑥 + 1 (Write your steps)

2. Examples

a. 𝑦 = 2𝑥2 + 8𝑥 − 9 b. 𝑓 𝑥 = 4𝑥2 + 24𝑥 + 11

c. 𝑦 = −2𝑥2 + 12𝑥 − 3 d. 𝑔 𝑥 = −3𝑥2 − 12𝑥 + 4

e. 𝑦 = 2𝑥2 + 20𝑥 − 14 f. 𝑦 = −5𝑥2 − 10𝑥 + 5

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3. What is the vertex of the function 𝑦 = −2𝑥2 + 6𝑥 − 9 ?

A. (−3

2 , −

9

2) B. (−

3

2 ,

27

2)

C. 3, −9 D. (3

2 , −

9

2)

B. SOLVING QUADRATIC EQAUTIONS BY GRAPHING 1. Write each equation in vertex from. Then solve the quadratic by graphing. In

addition, identify the vertex, axis of symmetry, the x-intercepts, and the maximum or minimum value.

a. 𝑦 = 2𝑥2 + 12𝑥 + 10

b. 𝑦 = −3𝑥2 – 18𝑥 – 24

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Intercept Form

𝒇 𝒙 = 𝒂 𝒙 − 𝒑 𝒙 − 𝒒

x-intercepts

𝒑 & 𝒒 → 𝒑, 𝟎 & 𝒒, 𝟎

Axis of Symmetry

𝒙 =𝒑 + 𝒒

𝟐

A. WRITING QUADRATIC EQUATION IN INTERCEPT FORM

1. Write each of the following quadratic equations or functions in intercept form using factoring.

a. 𝑦 = 𝑥2 + 6𝑥 + 5 b. 𝑓 𝑥 = 𝑥2 − 2𝑥 − 8 c. 𝑔 𝑥 = 6𝑥2 + 14𝑥 + 4 d. 𝑦 = −3𝑥2 + 24𝑥 − 45

3.4: Writing Quadratic Equation in Intercept Form

QUESTION How do you determine where the graph of an equation crosses the x-axis without writing

it in vertex form?

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e. 𝑦 = 𝑥2 + 4𝑥 f. 𝑦 = 4𝑥2 + 12𝑥 + 9

2. Write each equation / function in intercept form. Identify the y-intercept, x-intercepts and vertex of the functions graph. Then graph the function.

a. 𝑓 𝑥 = 𝑥2 − 2𝑥 − 3 b. 𝑦 = 𝑥2 + 2𝑥 − 8

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B. REVIEW OF THE DIFFERENT FORMS OF QUADRATIC EQUATIONS

FORMS of QUADRADIC FUNCTIONS

Forms of Quadratic Function What They Find Graph

Standard From y-intercept

𝒇 𝒙 = 𝒂𝒙𝟐 + 𝒃𝒙 + 𝒄 𝒇 𝟎 = 𝒄

Graphing Form vertex

𝒇 𝒙 = 𝒂 𝒙 − 𝒉 𝟐 + 𝒌 𝒉, 𝒌

Intercept/Factored Form x-intercept(s)

𝒇 𝒙 = 𝒂 𝒙 − 𝒑 𝒙 − 𝒒 𝒑, 𝟎 & 𝒒, 𝟎

1. Rewrite each equation in the indicated form.

a. Write 𝑓 𝑥 = 𝑥2 − 6𝑥 + 4 in vertex form. b. Write 𝑓 𝑥 = −3 𝑥 − 3 𝑥 − 5 in standard form. c. Write 𝑓 𝑥 = − 𝑥 + 3 2 + 4 in intercept form. d. Write 𝑓 𝑥 = 2 𝑥 + 1 𝑥 + 3 in vertex form.

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2. Which of following functions does NOT represent the parabola with a vertex at 1, 4 and x-intercepts −1, 0 and 3, 0 .

A. 𝑓 𝑥 = −𝑥2 + 𝑥 + 4 C. 𝑓 𝑥 = −𝑥2 + 2𝑥 + 3

B. 𝑓 𝑥 = − 𝑥 − 1 2 + 4 D. 𝑓 𝑥 = − 𝑥 + 1 𝑥 − 3

3. Translate the graph 𝑓 𝑥 = 𝑥2 + 6𝑥 + 5 four (4) units to the right. What is the

standard form of the resulting graph?

REFLECT What are the three forms of quadratic equations? What are some benefits & restrictions of each form?

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A. REVIEW OF SOLVING QUADRATIC EQUATIONS

Method Example When you can use When you can’t use

Square Roots

𝒚 = 𝒙𝟐 − 𝟑

Factoring

𝒚 = 𝒙𝟐 + 𝟕𝒙 + 𝟏𝟐

Quadratic Formula

𝒚 = 𝒙𝟐 + 𝟓𝒙 − 𝟒

Graphing

𝒚 = − 𝒙 − 𝟐 𝟐 + 𝟑

3.5: Review of Solving Quadratic Equations

WARM-UP 1. What are four ways to solve quadratic equations?

a. ________________________________ b. ________________________________ c. ________________________________ d. ________________________________

2. Solve each equation or functions using your preferred method (using a different method for each question). Why did you choose that method?

a. 𝑦 = 𝑥2 + 5𝑥 − 6 b. 𝑓 𝑥 = 2𝑥2 + 36

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B. EXAMPLES: Solve each quadratic equation using any method. Use the graphs on the

next page for any equation you choose to solve by graphing.

1. 𝑦 = 𝑥2 − 6𝑥 − 16 2. 𝑦 = 𝑥2 − 49 3. 2𝑥2 − 3𝑥 = 𝑥2 + 2 4. 𝑓 𝑥 = 𝑥 − 3 2 − 4 5. 2𝑥2 + 7𝑥 + 6 = 0 6. 𝑦 = 2𝑥2 + 4𝑥 + 7 7. 3𝑥2 + 10 = 70 8. 𝑓 𝑥 = − 2𝑥 + 3 𝑥 − 5 9. 𝑦 = −4𝑥2 − 10𝑥 10. 𝑓 𝑥 = −2𝑥2 + 4𝑥 + 5

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12. Which description explains how the graph of 𝑓 𝑥 = 𝑥2 + 6𝑥 + 5 is related to the

graph of 𝑔 𝑥 = 𝑥2 + 6𝑥 − 2.

A. 𝑓 𝑥 is vertically stretched to make 𝑔 𝑥 .

B. 𝑓 𝑥 is translated down seven (7) units to make 𝑔 𝑥

C. 𝑓 𝑥 is translated seven (7) units left to make 𝑔 𝑥

D. 𝑓 𝑥 is compressed vertically to make 𝑔 𝑥

13. What are the solutions to the quadratic equation 3𝑥2 + 7𝑥 + 11 = 5𝑥 + 7?

A. 𝑥 =

−2 ± 2𝑖√11

3 C. 𝑥 =

−1 ± 𝑖√11

3

B. 𝑥 =−1 ± 2𝑖√11

3 D. 𝑥 =

±𝑖√11

3

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1. The shape of a highway tunnel can be modeled by 𝒇 𝒙 = −𝟏

𝟏𝟎 𝒙 − 𝟐𝟎 𝟐 + 𝟒𝟎, where 𝒙 is

the horizontal distance in feet from the left end of the tunnel and 𝒇 𝒙 is the height in feet above the highway. Sketch the graph of the tunnel. Then determine the maximum height of the tunnel & the width of the tunnel at the highway.

a. What shape will the tunnel have? How do you know?

b. Why is the graph only in the first quadrant? c. Graph the functions.

Vertex: _____________ (What does the vertex represent in this situation?

Find the point at the left end of the support (where 𝑥 = 0).

Since 𝑓 0 = _______, the point ______ represents the left end.

Use symmetry to find the point at the right end of the tunnel. Since the left end is 20 feet to the left of the vertex, the right end will be 20 feet to the right of the vertex.

The point __________ represents the right end.

Find two other points on the support: 10, ________ 𝑎𝑛𝑑 30, _________ d. Determine the maximum height of the support.

The maximum of the function is ________.

So, the maximum height of the tunnel is _______________ . e. Determine the width of the tunnel at the level of the highway.

The distance from the left and to the right end is _________. So the width is _________________at the level of the highway.

3.6 Modeling with Quadratic Equations

QUESTION How can you model real world problems using quadratic equations & functions?

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2. The function 𝒉 𝒕 = −𝟏𝟔𝒕𝟐 + 𝟔𝟒𝒕 that gives the height 𝒉 in feet of a golf ball 𝒕 seconds after it is hit. The ball has a height of 48 feet after 1 second. Use the symmetry of the function’s graph to determine the other time at which the ball will have a height of 48 feet.

a. Write the function in vertex form. b. Use symmetry to sketch a graph of

the function and solve the problem.

The vertex is _________

The point _______0 is on the graph This point is 2 units to the left of the vertex. Based on symmetry, there is a point 2 units to the right of the vertex with the same 𝑦-coordinate at

______, 0

The point 1, 48 is on the graph. Based on symmetry, at the point _______, 48 is also on the graph. So, the ball will have a height of 48 feet after 1 second and again after ____________.

REFLECT: 2c. How can you check your answer to the problem? 2d. What is the maximum height that the ball reaches? How do you know? 2e. If you know the coordinates of a point to the left of the vertex for the graph of a quadratic

function, how can you use symmetry to find the coordinates of another point on the graph.

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3. The cross-sectional shape of the archway of a bridge is modeled by the function

𝒇 𝒙 = −𝟎. 𝟓𝒙𝟐 + 𝟐𝒙 where 𝒇 𝒙 is the height in meters of a point on the arch and x is the distance in meters from the left end of the arch’s base. How wide is the arch at its base? Will a wagon that is 2 meters wide and 1.75 meters tall fit under the arch.

a. Write the function intercept form. b. Identify the 𝑥-intercepts and the vertex.

The 𝑥-intercepts are _________ & __________.

The 𝑥-coordinate of the vertex is +

= ______

Find the 𝑦-coordinates of the vertex.

The vertex is ___________

c. Graph the function using the 𝑥-intercepts & the vertex.

d. Use the graph to solve the problem. The width of the arch at its base is

__________ meters. Sketch the wagon on your graph. Will the wagon fit under the arch?

Explain.

REFLECT: 3e. What do the x-intercepts represent in this situation? 3f. Explain how you would use the graph to find the width of the arch at its base. 3g. Explain how you modeled the shape of the wagon on the graph.