Algebra 1 Unit 9: Quadratics Intercept Formmathwithmills.weebly.com/uploads/8/6/0/2/86029964/key...1 Packet Score: Algebra 1 Unit 9: Quadratics – Intercept Form Note & Homework Packet

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Algebra 1

Unit 9: Quadratics – Intercept Form Note & Homework Packet

Date Topic/Assignment HW Page

Due Date Score (For Teacher Use Only)

9-A Graphing Parabolas in Intercept Form

9-B Solve Quadratic Equations Graphically - Intercept Form

9-C Find Zeros/Solve by Factoring

9-D Complete the Square – Standard Form to Vertex Form

9-E Solve Quadratic Equations by Completing the Square

9-F Solve Quadratic Equations by the Quadratic Formula

9-G Graph Quadratic Functions in Standard Form

9-H Graph Quadratic Functions in Standard Form Part 2

9-I Applications of Quadratic Functions

9-J Comparing Different Forms of Quadratic Equations & Transformations

Quizzes will be “pop”…2 or 3 per unit. You may use this packet to complete the quizzes.

This packet will be turned in on the day of the test for 100 points. Whenever you’re absent, you can get these notes filled out from a classmate or at my website www.skookummath.weebly.com. During the unit, I’ll check off homework each day to keep track of who is doing their homework, but your homework assignments won’t be scored and entered into IC until this packet is collected and graded at the end of the unit.

Name:

Period:

For Teacher Use

Packet Score:

2

Warm-Up Date:

Graph each parabola. State the domain and range.

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9-A: Graphing Parabolas in Intercept Form

Review Vertex Form

𝑓(𝑥) = 𝑎(𝑥 − ℎ)2 + 𝑘

Label each as either Intercept or Vertex Form.

𝑓(𝑥) = (𝑥 + 9)2 − 4 𝑓(𝑥) = − 13(𝑥 − 4)(𝑥 + 2) 𝑓(𝑥) = 1

2(𝑥 − 2)2

𝑓(𝑥) = −2(𝑥 − 3)(𝑥 − 5) 𝑓(𝑥) = (𝑥 + 9)2 − 4 𝑓(𝑥) = −3𝑥2 + 5

How is Intercept Form different from Vertex Form?

Intercept Form

𝑓(𝑥) = ±𝑎(𝑥 − 𝑟)(𝑥 − 𝑠)

Steps for graphing in Intercept Form:

Go back to the six functions above, state either the intercepts or the vertex for each.

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1. Example

Graph each. State the intercepts, vertex, AOS, domain and range.

a. 𝑓(𝑥) = (𝑥 + 3)(𝑥 − 1) b. 𝑓(𝑥) = (𝑥 + 1)(𝑥 − 3)

2. Guided Practice

a. 𝑓(𝑥) = (𝑥 − 1)(𝑥 − 5) b. 𝑓(𝑥) = (𝑥 − 2)(𝑥 − 4)

3. Example

a. 𝑓(𝑥) = −(𝑥 − 2)(𝑥 − 4) b. 𝑓(𝑥) = −2(𝑥 + 3)(𝑥 + 1)

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4. Guided Practice

a. 𝑓(𝑥) = −(𝑥 + 2)(𝑥 − 2) b. 𝑓(𝑥) = −2(𝑥 − 3)(𝑥 − 5)

5. Example 6. Guided Practice

𝑦 = 12(𝑥 − 2)(𝑥 − 6) 𝑦 = 1

2(𝑥 + 1)(𝑥 − 3)

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7. Example

Write the equation in INTERCEPT FORM for the parabola shown.

a. b.

8. Guided Practice

a. b.

c.

Which of these can you also write in VERTEX FORM? Do so on those that you can.

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9-B: Solve Quadratic Equations Graphically – Intercept Form

The solutions are the x values of the intersections of f(x) and g(x) when f(x) = g(x).

1. Example

Solve by graphing.

a. (𝑥 − 1)(𝑥 − 3) = 5 b. (𝑥 + 2)(𝑥 + 6) = −3

f(x) = f(x) =

g(x) = g(x) =

2. Guided Practice

a. (𝑥 − 1)(𝑥 + 3) = −3 b. (𝑥 − 3)(𝑥 − 5) = 3

f(x) = f(x) =

g(x) = g(x) =

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3. Example

a. 2(𝑥 − 1)(𝑥 − 3) = −2 b. 12 (𝑥 + 1)(𝑥 + 5) = −1.5

f(x) = f(x) =

g(x) = g(x) =

4. Guided Practice

a. 2(𝑥 + 1)(𝑥 + 3) = 6 b. 12 (𝑥 − 1)(𝑥 − 5) = 2.5

f(x) = f(x) =

g(x) = g(x) =

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5. Example

a. −(𝑥 − 2)(𝑥 − 6) = 3 b. −12(𝑥 + 2)(𝑥 + 6) = 1.5

f(x) = f(x) =

g(x) = g(x) =

6. Guided Practice

a. −2(𝑥 − 2)(𝑥 − 4) = −6 b. −(𝑥 − 1)(𝑥 + 3) = −5

f(x) = f(x) =

g(x) = g(x) =

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9-C: Find Zeros/Solve by Factoring

What are zeros?

Number of Solutions of a Quadratic Equations

1. Example

Find the roots/zeros/intercepts/solutions.

a. 𝑓(𝑥) = (𝑥 + 5)(𝑥 − 3) b. 𝑦 = −(𝑥 − 6)(𝑥 − 2) c. 𝑓(𝑥) = 12 (𝑥 − 1)(𝑥 + 7)

2. Guided Practice

a. 𝑓(𝑥) = (𝑥 − 10)(𝑥 + 1) b. 𝑦 = −(𝑥 + 5)(𝑥 − 2) c. 𝑓(𝑥) = 12 (𝑥 + 4)(𝑥 − 9)

3. Example

Convert from Standard Form to Intercept Form by factoring.

a. 𝑓(𝑥) = 𝑥2 + 7𝑥 + 10 b. 𝑓(𝑥) = 𝑥2 − 6𝑥 + 5 c. 𝑓(𝑥) = 𝑥2 + 3𝑥 − 18

Now name the roots/intercepts/zeros/solutions of each.

a. b. c.

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4. Guided Practice

Convert to Intercept Form by factoring.

a. 𝑓(𝑥) = 𝑥2 + 9𝑥 + 18 b. 𝑓(𝑥) = 𝑥2 − 9𝑥 + 14 c. 𝑓(𝑥) = 𝑥2 + 3𝑥 − 10

Now name the roots/intercepts/zeros/solutions of each.

a. b. c.

5. Example

Find the zeros.

a. 𝑓(𝑥) = 2𝑥2 + 7𝑥 + 3 b. 𝑓(𝑥) = 3𝑥2 − 16𝑥 + 5 c. 𝑓(𝑥) = 2𝑥2 + 𝑥 − 6

6. Guided Practice

a. 𝑓(𝑥) = 3𝑥2 + 5𝑥 + 2 b. 𝑓(𝑥) = 2𝑥2 − 7𝑥 + 3 c. 𝑓(𝑥) = 3𝑥2 − 𝑥 − 2

7. Example

Solve each equation by factoring.

a. 𝑥2 + 13𝑥 + 36 = 0 b. 𝑥2 − 7𝑥 + 12 = 0 c. 2𝑥2 + 9𝑥 − 5 = 0

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8. Guided Practice

Solve each equation by factoring.

a. 𝑥2 − 8𝑥 + 15 = 0 b. 𝑥2 − 2𝑥 − 15 = 0 c. 2𝑥2 − 5𝑥 + 3 = 0

9. Example

Solve by factoring.

a. 3𝑥2 + 13𝑥 = −4 b. 2𝑥2 − 7𝑥 = −3

10. Guided Practice

a. 𝑥2 − 7𝑥 = −12 b. 𝑥2 − 2𝑥 = 15

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9-D: Complete the Square – Standard Form to Vertex Form

Vertex Form Intercept Form Standard Form

𝑦 = 𝑎(𝑥 − ℎ)2 + 𝑘 𝑦 = 𝑎(𝑥 − 𝑟)(𝑥 − 𝑠) 𝑦 = 𝑎𝑥2 + 𝑏𝑥 + 𝑐

How do we convert Vertex or Intercept form into Standard form?

1. Example

Convert each quadratic equation to Standard Form.

a. 𝑓(𝑥) = (𝑥 + 2)2 − 3 b. 𝑦 = (𝑥 − 3)2 + 1 c. 𝑓(𝑥) = (𝑥 + 2)(𝑥 − 4)

2. Guided Practice

a. 𝑓(𝑥) = (𝑥 − 4)2 − 2 b. 𝑓(𝑥) = (𝑥 + 1)2 − 5 c. 𝑦 = (𝑥 + 5)(𝑥 + 2)

How do we convert Standard Form to Intercept Form?

Now we’re going to convert Standard Form to Vertex Form. First, we need to review Perfect Square Trinomials.

Perfect Square Trinomials Factored Form

𝑥2 + 8𝑥 + 16

𝑥2 − 10𝑥 + 25

4𝑥2 + 12𝑥 + 9

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3. Example

Find the value of c that makes the expression a perfect square trinomial. Then write the expression as a binomial squared (factored form).

a. 𝑥2 + 6𝑥 + 𝑐 b. 𝑥2 + 8𝑥 + 𝑐 c. 𝑥2 − 10𝑥 + 𝑐

4. Guided Practice

a. 𝑥2 + 2𝑥 + 𝑐 b. 𝑥2 − 8𝑥 + 𝑐 c. 𝑥2 + 6𝑥 + 𝑐

Why is c always positive in a perfect square trinomial?

5. Example

Convert from Standard Form to Vertex Form by Completing the Square.

a. 𝑦 = 𝑥2 + 6𝑥 + 3 b. 𝑦 = 𝑥2 + 4𝑥 − 5 c. 𝑓(𝑥) = 𝑥2 − 8𝑥 − 1

5. Guided Practice

a. 𝑦 = 𝑥2 + 10𝑥 + 3 b. 𝑓(𝑥) = 𝑥2 − 6𝑥 + 2 c. 𝑦 = 𝑥2 − 2𝑥 − 7

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What if a > 1?

6. Example

a. 𝑓(𝑥) = 2𝑥2 + 8𝑥 + 5 b. 𝑦 = 3𝑥2 − 12𝑥 + 3

7. Guided Practice

a. 𝑦 = 2𝑥2 − 10𝑥 − 7 b. 𝑓(𝑥) = 3𝑥2 + 15𝑥 − 2

What is the extra step we need to remember when a > 1?

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9-E: Solve Quadratic Equations by Completing the Square

1. Example

Solve the quadratic equation by completing the square.

a. 𝑥2 + 14𝑥 − 51 = 0 b. 𝑥2 − 12𝑥 + 10 = 0

2. Guided Practice

a. 𝑥2 + 6𝑥 + 8 = 0 b. 𝑥2 − 2𝑥 − 4 = 0

Remember what to do when a > 1?

3. Example

a. 5𝑥2 = 60 − 20𝑥 b. 6𝑥2 − 48 = −12

4. Guided Practice

a. 8𝑥2 + 16𝑥 = 42 b. 3𝑥2 = −4 + 8𝑥

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9-F: Solve Quadratic Equations by the Quadratic Formula

What are all of the ways that we can solve quadratic equations?

Here’s another…

1. Example

Solve using the quadratic formula.

a. b.

2. Guided Practice

a. b.

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3. Example

a. b.

4. Guided Practice

a. b.

c. d.

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9-G: Graph Quadratic Functions in Standard Form

Name all different graphing forms of quadratic functions.

Here’s another way to graph parabolas…

Standard Form Step 1: Find the Vertex and plot it on the grid.

Axis of Symmetry

Step 2: Use a to draw the legs

Example

1. 𝑦 = 𝑥2 − 4𝑥 + 5 2. 𝑦 = 𝑥2 + 4𝑥 + 1

Guided Practice

3. 𝑦 = 𝑥2 − 6𝑥 + 8 4. 𝑦 = 𝑥2 − 6𝑥 + 11

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Example Guided Practice

5. 𝑦 = −𝑥2 + 8𝑥 − 16 6. 𝑦 = −𝑥2 + 4𝑥 − 5

Example

Write the equation in standard form for the parabola shown.

7. 8.

Guided Practice

9.

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9-H: Graph Quadratic Functions in Standard Form Part 2

Example

1. 𝑦 = 2𝑥2 + 4𝑥 − 4 2. 𝑦 = 12 𝑥2 + 2𝑥

Guided Practice

3. 𝑦 = 2𝑥2 − 8𝑥 + 3 4. 𝑦 = 12 𝑥2 − 4𝑥 + 9

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Example Guided Practice

5. 𝑓(𝑥) = −2𝑥2 − 4𝑥 + 2 6. 𝑓(𝑥) = −12 𝑥

2 − 4𝑥 − 6

Example

Write the equation for the quadratic function shown.

7. 8.

Guided Practice

9.

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9-I: Applications of Quadratic Functions

1. Below is a graph for the P profits for various selling prices s of a skateboard.

2.

3.

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4.

5.

Find the value of x.

6. 7.

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9-J: Comparing Different Forms of Quadratic Equations & Transformations

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Write the equation for the parabola described. A graph is provided if needed.

3. Translate the graph 𝑓(𝑥) = 𝑥2 up three

units and two units right. What is the new

equation?

5. Translate the graph 𝑦 = (𝑥 − 1)(𝑥 − 5) nine units up and three units left. What is the new equation?

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9-A Assignment

Graph each. State the x-intercepts and vertex for each.

1. 𝑦 = (𝑥 − 1)(𝑥 − 3) 2. 𝑦 = (𝑥 + 3)(𝑥 + 1) 3. 𝑦 = (𝑥 − 1)(𝑥 − 3)

4. 𝑦 = −(𝑥 + 1)(𝑥 − 3) 5. 𝑦 = −2(𝑥 − 1)(𝑥 − 3) 6. 𝑦 = 2(𝑥 − 1)(𝑥 − 3)

Write the equation in Intercept Form AND Vertex Form for the parabola shown.

7. 8.

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9-B Assignment

Solve each quadratic equation graphically.

1. −(𝑥 + 3)(𝑥 + 5) = −3 2. (𝑥 − 2)(𝑥 − 4) = 3

3. −2(𝑥 + 2)(𝑥 + 4) = −6 4. 12 (𝑥 + 3)(𝑥 + 7) = 2.5

5. (𝑥 − 3)(𝑥 − 5) = 3 6. 2(𝑥 + 2)(𝑥 + 4) = 6

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9-C Assignment

Find the roots of each quadratic function.

1. 𝑦 = (𝑥 − 4)(𝑥 = 7) 2. 𝑓(𝑥) = 𝑥2 + 8𝑥 + 7 3. 𝑓(𝑥) = 2𝑥2 + 13𝑥 + 15

Convert to Intercept Form by factoring.

4. 𝑓(𝑥) = 𝑥2 − 6𝑥 − 7 5. 𝑓(𝑥) = 𝑥2 + 3𝑥 − 10 6. 𝑦 = 3𝑥2 − 11𝑥 + 6

Now name the roots for problems 4 – 6.

4. 5. 6.

Solve each equation by factoring.

7. 𝑥2 + 3𝑥 − 18 = 0 8. 𝑥2 − 7𝑥 + 12 = 0 9. 2𝑥2 + 𝑥 − 6 = 0

10. 3𝑥2 − 11𝑥 = −6 11. 2𝑥2 + 9𝑥 = 5 12. 𝑥2 − 2𝑥 = 15

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9-D Assignment

Convert each quadratic equation to Standard Form.

1. 𝑓(𝑥) = (𝑥 + 4)2 − 1 2. 𝑦 = (𝑥 − 5)2 + 2 3. 𝑓(𝑥) = (𝑥 + 3)(𝑥 − 5)

Find the value of c that makes the expression a perfect square trinomial. Then write the expression as a binomial squared (factored form).

4. 𝑥2 + 12𝑥 + 𝑐 5. 𝑥2 + 2𝑥 + 𝑐 6. 𝑥2 − 14𝑥 + 𝑐

Convert from Standard Form to Vertex Form by Completing the Square.

7. 𝑦 = 𝑥2 + 4𝑥 + 7 8. 𝑦 = 𝑥2 + 12𝑥 − 2 9. 𝑓(𝑥) = 𝑥2 − 14𝑥 − 3

10. 𝑦 = 𝑥2 − 12𝑥 + 3 11. 𝑦 = 2𝑥2 + 8𝑥 − 3 12. 𝑓(𝑥) = 3𝑥2 − 15𝑥 + 5

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9-E Assignment

Solve each equation by completing the square.

1. 𝑥2 + 14𝑥 − 15 = 0 2. 𝑥2 = 18𝑥 + 40 3. 2𝑥2 = −6 + 8𝑥

4. 𝑥2 + 2𝑥 − 3 = 0 5. 𝑥2 − 2𝑥 − 7 = 0 6. 𝑥2 + 8𝑥 + 12 = 0

7. 𝑥2 − 2𝑥 − 48 = 0 8. 𝑥2 + 12𝑥 + 21 = 0 9. 𝑥2 − 8𝑥 − 48 = 0

10. 7𝑥2 − 14𝑥 − 56 = 0 11. 2𝑥2 + 12𝑥 = −10 12. 𝑥2 − 6𝑥 = 7

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9-F Assignment

Solve by using the quadratic formula.

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9-G Assignment

Graph. Label the vertex and axis of symmetry.

1. 𝑓(𝑥) = 𝑥2 − 8𝑥 + 16 2. 𝑓(𝑥) = 𝑥2 − 2𝑥 − 3 3. 𝑓(𝑥) = −𝑥2 − 4𝑥

4. 𝑓(𝑥) = −𝑥2 + 6𝑥 − 4 5. 𝑓(𝑥) = 𝑥2 + 4𝑥 + 4 6. 𝑓(𝑥) = −𝑥2 + 4𝑥 + 2

Write the equation for the parabola shown.

7. 8.

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9-H Assignment

Graph. Label the vertex and axis of symmetry.

1. 𝑦 = 2𝑥2 − 4𝑥 − 4 2. 𝑦 = 12 𝑥

2 + 4𝑥 + 9

3. 𝑦 = −12 𝑥

2 + 2𝑥 + 4 4. 𝑦 = −2𝑥2 + 8𝑥 − 4

Write the equation for the parabola shown.

7. 8.

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9-I Assignment

1. Given the graph of a ball being thrown from a catapult, answer the following questions. The vertical axis represents the height of the ball, the horizontal axis represents the distance from the catapult.

2. You throw a basketball whose path can be modeled by the graph of 𝑦 = −16𝑥2 + 19𝑥 + 6 where x is the time (in seconds) and y is the height (in feet) of the basketball.

3.

a. How far away from the catapult does the ball land? What part of the graph tells you this?

b. What is the maximum height the ball goes? What part of the graph tells you this?

c. Is the value of “a” positive or negative? How do you know?

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4.

5.

Find the value of x.

6. 7.

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9-J Assignment

3. Translate the graph 𝑓(𝑥) = 𝑥2 up 5 units and 3 units right. What is the new equation?

5. Translate the graph 𝑦 = (𝑥 + 1)2 − 2 down 7 units and 4 units left. What is the new equation?