Topic: Geometric definitions



Name: _________________________________ Date: _________________

Lesson 9.1: Solving Quadratic Equations Algebraically (Day 1) Algebra I

EX 1: Consider the quadratic function [pic]. It’s graph is shown below.

a) What are the zeros of the function? Write their x – values and circle them on the graph.

b) Verify that the positive zero is correct by showing that y = 0.

c) Factor the expression [pic]. How do these factors compare to the zeros.

d) Based on part c, determine where the zeros of [pic]are algebraically. Verify using a table on your graphing calculator.

EX 2: Use the zero product law to find all solutions to each of the following equations.

a) [pic]

b) [pic]

c) [pic]

Steps to solve a quadratic Equation:

1.

2.

3.

4.

EX 3: Solve each of the quadratic equations below.

a) x2 – 11x – 12 = 0

b) x2 + 2x = 24

c) 13z + 30 = –z2

d) a2 = 6 – a

e) 2m2 – 10m + 8 = 0

f) [pic]

g) x(x – 6) = 7

h) (x + 1)(x – 2) = 0

EX 4: The roots of [pic] can be found by factoring as

1) {6, –2} (3) {3–4}

2) {–6, 2} (4) {–3, 4}

Name: _________________________________ Date: _________________

Homework 9.1: Solving Quadratic Equations Algebraically (Day 1) Algebra I

Solve each of the quadratic equations below.

1. [pic]

2. [pic]

3. [pic]

4. [pic]

5. [pic]

6. [pic]

7. [pic]

8. [pic]

9. The equation [pic] has roots of

1) {–7, 1½ } (3) {–7, 3}

2) {3, 7} (4) {½, –3}

Name: _________________________________ Date: _________________

Lesson 9.1: Solving Quadratic Equations Algebraically (Day 2) Algebra I

EX 1: Solve the following quadratic equations:

a) x2 – 16 = 0

b) 7x + x2 = 0

c) 5g2 – 20 = 0

d) 15y = 5y2

e) [pic]

f) [pic]

g) [pic]

EX 2: Solve the following proportions:

a) [pic]

b) [pic]

c) [pic]

EX 3: If x2 y3 = 5184, y = 4 and x is an integer, then find all possible values of x.

EX 4: Find all the zeros of the quadratic function [pic] algebraically. Then verify your answer by using your calculator to sketch a graph of the parabola using the window indicated on the axes below. Clearly mark the zeros on the graph.

Name: _________________________________ Date: _________________

Homework 9.1: Solving Quadratic Equations Algebraically (Day 2) Algebra I

Solve each of the quadratic equations below.

1. [pic]

2. [pic]

3. [pic]

4. 2x2 – 20 = 52

5. [pic]

6. [pic]

7. [pic]

8. If [pic], y = 4 and x is an integer, then find all possible values of x.

Find the roots of each of the following quadratic equations by factoring:

9. [pic]

10. [pic]

11. [pic]

12. [pic]

13. [pic]

14. [pic]

15. [pic]

16. [pic]

Name: _________________________________ Date: _________________

Lesson 9.1: Solving Quadratic Equations Algebraically (Day 3) Algebra I

EX 1: Solve the following quadratic equations:

a) [pic]

b) [pic]

c) [pic]

d) [pic]

e) [pic]

f) [pic]

EX 2: James graphed the quadratic [pic] using tables on his calculator and found the graph shown below. He can tell from his graph and table that x = –2 is one of the two zeros. But, he couldn’t tell what the other was because it did not fall on an integer location.

a) Write down an equation that would allow you to solve for the zeros of this function.

b) How does knowing that x = –2 is a zero help you to factor the trinomial [pic]?

c) Solve the equation in part a using factoring to find the other zero of this function.

EX 3: Consider the cubic function [pic]

a) Find the zeros of this function algebraically by factoring.

b) Use your calculator to sketch a graph of this function. Circle the zeros on the graph.

EX 4: Consider the quadratic [pic]

a) Find the zeros of this function algebraically using factoring.

b) Write the quadratic function in vertex form and identify the coordinates of its turning point.

c) What is true about the x – coordinate of the turning point compared to the zeros you found in part a?

d) Without using your calculator sketch a graph of this quadratic on the axis below.

[pic]

EX 5: A quadratic function can be written in factored form as [pic]. Which of the following would be the x – coordinate of its turning point?

1) x = 6 (3) x = 5

2) x = 2 (4) x = 4

Name: _________________________________ Date: _________________

Homework 9.1: Solving Quadratic Equations Algebraically (Day 3) Algebra I

1. Solve each of the following:

a) [pic]

b) [pic]

2. Solve each of the following by factoring:

a) [pic]

b) [pic]

c) [pic]

d) [pic]

e) [pic]

3. Consider the polynomial [pic]

a) Find the three zeros of this function algebraically by factoring.

b) Use your calculator to sketch a graph of the cubic on the axis below. Mark your answers from part a.

4. Consider the quadratic function [pic]

a) Find its zeros algebraically.

b) Using your calculator, sketch a graph of the function on the axes given.

c) Find the zeros of [pic] algebraically.

d) Using your calculator, sketch a graph of this function on the same axes. How does the second graph compare to the first that your drew?

Name: _________________________________ Date: _________________

Lesson 9.2: Solving Quadratic Equations Using Inverse Operations Algebra I

EX 1: Solve each of the following equations for all values of x. Write your answers in simplest radical form.

a) [pic]

b) [pic]

c) [pic]

EX 2: Solve each of the following equations for all values of x by using inverse operations. In each case, your final answer will be rational numbers.

a) [pic]

b) [pic]

c) [pic]

d) [pic]

e) [pic]

f) [pic]

EX 3: Solve each of the following quadratic equations by using inverse operations. Express all of your answers in simplest radical form.

a) [pic]

b) [pic]

EX 4: Francis graphs the parabola [pic] on the grid below. He believes that the quadratic has zeros of [pic] and [pic].

a) Find the zeros of this function in simplest radical form and explain why Francis must be incorrect.

b) Francis was incorrect based on part a, but not too far off. How can you tell how good his estimate was?

EX 5: Find the zeros of the function [pic] in simplest radical form. Then, express them in terms of a decimal rounded to the nearest hundredth.

Name: _________________________________ Date: _________________

Homework 9.2: Solving Quadratic Equations Using Inverse Operations Algebra I

1. Solve each of the following quadratic equations by using inverse operations. Express all of your answers in simplest radical form.

a) [pic]

b) [pic]

c) [pic]

d) [pic]

e) [pic]

f) [pic]

g) [pic]

2. Which of the following is the solution set to the equation [pic]

1) {0, 12} (3) {–2, 16}

2) {–4, 8} (4) {–2, 14}

3. Solve each of the following quadratic equations by using inverse operations. Express each of your answers in simplest radical form.

a) [pic]

b) [pic]

c) [pic]

d) [pic]

e) [pic]

4. The height, h, of an object above the ground in feet, can be modeled as a function of time, t, in seconds using the equation: [pic], for [pic]

a) Find the time, in seconds, when the object reaches the ground, h = 0

b) Find all the times when the object is at a height of 300 feet. Round your answer to the nearest tenth of a second.

5. Which of the following choices represent the zeros of the function [pic]?

1) [pic] (3) [pic]

2) [pic] (4) [pic]

Name: _________________________________ Date: _________________

Lesson 9.3: Solving Quadratic Equations by Completing the Square Algebra I

EX 1: For the quadratic function [pic]

a) Find the zeros in simplest radical form.

b) Find the zeros to the nearest tenth.

c) State the vertex of this parabola.

EX 2:Consider the quadratic [pic].

a) Find the zeros by factoring.

b) Find the zeros by Completing the Square.

EX 3: Consider the quadratic [pic].

a) Find the zeros by Completing the Square.

b) Try to factor [pic].

EX 4: Consider the function [pic].

a) Write the function in vertex form and state the coordinates of the turning point.

b) Using your answer from part a find the zeros of the function.

c) Determine the function’s y – intercept.

d) Draw a rough sketch of the function on the axes below. Label all quantities in parts a through c.

EX 5: The quadratic function pictured has a leading coefficient equal to 1. Answer the following questions based on your previous work.

a) Write the equation of this quadratic equation in vertex form.

b) Write the equation of this quadratic in factored form.

c) How could you establish that these were equivalent functions?

Name: _________________________________ Date: _________________

Homework 9.3: Solving Quadratic Equations by Completing the Square Algebra I

1. Solve the equation [pic]two ways:

a) By Factoring

b) By Completing the Square

2. Solve the equation [pic]two ways:

a) By Factoring

b) By Completing the Square

3. Find the solutions to the following equation in simplest radical form using the technique of Completing the Square.

[pic]

4. Using the method of Completing the Square, find the zeros of the following function to the nearest hundredth.

[pic]

5. Consider the quadratic function shown below whose leading coefficient is equal to 1.

[pic]

a) Write the equation of this quadratic in [pic] form.

b) Find the zeros of this quadratic in simplest radical form.

c) Write the equation of this quadratic function in standard form, [pic]

6. Find the zeros of the function [pic]in simplest radical form. Based on this answer, how do you know that you could not use factoring to find these zeros?

7. Consider the quadratic function [pic] written in standard form.

a) Write the quadratic function in its vertex form and state the coordinates of its turning point.

b) Find the zeros of the function algebraically by setting your equation from part a equal to zero.

c) State the range of this quadratic function. Justify your answer by creating a sketch of the function from what you found in parts a and b.

d) This quadratic can also be written

in equivalent factored form as [pic]. What graphical features are easy to determine when the function is written in this form?

Name: _________________________________ Date: _________________

Lesson 9.4: The Quadratic Formula Algebra I

EX 1: Solve the equation [pic] by Completing the Square. What type of numbers do your answers represent?

EX 2: For the previous quadratic [pic] identify the following:

a) The values of a, b, and c in the quadratic formula.

b) Carefully substitute these values in the quadratic formula and simplify your expression. Compare your result to EX 1.

EX 3: Consider the quadratic equation [pic]

a) Find the solutions to this equation by factoring.

b) Find the solutions to this equation using the Quadratic Formula.

EX 4: For each of the following quadratic equations, find the solutions using the Quadratic Formula and express your answers in simplest radical form.

a) [pic]

b) [pic]

EX 5: A projectile is fired vertically from the top of a 60 foot tall building. It’s height in feet above the ground after t seconds is given by the formula

[pic]

Using your calculator, sketch a graph of the projectile’s height, h, using the indicated window. At what time, t, does the ball hit the ground? Solve by using the quadratic formula to the nearest tenth of a second.

[pic]

Name: _________________________________ Date: _________________

Homework 9.4: The Quadratic Formula Algebra I

1. Solve the equation [pic] two ways.

a) By Factoring

b) By the Quadratic Formula

2. Solve the equation [pic] two ways. Express your answers both times in simplest radical form.

a) By Completing the Square

b) By the Quadratic Formula

3. Solve the equation [pic] two ways.

a) By Factoring

b) By the Quadratic Formula

4. If the quadratic formula is used to solve the equation [pic], the correct roots are

1) [pic] (3) [pic]

2) [pic] (4) [pic]

5. The quadratic function [pic] is shown below.

a) Find the zeros of this function in simplest radical form by using the quadratic equation.

b) Write this function in vertex form by completing the square. Based on this, what are the coordinates of its turning point? Verify on your graph.

6. The flow of oil in a pipe, in gallons per hour, can be modeled using the function [pic]

a) Using your calculator, graph the function on the axes provided.

[pic]

b) Using the quadratic formula, find, to the nearest tenth of an hour, the time when the flow stops (is zero). Show your work.

c) Use the process of completing the square to write [pic] in its vertex form. Then, identify the peak flow and the time at which it happens.

Name: _________________________________ Date: _________________

Lesson 9.5: Mixed Methods for Solving Quadratic Equations Algebra I

You now have several methods to solve quadratic equations. These techniques include factoring, completing the square, and the quadratic formula. In each method it is important that the equation is equal to zero. If it isn’t then some minor manipulation might be needed.

EX 1: Solve each of the following quadratic equations using the required method.

a) Solve by factoring: [pic]

b) Solve by Completing the Square: [pic]

c) Solve by the Quadratic Formula. Express answers to the nearest tenth: [pic]

d) Solve by the Quadratic Formula. Express answers in simplest radical form: [pic]

EX 2: The quadratic [pic] is shown graphed below.

a) What are the values of h and k?

b) What happens when you try to solve for the zeros of f given the values of h and k from part a? Why can’t you find solutions?

c) How does what you found in part b show up in the graph to the right?

EX 3: Which of the following three quadratic functions has no real zeros (there may be more than one). Determine by using the quadratic formula. Verify each answer by graphing in the standard viewing window.

a) [pic]

b) [pic]

c) [pic]

Name: _________________________________ Date: _________________

Homework 9.5: Mixed Methods for Solving Quadratic Equations Algebra I

1. Solve each of the following equations using the method listed. Place your final answers in the form asked for.

a) Solve by factoring (Answers are exact)

[pic]

b) Solve by factoring (Answers are exact)

[pic]

c) Solve by Completing the Square (Round answers to the nearest tenth)

[pic]

d) Solve using the Quadratic Formula (Express answers in Simplest Radical form)

[pic]

2. Which of the following represents the zeros of the function [pic]?

1) {–1, 2} (3) [pic]

2) [pic] (4) {–1, 4}

3. The percent of popcorn kernels that will pop, P, is modeled using the equation:

[pic],

where T is the temperature in degrees Fahrenheit.

Determine the two temperatures, to the nearest degree Fahrenheit, that result in zero percent of the kernels popping. Use the Quadratic Formula. Show work that justifies your answer. The numbers here will be messy. Use your calculator to help you and carefully write out your work.

4. Find the zeros of the function [pic] by Completing the Square. Express answers in Simplest Radical form

5. Explain how you can tell that the quadratic function [pic]has no real zeros without graphing the function.

6. Use the Quadratic Formula to determine which of the two functions below would have real zeros and which would not, then verify by graphing on your calculator using a standard viewing window.

a) [pic]

b) [pic]

Name: _________________________________ Date: _________________

Lesson 9.6: Modeling with Quadratic Equations (Day 1) Algebra I

EX 1: A rock is thrown vertically from the ground. It’s path can be modeled by the equation [pic], where h is the height in meters and t is the time in seconds.

a) How many seconds will it take for the rock to reach its maximum height? Round your answer to the nearest tenth of a second.

b) What is the maximum height the rock will reach? Round your answer to the nearest tenth of a meter.

EX 2: An athlete dives from the 3 meter springboard. Her height, y, at horizontal distance, x, can be approximated by the function [pic].

a) What is the maximum height she will reach, rounded to the nearest tenth of a meter?

b) How far will she be from the springboard when she reaches her maximum height? Round to the nearest tenth of a meter.

EX 3: The base of a rectangle measures 7 cm more than its height. If the area of the rectangle is 30 sq. cm, find the measure of its base and the measure of its height.

EX 4: The altitude of a triangle measures 5 cm less than its base. The area of the triangle is 42 cm2. Find the lengths of the base and the altitude.

EX 5: Alvin is building a rectangular pasture. He needs the length to be five feet more than three times the width, and the area must be 288 square feet. Find the length and width of his pasture.

EX 6: Deborah built a box by cutting 3-inch squares from the corners of a rectangular sheet of cardboard, as shown in the accompanying diagram, and then folded the sides up. The volume of the box is 150 cubic inches, and the longer side of the box is 5 inches more than the shorter side. Find the number of inches in the shorter side of the original sheet of cardboard.

Name: _________________________________ Date: _________________

Homework 9.6: Modeling with Quadratic Equations (Day 1) Algebra I

1. The price of a stock, A(x), over a 12-month period decreased and then increased according to the equation [pic], where x is the number of months. What was the lowest price of this stock?

2. The area of a rectangle is 40. If the length is 6 more than the width, find the dimensions of the rectangle.

3. A garden is in the shape of a square. The length of one side of the garden is increased by 3 feet and the adjacent side is increased by 2 feet. The garden now has an area of 72 square feet. What is the measure of a side of the original square garden.

4. In the accompanying diagram, ABCD is a rectangle and DEFG is a square. The area of ABCD is 80, CG = 8, and AE = 6. Find the length of the side of square DEFG.

[pic]

Name: _________________________________ Date: _________________

Lesson 9.6: Modeling with Quadratic Equations (Day 2) Algebra I

EX 1: The Perris Pandas baseball team has a new promotional activity to encourage fans to attend games: launching free T–shirts! They can launch a T–shirt in the air with an initial velocity of 91 feet per second from 5 ½ feet off the ground (the height of the team mascot).

A T–shirt’s height can be modeled with the quadratic function [pic], where t is the time in seconds and h(t) is the height of the launched T–shirt in feet. Determine how long it will take for a T–shirt to land back on the ground after being launched.

EX 2: A friend of yours is working on a project that involves the path of a kicked soccer ball. After numerous test kicks, she modeled the general path of the ball using a quadratic function [pic], where h is the horizontal distance the ball traveled in meters, and [pic]is the vertical distance the ball traveled in meters.

a) Determine the horizontal distance the ball traveled before it hit the ground.

b) Determine the horizontal distance the ball traveled when it reached a height of 6 meters. Round your answers to the nearest hundredth.

EX 3: Solve the system of two quadratic equations: [pic]

EX 4: Amy tossed a ball in the air in such a way that the path of the ball was modeled by the equation [pic]. In the equation y represents the height of the ball in feet and x is the time in seconds. Determine the maximum height of the ball and how long it took the ball to reach that height.

EX 5: Jason jumped off of a cliff into the ocean in Acapulco while vacationing with some friends. His height as a function of time could be modeled by the function [pic]where t is the time in seconds and h is the height in feet.

a) How long did it take for Jason to reach his maximum height?

b) What was the highest point that Jason reached?

c) Jason hit the water after how many seconds?

EX 6: You are trying to dunk a basketball. You need to jump 2.5 ft. in the air to dunk the ball. The height that your feet are above the ground is given by the function [pic]. What is the maximum height your feet will be above the ground? Will you be able to dunk the basketball?

EX 7: If a toy rocket is launched vertically upward from ground level with an initial velocity of 128 feet per second, then its height h after t seconds is given by the equation [pic] (if air resistance is neglected).

a) How long will it take for the rocket to return to the ground?

b) After how many seconds will the rocket be 112 feet above the ground?

c) How long will it take the rocket to hit its maximum height?

d) What is the maximum height?

EX 8: A ship drops anchor in a harbor. The anchor is 49 ft. above the surface of the water when it is released. Use the vertical motion formula [pic]to answer the following questions.

a) What is the value of s, the starting height?

b) What is the value of h when the anchor hits the water?

c) The starting velocity is zero. After how many seconds will the anchor hit the water?

EX 9: A bus company charges $2 per ticket but wants to raise the price. The daily revenue is modeled by [pic], where x is the number of $0.15 price increases and R(x) is the revenue in dollars. What should the price of the tickets be if the bus company wants to collect daily revenue of $30,000?

Name: _________________________________ Date: _________________

Homework 9.6: Modeling with Quadratic Equations (Day 2) Algebra I

1. Solve the system of equations: [pic]

2. The area of a rectangle is 560 cm2. The length is 3 more than twice the width. Find the dimensions of the rectangle.

3. A diver is standing on a platform 24 ft. above the pool. He jumps form the platform with an initial upward velocity of 8 ft/s. Use the formula [pic], where h is his height above the water, t is the time, v is his starting upward velocity, and s is his starting height. How long will it take for him to hit the water?

4. You and a friend are hiking in the mountains. You want to climb to a ledge that is 20 ft. above you. The height of the grappling hook you throw is given by the function [pic]. What is the maximum height of the grappling hook? Can you throw it high enough to reach the ledge?

5. An architect is designing a museum entranceway in the shape of a parabolic arch represented by the equation [pic], where x is a measurement in feet and y is the height of the arch in feet. Determine the maximum height of the arch.

-----------------------

9.1 Day 1 Notes

Zero Product Law

If two or more quantities have a product of zero, then at least one of them must be ____________. In symbolic form:

If ________________ then either ____________ or ___________(or both are zero)

9.1 Day 1 HW

9.1 Day 2 Notes

9.1 Day 2 HW

9.1 Day 3 Notes

9.1 Day 3 HW

9.2 Notes

9.2 HW

9.3 Notes

9.3 HW

9.4 Notes

The Quadratic Formula

For the quadratic equation [pic], the zeros can be found by

9.4 HW

9.5 Notes

9.5 HW

9.6 Day 1 Notes

Keystrokes for finding the vertex:

2nd trace (calc)

3: minimum OR

4: maximum

x + 5

x

3 in

3 in

3 in

3 in

3 in

3 in

3 in

3 in

9.6 Day 1 HW

9.6 Day 2 Notes

9.6 Day 2 Notes

9.6 Day 2 HW

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