Unit 4QUADRATIC FUNCTIONS AND FACTORING!!!

Unit Essential Question: What are the different ways to graph a quadratic function and to solve quadratic equations?

Lesson 4.1GRAPHING QUADRATIC FUNCTIONS IN STANDARD FORM

Lesson Essential Question: How do we graph a quadratic function in standard form?

Standard Form of a Quadratic Function:

This is the general form for a parabola!

To graph: 1) Find the vertex. The x-value of the vertex = , then use this x-value to find the y-value.

2) Find the y-intercept. Plug 0 in for x, and solve for y.

3) Use the axis of symmetry to reflect the y-intercept to find a third point.

Pay attention to the value of a. If a > 0, then the graph opens up.

If a < 0, then the graph opens down.

What if the vertex is on the y-axis?

Then find another point to use instead of the y-intercept. Plug another number in for x, and solve for y. Then reflect this point across the axis of symmetry!!!

Minimum/Maximum Values A parabola can have either a minimum or maximum value depending upon the value in from the x² term. If a is positive, then the function will have a minimum. If a is negative, then it will have a maximum.

The minimum or maximum value of a given quadratic function will be equal to the y-value in the vertex!

Example: Latricia is selling sticky buns at the Bloomsburg Fair to raise money for the soccer team. She charges $5 per sticky bun, and sells about 800 a day. She found that for everytime she raises the price by $0.25, she sells 20 less sticky buns. Write a quadratic function and find out how she can maximize her revenue.

Homework: Pages 240 – 242 #’s 21 – 39 odds, 53 – 59 all (59 only part A)

Bell Work: 1) Papa John’s in Bloomsburg sells about 500 pizzas in a day when they charge $12. For every time they decrease the price by $0.50, they sell 60 more pizzas. Find the price per pizza and number of pizzas sold each day that will maximize their profit.

Lesson 4.2GRAPHING QUADRATIC FUNCTIONS IN VERTEX FORM AND INTERCEPT FORM

Lesson Essential Question: What is vertex form and intercept form for a quadratic function and how can we use it?

Vertex Form

Where: The vertex is (h,k). The axis of symmetry is x = h. The graph opens up if a > 0 and down if a < 0.

Intercept Form

Where: p and q are the x-intercepts. The axis of symmetry is halfway between p and q. The graph opens up if a > 0 and down if a < 0.

Homework: Pages 249 – 251 #’s 3 – 21 odds, 22, 25 – 41 odds, 51 – 54, 56

Bell Work: Graph the following quadratic functions using the given form:

1)

2)

3)

Examples: Sketch the graphs of the following functions:

Example: Find the equation for a quadratic function if it has a vertex of (2,-4) and passes through the point (4,12).

Example: Find the equation of the quadratic function if is has x-intercepts at -2 and 4, and also passes through the point P(6,-32).

Example: The path that a football travels on the opening kickoff is given as where x is the horizontal distance the ball traveled in yards, and y is height in yards.

A) How far was the football kicked?

B) What was the maximum height of the football?

Example: The path of a frog jumping can be modeled by the equation where x is the horizontal distance in feet and y is the vertical distance in feet from its starting point.

A) How far can the frog jump?

B) How high can the frog jump?

Bell Work: Megan sells lemonade to save money for a trip to the Bahamas. She sells lemonade for $1 a cup, and sells about 100 cups a week. When she raises the price by $0.10, she sells 4 less cups each week. Write a quadratic function to model Megan’s income. How can Megan maximize her revenue???

Examples: Sketch the graphs of the following functions:

Examples: A quadratic function has an x-intercept at -2 and a vertex of (2,64). Write the equation for this function in standard form.

A quadratic function has x-intercepts at -3 and -9, and it passes through the point (2,165). Write the equation for this function in intercept form.

A quadratic function has a vertex V(4,8) and passes through the point (8,4). Write the equation for this function in vertex form.

Example: Abby jumps from point A to point B 12 feet away. Her maximum height during the jump was 6 feet. Write the equation for a quadratic function that describes her jump.

The equation for how Abby the prairie dog jumps is given as where x is the horizontal distance traveled in inches, and y is the vertical distance traveled in inches. Find the maximum height that Abby the prairie dog can jump, and how far she can jump.

Example: Abby was walking underneath an arch, and found that the equation that models the arch is where x is the horizontal distance in feet and y is the vertical distance in feet. If Abby is standing directly in the middle of the arch, how high would she have to jump to reach the arch?

Why would the height to top of the arch be same from ¼ of the way across the arch as ¾? Prove that it is the same.

How wide is the arch?

Small Quiz Tomorrow! Quadratic Functions: Standard Form Vertex Form Intercept Form Minimum/Maximum Word Problems You need to have the forms memorized!!!

Bell Work: Factor:

1)

2)

3)

4)

Lesson 4.3SOLVING QUADRATIC EQUATIONS BY FACTORING WHEN A=1.

Lesson Essential Question: How do we factor trinomials and binomials, and how will this help us solve quadratic equations?

Factoring:

Solving Quadratic Equations: Three Easy Steps: 1) Make sure the equation is set equal to zero.

2) Factor the equation as much as possible.

3) Set each factor equal to zero and solve.

Your answers to each equation are the roots of the equation.

Examples: GCF Solving

Regular Solving

Homework: Pages 255 – 256 #’s 3 – 40 ALL

YOU NEED TO KNOW HOW TO FACTOR!!!

Bell Work: Solve each quadratic equation by factoring:

1)

2)

3)

4)

Finding Zeros A regular quadratic function has zeros if it crosses the x-axis. The zeros are x-intercepts!!!

How would we find the zeros of this function?

To find zeros/x-intercepts: Set the function equal to zero, then factor and solve!!!

Special Cases: If the function has only one zero, then that x-intercept is the vertex!!!

What if the function is unfactorable? What does that tell us about the function?

Homework: Pages 256 – 257 #’s 42 – 63, 65 – 67, and 69 – 71

Bell Work: Page 258 # 72

Lesson 4.4SOLVING QUADRATIC EQUATIONS BY FACTORING WHEN A≠1.

Lesson Essential Question: How do we factor trinomials differently when the a value does not equation 1???

Master Product Method (Factoring)

When factoring a trinomial in the form , where a ≠ 1, we must use a different form of factoring.

Examples:

Homework: Page 263 #’s 3 – 31 odds

Bell Work: Factor each expression:

Solving Quadratic Equations: Solve:

Example: A 10x12 inch picture is to be framed. The thickness of the frame on all sides will be the same. If the area of just the frame is 75 square inches, find the thickness of the frame.

Homework: Pages 263 – 265 #’s 33 – 61 odds and 62– 67

Quiz Tomorrow! Factoring when a does and does not equal 1 Finding zeros Word Problems

Bell Work: 1) Best Buy charges $1000 for their 60 inch HD TV’s and they sell 50 per week. When they decrease the price by $25, they sell 2 more each week. What price should they charge for their HDTV’s to maximize their weekly revenue?

2) A suspension bridge has supports that rise up 90 feet from the bridge, and there is 400 feet between each support. The lowest point of the suspension cable connecting the two supports is 10 feet above the road. Use the figure on the board to answer the following:

A) Write an equation in vertex form for the parabolic shape of the suspension cable.

B) What is the total length of steel cable needed to connect the bridge to the large suspension cable connecting the two supports assuming they are equally spaced?

Examples: 3) Gage is quitting school to join the circus and become a human cannonball. In his first trial, he is shot out of a cannon that is 15 feet high. He travels a total distance of 175 feet. Because he was shot at a 45 degree angle, we can model his parabolic shape using the equation Find an equation in standard form that models Gage’s flight. What is the maximum height he reaches?

4) J-Yo has been spending time learning how to carve wood. She carves a wooden door that models a parabolic arch. The base of the door is 6 feet wide, and the height of the door is 9 feet. Write an equation is intercept form that models J-Yo’s door.

Examples: 5) Eliana has recently purchased a shop that sells scarfs. The building measures 50 by 80 feet. The parking lot wraps around two sides of the building with uniform width on both sides being 40 feet. The scarf business really picked up, and now Eliana has to increase her parking lot. By how much should she increase her parking lot if she wants to increase her parking lot area by 14,575 square feet?

6) Gina throws a softball straight up in the air. The flight of the softball can be measured by where t is the time in seconds, v is the initial velocity of the softball, and s(t) is the height of the softball. Find the following:

A) What is the initial velocity of the softball if it was in the air for 12 seconds?

B) What was the maximum height of the softball?

C) Explain why the softball would have the same height at 4 and 8 seconds.

Examples: 7) The path at which Hayden throws a bowling ball can be modeled by the function where x is the horizontal and y is the vertical distance measured in yards. How far can she throw the bowling ball? How high?

8) A picture measuring 36 by 10 inches has a mat placed around that is three times as wide on the left and right as it is on the top and bottom. Find the width of the mat placed around the picture if the mat has an area of 504 square inches.

Examples: Sketch the graphs of the following quadratic functions:

Examples: Find the zeros for each quadratic function:

Examples: Solve each quadratic equation:

Bonus for Unit 4 Test 1: For what values of c can the following expression be factored if we know that c is positive?

You must have all values to receive credit!

Bell Work: Simplify:

1)

2)

3)

Review of Simplifying Radicals!!! Let’s review properties of radicals and how to simplify them!

Rationalizing a Denominator We are not allowed to leave a radical in the denominator of an expression, so we “rationalize” it by multiplying the numerator and denominator by that same radical to cancel it out!

Conjugates If a radical is in the denominator with another constant, then we must multiply the numerator and denominator by the conjugate!

Homework: Pages 269 - 270 #’s 3 – 20 all

Bell Work: Simplify each radical expression:

Bell Work: Solve the following equations by factoring: 1)

2)

3)

Lesson 4.5SOLVING QUADRATIC EQUATIONS WITH SQUARE ROOTS

Lesson Essential Question: How do square roots help us solve quadratic equations?

Solving by Square Rooting…

Homework: Pages 270 – 271 #’s 21 – 43 odds

Bell Work: Simplify the following radicals:

1) 2) 3)

Solve each equation:

4) 5) 6)

Classwork/Homework: Pages 270 – 271 #’s 22 – 42 evens

This assignment will be collected at the beginning of class tomorrow!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

Bell Work: Solve the equation

Solve the equation

Solve the equation

Lesson 4.6COMPLEX NUMBERS

Lesson Essential Question: How do complex numbers give us imaginary solutions for quadratic equations?

Complex Numbers A complex number is when you take the square root of a negative number. To simplify complex numbers, we use the imaginary unit (i). This allows us to rewrite a negative square root.

So,

Standard Form of Complex Numbers:

𝑎+𝑏𝑖 , h𝑤 𝑒𝑟𝑒𝑎𝑎𝑛𝑑𝑏𝑎𝑟𝑒𝑟𝑒𝑎𝑙𝑛𝑢𝑚𝑏𝑒𝑟𝑠 .

Add/Subtract/Multiply/Divide:

Complex Conjugates:

Solving Quadratic Equations with Complex Solutions:

Homework: Pages 279 – 280 #’s 3 – 33 odds

Bell Work: Solve the following equations: 1) 2)

Simplify each complex expression: 3) 4)

Pop Quiz: Solve 1 – 3 by square rooting:

1) 2) 3)

For 4 – 6, simplify the expression as much as possible: 4) 5) 6)

Quiz Monday We will be having a quiz Monday on:

Solving Quadratic Equations by Square Rooting (Real and Complex Solutions)

Operations with Complex Numbers

Review the word problems from Lesson 4.5

Bell Work: Factor:

Lesson 4.7COMPLETING THE SQUARE!!!

Lesson Essential Question: How do we complete the square and why do we use it with quadratic functions?

Completing the Square: This is when you add a value to both sides of an equation to make one side a perfect square trinomial that will factor into a binomial squared.

How do we do it???

Simple…

Homework: Pages 288 – 289 #’s 3 – 49 odds

Bell Work:Find the zeros for each quadratic function:

Lesson 4.8QUADRATIC FORMULA

Lesson Essential Question: What is the quadratic formula and how does it help us find zeros for quadratic functions?

Quadratic Formula: When a quadratic function is in standard form, we can find the zeros for the function by using this formula:

What is so special about this??? If used properly, it works 100% of the time for any quadratic function, whether it has two zeros, one zero, or two imaginary zeros!!!

What is the Discriminant? What does it tell us?

Homework: Page 296 #’s 3 – 47 every other odd

Bell Work: 1) Solve by factoring:

2) Solve by completing the square:

3) Solve by using the quadratic formula:

Bell Work: Solve by completing the square:

1)

2)

Solve by using the quadratic formula:

3)

4)

Class Work: Page 298 #’s 68 – 72

Homework: Pages 288 - 289 #’s 24, 26, 32, 44, 48

Page 296 #’s 4, 7, 19, 43, 45

Bell Work: Sketch the graphs of the following quadratic functions:

1)

2)

3)

Lesson 4.9QUADRATIC INEQUALITIES

Lesson Essential Question: How do we graph and solve quadratic inequalities?

Graphing:

Solving:

Homework: Pages 304 – 307 #’s 3 – 13 odds, 47 – 57 odds, 72, 73, 75

Bell Work: Page 305 # 69

Classwork/Homework: Pages 304 – 307 #’s 6, 10, 12, 18, 20, 48, 60, 64, 66, 76

This will be collected!!!

Bell Work: Grab a small piece of paper and be ready for a pop quiz!

We will go over the homework and answer questions first, so have your work out and be ready to ask questions.

Pop Quiz: 1) Graph:

2) Solve:

3) A truck with a height of 14 feet and width of 8 feet is passing under a bridge. The bottom of the bridge models a parabolic arch . Will the truck make it under the bridge? Explain. Use the sketch on the board to help. (Show your work!)

Upcoming: Unit 4 Test Part 2 Solving Quadratic Functions (Factoring, Square Rooting, Completing the Square, Quadratic Formula)

Solving and Graphing Quadratic Inequalities

Word Problems

Review Assignment: Pages 318 – 322 #’s 6 – 20 evens, 21 – 44

Extra Practice if you need it: Page 323 #’s 1 – 27