# Unit 4 QUADRATIC FUNCTIONS AND FACTORING Unit Essential

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Unit 4 QUADRATIC FUNCTIONS AND FACTORING!!!

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

Lesson 4. 1 GRAPHING QUADRATIC FUNCTIONS IN STANDARD FORM

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

Standard Form of a Quadratic Function:

To graph:

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. 2 GRAPHING 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

Intercept Form

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

Bell Work:

Examples:

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:

Example:

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:

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:

Example:

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

Bell Work:

Lesson 4. 3 SOLVING 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:

Finding Zeros

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. 4 SOLVING 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)

Homework: Page 263 #’s 3 – 31 odds

Bell Work:

Solving Quadratic Equations:

Example: A 10 x 12 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:

Examples:

Examples:

Examples:

Examples:

Examples:

Bonus for Unit 4 Test 1:

Bell Work:

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:

Bell Work:

Lesson 4. 5 SOLVING 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:

Classwork/Homework: Pages 270 – 271 #’s 22 – 42 evens This assignment will be collected at the beginning of class tomorrow!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

Bell Work:

Lesson 4. 6 COMPLEX NUMBERS

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

Complex Numbers

Standard Form of Complex Numbers:

Add/Subtract/Multiply/Divide:

Complex Conjugates:

Solving Quadratic Equations with Complex Solutions:

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

Bell Work:

Pop Quiz:

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:

Lesson 4. 7 COMPLETING 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:

Lesson 4. 8 QUADRATIC FORMULA

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

Quadratic 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:

Bell Work:

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:

Lesson 4. 9 QUADRATIC 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:

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