Rationalize the Denominator: Get the root the denom. Multiply by more roots to cancel. w/ and w/

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1 Name Unit 2 Day 1 Simplifying Square Roots Properties: 1. = Examples: 2. = Rationalize the Denominator: Get the root the denom. Multiply by more roots to cancel. w/ and w/ Conjugate: Opposite Examples: Rationalize the following. A. B. C. D.

2 Simplifying roots worksheet - Simplify the following

3 Name Unit 2 Day 2 Solving Quadratic Functions by Square Rooting Examples: Solve the following by square rooting both sides. A) ( 3) = 7 B) (2 3) 16 = 0 C) ( + 5) + 4 = 0 Square root solving worksheet - Solve the following by square rooting both sides. 1. = = = ( 5) = ( + 2) = ( 4) = ( + 3) 18 = 0 8. ( 6) 4 = ( + 6) + 60 = ( + 4) = ( 2) + 15 = ( 2) 45 = 0

4 Steps: Name Unit 2 Day 3 Solving Quadratic Functions by Factoring Examples: A) = 0 B) 9 8 = Non-Quadratic Functions: Highest variable will equal (can be real or imaginary roots!) Example: C) = 0 D) ( 5) 16 = 0 (hint: grouping)

5 Solve by Factoring Worksheet Solve each of the following by factoring = = = = = = = = 8 9. ( 5) 9 = (3 + 4) 16 = 0

6 Name Unit 2 Day 4 Solving Quadratic Functions by Completing the Square To solve an equation in looks like ( ± #) = # + + = 0 form, you can change it so it Steps for Completing the Square: 1. Move the constant to the right side 2. Put a + on each side 3. Take ½ of b 4. Square the answer to step 4 5. Add the answer to step 5 in both + 6. Factor and simplify 7. Solve SHORTCUT: Example: Solve by Completing the Square D) 2 = 15 E) CTS worksheet Solve the following by completing the square = = 0 6 3= = = = = = = = = 0

7 Name Unit 2 Day 5 Solving Quadratic Functions by the Quadratic Formula and the Discriminant Discriminant: Determines what type of root you ll get = ( ) 4 1. If > 0, perfect square = 2 rational Non-perfect square = 2 irrat. 2. If < 0, If = 0, 5. Examples: Determine the type of root only! C) = 0 D) = 0 E) = 0 F) + = 0 Quadratic Formula: Used to solve equations that can t be factored = ± ( ) 4 2 Recall: 1 = Examples: A) = 0 B) 5 = 6 3 Exact vs. Approximate:

8 Quadratic Formula and Discriminant Worksheet Solve the following by using the quadratic formula = = = 2 4. = = = = = = 0 Determine the value of the discriminant and the type of root(s). DO NOT SOLVE!!! = = = 0 Disc = Disc = Disc = Type: 2 Real 1 Real 2 Complex Type: 2 Real 1 Real 2 Complex Type: 2 Real 1 Real 2 Complex Circle one Circle one Circle one = = = 7 Disc = Disc = Disc = Type: 2 Real 1 Real 2 Complex Type: 2 Real 1 Real 2 Complex Type: 2 Real 1 Real 2 Complex Circle one Circle one Circle one

9 Recall: Parabola: Name Unit 2 Day 6 Solving Quadratic Functions by Graphing Zeros/Roots (solve): Steps for graphing and solving: 1. Go to Y = (top left button) and plug in equation use the,,, button for x 2. Press GRAPH (top right button) 3. Press 2 nd TRACE and choose option 2 ZERO 4. Calculator will ask for Left Bound, Right Bound and Guess you will need to use the arrows to move your cursor to the left and right of the zero, each time you get to the left or right, press enter, then press enter once more for the guess 5. You may have to Zoom In or Out in order to see the whole graph go to ZOOM (top middle button) and choose in or out and hit enter (sometimes you have to enter twice) Examples: Graph and find all the zeros. A) ( ) = B) ( ) = C) = 2( 2) + 1 D) = 4

10 Solve by Graphing Worksheet Given the graph, estimate the solutions (zeros/roots) Graph the following quadratic equations and estimate the solutions (zeros/roots). 7. = 3 8. = = = =2 =

11 Name Unit 2 Day 7 CTS with Parabolas Graphing Form (Vertex/Completed Square): = ( h) Vertex = (h, ) (always opposite signs) Axis: = h Graph the following: Use 2 nd Table to get some of the points. Find the vertex and axis of symmetry. A) = B) 2 = C) = ( + 2) Vertex: Vertex: Vertex: Axis: Axis: Axis: Use CTS to put an equation of a parabola in graphing form. Examples: Change to graphing form and find the vertex and axis of symmetry for each. D) ( ) = E) = 2 3 Equation in graphing form: Equation in graphing form: Vertex: Axis: Vertex: Axis:

12 CTS with Parabolas Worksheet Use CTS to find the graphing form of the quadratic function. State the vertex and axis of symmetry of the function. 1. = = = Equation in graphing form: Equation in graphing form: Equation in graphing form: Vertex: Vertex: Vertex: Axis of Symmetry: Axis of Symmetry: Axis of Symmetry: 4. = = = Equation in graphing form: Equation in graphing form: Equation in graphing form: Vertex: Vertex: Vertex: Axis of Symmetry: Axis of Symmetry: Axis of Symmetry: 7. = = Equation in graphing form: Vertex: Axis of Symmetry: Equation in graphing form: Vertex: Axis of Symmetry:

13 Name Unit 2 Day 8 Standard Form of Parabolas Standard Form (General): = + + Vertex = (, ) Axis: = Examples: Find the vertex and axis of symmetry of each. A) ( ) = B) ( ) = Vertex: Vertex: Axis: Axis: Standard Form Parabolas Worksheet Use the vertex formula to find the vertex of the parabola. Then find the axis of symmetry. 1. = = 6 3. = 10 Vertex: Vertex: Vertex: Axis of Symmetry: Axis of Symmetry: Axis of Symmetry: 4. = 4 5. = = Vertex: Vertex: Vertex: Axis of Symmetry: Axis of Symmetry: Axis of Symmetry: 7. ( ) = ( ) = ( ) = Vertex: Vertex: Vertex: Axis of Symmetry: Axis of Symmetry: Axis of Symmetry:

14 Unit 2 Day 9 Max/Min and Story Problems Max/Min ( ): the highest (max) or lowest (min) point of a parabola Max/Min are determined by a. If = +# = + + or If = # = ( ℎ) Examples: Determine if the parabola has a max/min. Use = or a calculator to determine the max/min value. A) = B) ( ) = Story Problem Steps: 1. Read the problem. Decide what is and what you are looking 2. Choose your 3. Write your 4. Solve your equation(s) and find what you were looking for. Maximum/Minimum: Max of Min will be between your zeros. Solve Mathematically: find midpoint of zeros, plug in for x and solve for y. Solve Graphically: You will have to change your window for most. 5. Check your answer. Does it make sense????? Example: A ball is thrown vertically upward with an initial velocity of 48 f/s. If the ball started its flight from 8 ft, then its height h at time t can be determined by ℎ( ) = Determine: A) The time it takes to reach the ground. B) The time it takes to reach its max height. C) The max height.

15 Max/Min and Story Problems Worksheet Determine if the parabola has a max/min. Use 1. ( )= = = or a calculator to determine the max/min value ( )= Circle one: Max Min Circle one: Max Min Circle one: Max Min Value = Value = Value = 4. The height ℎ( ), in meters, above the ground of a certain soccer ball kick t seconds after the ball is kicked can be approximated by ℎ( ) = Determine the time for which the ball is in the air. Round to the nearest tenth of a second. 5. The temperature ( ), in degrees Fahrenheit, during the day can be modeled by the equation ( ) = , where t is the number of hours after 6:00 A.M. A) After how many hours is the temperature a maximum? Round to the nearest tenth of an hour. B) What is the maximum temperature? Round to the nearest degree. 6. If the initial velocity of a projectile is 128 feet per second, then its height h, in feet, is a function of time t, in seconds, given by the equation ℎ( ) = A) Find the time t when the projectile achieves its maximum height. B) Find the maximum height of the projectile. C) Find the time t when the projectile hits the ground.

WHAT ARE THE PARTS OF A QUADRATIC?

WHAT ARE THE PARTS OF A QUADRATIC? 4.1 Introduction to Quadratics and their Graphs Standard Form of a Quadratic: y ax bx c or f x ax bx c. ex. y x. Every function/graph in the Quadratic family originates from the parent function: While