Math 2 Warm Up Simplify each expression: 1. 2. 3. 4.

5. 6. 7. 8. 9. 2x2 4x(3x 5)

3x(x 2) (x 2)(x + 5) (-4x + 3)(2x 7) 3x(2x 7) + 6x(4x + 5) x(1 x) (1 2x2) (5x + 3) 2 -7x(5x2 4x)

(4x 5) (-2x2 + 3x 9) Unit 5: Quadratic Functions Lesson 1 - Properties of Quadratics Objective: To find the vertex & axis of symmetry of a quadratic function then graph the function. quadratic function is a function that can be written in

the standard form: y = ax2 + bx + c, where a 0. Examples: y = 5x2 y = -2x2 + 3xy = x2 x 3

Properties of Quadratics parabola the graph of a quadratic equation. It is in the form of a U which opens either upward or downward. vertex the maximum or minimum point of a parabola.

Properties of Quadratics axis of symmetry the line passing through the vertex about which the parabola is symmetric (the same on both sides).

Properties of Quadratics Find the coordinates of the vertex, the equation for the axis of symmetry of each parabola. Find the coordinates points corresponding to P and Q. Graphing a Quadratic Equation y = ax2 + bx + c

1) Direction of the parabola? If a is positive, then the graph opens up. If a is negative, then the graph opens down. Graphing a Quadratic Equation

y = ax2 + bx + c 2) Find the vertex and axis of symmetry. The x-coordinate of the vertex is (also the equation for the axis of symmetry). Substitute the value of x into the quadratic equation and solve for the y-coordinate. Write vertex as an ordered pair (x , y).

Graphing a Quadratic Equation y = ax2 + bx + c 3) Table of Values. Choose two values for x that are one side of the vertex (either right or left). Substitute those values into the quadratic equation to find y

values. Graph the two points. Graph the reflection of the two points on the other side of the parabola (same y-values and same distance away from the axis of symmetry). Find the vertex and axis of symmetry of the following quadratic equation. Then, make a table of values and

graph the parabola. y = 2x2 + 4x + 3 Direction: _____ Vertex: ______ Axis: _______

Find the vertex and axis of symmetry of the following quadratic equation. Then, make a table of values and graph the parabola. y = x2 + 3x 1 Direction: _____

Vertex: ______ Axis: _______ Find the vertex and axis of symmetry of the following quadratic equation. Then, make a table of values and graph the parabola.

y = x2 + 2x + 5 Direction: _____ Vertex: ______ Axis: _______

Find the vertex and axis of symmetry of the following quadratic equation. Then, make a table of values and graph the parabola. y = 3x2 4 Direction: _____ Vertex: ______

Axis: _______ Apply! The number of widgets the Woodget Company sells can be modeled by the equation -5p2 + 10p + 100, where p is the selling price of a widget. What price for a widget

will maximize the companys revenue? What is the maximum revenue? End of Day 1 P 244 #10-21, 28-32

Math 2 Unit 5 Lesson 2 Unit 5:"Quadratic Functions" Title: Translating Quadratic Functions

Objective: To use the vertex form of a quadratic function. y = a(x h)2 + k where (h, k) is the vertex. Example 1: Graphing from Vertex Form

y = 2(x 1) 2 + 2 Direction: _____ Vertex: ______ Axis: _______

Example 2: Graphing from Vertex Form y = (x + 3) 2 1 Direction: _____ Vertex: ______ Axis: _______

Example 3: Graphing from Vertex Form y = (x 3) 2 2 Direction: _____ Vertex: ______

Axis: _______ Example 4: Write quadratic equation in vertex form. Example 5: Write quadratic equation in vertex form. Example 6: Converting Standard Form to Vertex Form.

Step 1: Find the Vertex y= x = -b = x2 - 4x + 6 2a

y = Step 2: Substitute into Vertex Form:

Example 7: Converting Standard Form to Vertex Form. Step 1: Find the Vertex y= x = -b = 6x2 10

2a y = Step 2: Substitute into Vertex Form:

Example 8: Converting Vertex Form to Standard Form. Step 1: Square the Binomial. y = 2(x 1) 2 + 2 Step 2: Simplify to

Example 9: Converting Vertex Form to Standard Form. Step 1: Square the Binomial. y = (x 3) 2 2 Step 2: Simplify to

Honors Math 2 Assignment: In the Algebra 2 textbook: pp. 251-253 #3, 6, 9, 17-20, 25, 27, 31, 34, 52, 54 End of Day 2 P 251

#3, 6, 17-19, 27, 31, 34, 43, 45, 52, 54 Factoring Quadratic Expressions Objective: To find common factors and binomial factors of quadratic expressions. factor if two or more polynomials are multiplied together, then each polynomial is a factor of the product.

(2x + 7)(3x 5) = 6x2 + 11x 35 FACTORS PRODUCT (2x 5)(3x + 7) = 6x2 x 35 FACTORS PRODUCT

factoring a polynomial reverses the multiplication! Finding Greatest Common Factor greatest common factor (GCF) the greatest of the common factors of two or more monomials. + 20x 12

24x Finding Binomial Factors + 14x 40 Finding Binomial Factors

+ 12x 32 Finding Binomial Factors 11x 24 Finding Binomial Factors 17x 72

Finding Binomial Factors 14x 32 Finding Binomial Factors 3x 28

Finding Binomial Factors 11x 12 Finding Binomial Factors 31x 35 Finding Binomial Factors

32x 35 Finding Binomial Factors 16x 12 Finding Binomial Factors* 35x 45

Finding Binomial Factors* 42x Finding Binomial Factors* 90x

Factoring Special Expressions* 49 9 192

36 Math 2 Assignment pp. 259-260 #7-21 odd, 35-45 odd, 48 End of Day 3

Factor. 35x 45 36 16x 12

Solving Quadratics Equations: Factoring and Square Roots Objective: To solve quadratic equations by factoring and by finding the square root. Solve by Factoring 7x 18 = 0

Solve by Factoring 20x 7 = 0 Solve by Factoring 5 = 6x

Solve by Factoring = Solve by Factoring* 16x = 10x

Solve by Factoring* Solve Using Square Roots Quadratic equations in the form can be solved by finding square roots. = 243

Solve Using Square Roots 200 = 0 Solve Using Square Roots* 25 = 0

Math 2 Assignment p. 266 #1-19 End of Day 4 Unit 4, Lesson 5:

Complex Numbers Objective: To define imaginary and complex numbers and to perform operations on complex numbers Introducing Imaginary Numbers

Find the solutions to the following equation: Introducing Imaginary Numbers Find the solutions to this equation:

Imaginary numbers offer solutions to this problem! i1 = i i2 = -1 i3 = -i i4 = 1

Simplifying Complex Numbers i 21 Adding/Subtracting Complex Numbers

(8 + 3i) (2 + 4i) 7 (3 + 2i) (4 - 6i) + (4 + 3i)

Multiplying Complex Numbers (12i) (7i) (3 - 7i)(2 - 4i)

(6 - 5i)(4 3i) (4 - 9i)(4 + 3i) Now we can finally find ALL solutions to

this equation! Complex Solutions 3x + 48 = 0

-5x - 150 = 0 8x + 2 = 0 9x + 54 = 0

Math 2 Assignment P. 274-275 # 1-17 odd, 29-39 odd, 41-46

End of Day 5 Completing the Square 1.) Move the constant to opposite side of the equation as the terms with variables in them. 2.) Take half of the coefficient with the x-term

and square it 3.) Add the number found in step 2 to both sides of the equation. 4.) Factor side with variables into a perfect square. 5.) Square root both sides (put + in front of square root on side with only constant)

6.) Solve for x. Solve the following, using completing the square 1.) x2 3x 28 = 0 2.) x2 3x = 4

3.) x2 + 6x + 9 = 0 If a 1, then divide all the term by a. 1.) 2x2 + 6x = -6 2.) 3x2 12x + 7 = 0

3.) 5x2 + 20x + -50 Math 2 Assignment P 281-283 # 15 25, 37, 39

End of Day 6 Solve using Completing the square x2 + 4x = 21 x2 8x 33 = 0 4x2 + 4x = 3

Solving Quadratic Equations: Quadratic Formula Objective: To solve quadratic equations using the Quadratic Formula. 5x = 0

Not every quadratic equation can be solved by factoring or by taking the square root! Solve using Quadratic Formula 5x 8 = 0 Solve using Quadratic Formula

23x 40 = 0 Solve using Quadratic Formula + =

Solve using Quadratic Formula* = Solve using Quadratic Formula

+= Solve using Quadratic Formula + =

Solve using Quadratic Formula = -6x 7 Math 2 Assignment P 289 #1, 2, 22-30

End of day 7 Solving Quadratic Equations: Graphing Objective: To solve quadratic equations and systems that contain a quadratic equation by graphing.

When the graph of a function intersects the x-axis, the y-value of the function is 0. Therefore, the solutions of the quadratic equation ax2 + bx + c = 0 are the x-intercepts of the graph. Also known as the zeros of the function or the roots of the function.

Solve Quadratic Equations by Graphing Solution Solution Solve Quadratic Equations by Graphing

Step 1: Quadratic equation must equal 0! ax2 + bx + c =0 Step 2: Press [Y=]. Enter the quadratic equation in Y1. Enter 0 in Y2. Press [Graph]. MAKE SURE BOTH X-INTERCEPTS ARE ON SCREEN! ZOOM IF NEEDED! Step 3: Find the intersection of ax2 + bx + c and 0.

Press [2nd] [Trace]. Select [5: Intersection]. Press [Enter] 2 times for 1st and 2nd curve. Move cursor to one of the x-intercepts then press [Enter] for the 3rd time. Repeat Step 3 for the second x-intercept! Solve by Graphing

6x 4 = 0 Solve by Graphing 4x 7 = 0 Solve by Graphing 5x = 20

Solve by Graphing = 19x Solve by Graphing = -2x + 7

Solve by Graphing 2x 6 = 0 Solve by Graphing + 16 = 0 P 266

#20-31, 54-56 End of Day 8 Solving Systems of Equations Solve a System with a Quadratic Equation

x = + Solve a System with a Quadratic Equation x

= + Solve a System with a Quadratic Equation =

Solve a System with a Quadratic Equation x = Solve a System with Quadratic Equations

= + Solve a System with Quadratic Equations

= = +

Math 2 Assignment Worksheet Solve each quadratic equation or system by graphing. End of Day 9

Finding a Quadratic Model 1) Turn on plot: Press [2nd] [Y=], [ENTER], Highlight On, Press [ENTER] 2) Turn on diagnostic: Press [2nd] [0] (for catalog),

Scroll down to find DiagonsticOn. Press [ENTER] to select. Press [ENTER] again to activate. Finding a Quadratic Model 3) Enter data values: Press [STAT], [ENTER] (for EDIT),

Enter x-values (independent) in L1 Enter y-values (dependent) in L2 Clear Lists (if needed): Press [STAT], [ENTER] (for EDIT), Highlight L1 or L2 (at top) Press [CLEAR], [ENTER].

Finding a Quadratic Model 4) Graph scatter plot: Press [ZOOM], 9 (zoomstat) 5) Find quadratic equation to fit data: Press [STAT], over to CALC, For Quadratic Model - Press 5: QuadReg Press [ENTER] 4 times, then Calculate.

Write quadratic equation using the values of a, b, and c rounded to the nearest thousandths if needed. Write down the R2 value! Find a quadratic equation to model the values in the table.

X -1 2 3 Y -8

1 8

is a measure of the goodness-of-fit of a regression model. the value of R2 is between 0 and 1 (0 R2 1) R2 = 1 means all the data points fit the model (lie exactly on the graph with no scatter) knowing x lets you predict y perfectly! R2 = 0 means none of the data points fit the model

knowing x does not help predict y! An R2 value closer to 1 means the better the regression model fits the data. Find a quadratic equation to model the values in the table. X

2 3 4 Y 3 13

29 Find a quadratic equation to model the values in the table. X -5 0

2 Y -18 -4 -14

Find a quadratic equation to model the values in the table. X -2 1 5 7

Y 27 10 -10 12

Apply! The table shows data about the Wavelength Wave Speed wavelength (in meters) and the (m) (m/s) wave speed (in meters per

3 6 second) of the deep water ocean 5 16 waves. Model the data with a quadratic function then use the

7 31 model to estimate: 8 40 a) the wave speed of a deep water wave that has a

wavelength of 6 meters. b) the wavelength of a deep water wave with a speed of 50 meters per second. Apply! The table at the right shows the

height of a column of water as it drains from its container. Model the data with a quadratic function then use the model to estimate: a) b) c)

d) the water level at 35 seconds. the waver level at 80 seconds. the water level at 3 minutes. the elapsed time for the water level to reach 20 mm.

Math 2 Assignment p 237 #16-22, 31 Write down the R value for each equation!

End of Day 10