Math 2 Warm Up Simplify each expression 1

Math 2 Warm Up Simplify each expression: 1. 2. 3. 4. 5. 6. 7. 8. 9. 2 x 2 – 4 x(3 x – 5) 3 x(x – 2)(x + 5) (-4 x + 3)(2 x – 7) 3 x(2 x – 7) + 6 x(4 x + 5) x(1 – x) – (1 – 2 x 2) (5 x + 3) 2 -7 x(5 x 2 – 4 x) (4 x – 5) (-2 x 2 + 3 x – 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 = ax 2 + bx + c, where a ≠ 0. Examples: y = 5 x 2 y = -2 x 2 + 3 x y = x 2 – 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 = ax 2 + 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 = ax 2 + bx + c •

Graphing a Quadratic Equation y = ax 2 + 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 = 2 x 2 + 4 x + 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 = – x 2 + 3 x – 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. •

Find the vertex and axis of symmetry of the following quadratic equation. Then, make a table of values and graph the parabola. y = 3 x 2 – 4 Direction: _____ Vertex: ______ Axis: _______

Apply! The number of widgets the Woodget Company sells can be modeled by the equation -5 p 2 + 10 p + 100, where p is the selling price of a widget. What price for a widget will maximize the company’s 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 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 x = -b = y = x 2 - 4 x + 6 2 a y = Step 2: Substitute into Vertex Form:

Example 7: Converting Standard Form to Vertex Form. Step 1: Find the Vertex x = -b = y = 6 x 2 – 10 2 a 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. 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. (2 x + 7)(3 x – 5) = 6 x 2 + 11 x – 35 FACTORS PRODUCT (2 x – 5)(3 x + 7) = 6 x 2 – 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.

Finding Binomial Factors

Finding Binomial Factors

Finding Binomial Factors

Finding Binomial Factors

Finding Binomial Factors

Finding Binomial Factors

Finding Binomial Factors

Finding Binomial Factors

Finding Binomial Factors

Finding Binomial Factors

Finding Binomial Factors*

Finding Binomial Factors*

Finding Binomial Factors*

Factoring Special Expressions*

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

Factor.

Solving Quadratics Equations: Factoring and Square Roots Objective: To solve quadratic equations by factoring and by finding the square root.

Solve by Factoring

Solve by Factoring

Solve by Factoring

Solve by Factoring

Solve by Factoring*

Solve by Factoring*

Solve Using Square Roots •

Solve Using Square Roots

Solve Using Square Roots*

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! i 1 = i i 2 = -1 i 3 = -i i 4 = 1

Simplifying Complex Numbers 21 i

Adding/Subtracting Complex Numbers (8 + 3 i) – (2 + 4 i) 7 – (3 + 2 i) (4 - 6 i) + (4 + 3 i)

Multiplying Complex Numbers (12 i)(7 i) (3 - 7 i)(2 - 4 i) (6 - 5 i)(4 - 3 i) (4 - 9 i)(4 + 3 i)

Now we can finally find ALL solutions to this equation!

Complex Solutions 3 x² + 48 = 0 -5 x² - 150 = 0 8 x² + 2 = 0 9 x² + 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. ) x 2 – 3 x – 28 = 0 2. ) x 2 – 3 x = 4 3. ) x 2 + 6 x + 9 = 0

If a ≠ 1, then divide all the term by “a”. 1. ) 2 x 2 + 6 x = -6 2. ) 3 x 2 – 12 x + 7 = 0 3. ) 5 x 2 + 20 x + -50

Math 2 Assignment P 281 -283 # 15 – 25, 37, 39 End of Day 6

Solve using Completing the square x 2 + 4 x = 21 x 2 – 8 x – 33 = 0 4 x 2 + 4 x = 3

Solving Quadratic Equations: Quadratic Formula Objective: To solve quadratic equations using the Quadratic Formula. Not every quadratic equation can be solved by factoring or by taking the square root!

Solve using Quadratic Formula

Solve using Quadratic Formula

Solve using Quadratic Formula

Solve using Quadratic Formula*

Solve using Quadratic Formula

Solve using Quadratic Formula

Solve using Quadratic Formula

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. v When the graph of a function intersects the x-axis, the y-value of the function is 0. v Therefore, the solutions of the quadratic equation ax 2 + bx + c = 0 are the x-intercepts of the graph. v Also known as the “zeros of the function” or the “roots of the function”.

Solve Quadratic Equations by Graphing Solution

Solve Quadratic Equations by Graphing q Step 1: Quadratic equation must equal 0! ax 2 + bx + c = 0 q Step 2: Press [Y=]. Enter the quadratic equation in Y 1. Enter 0 in Y 2. Press [Graph]. MAKE SURE BOTH X-INTERCEPTS ARE ON SCREEN! ZOOM IF NEEDED! Step 3: Find the intersection of ax 2 + bx + c and 0. Press [2 nd] [Trace]. Select [5: Intersection]. Press [Enter] 2 times for 1 st and 2 nd curve. Move cursor to one of the x-intercepts then press [Enter] for the 3 rd time. Repeat Step 3 for the second x-intercept!

Solve by Graphing

Solve by Graphing

Solve by Graphing

Solve by Graphing

Solve by Graphing

Solve by Graphing

Solve by Graphing

P 266 #20 -31, 54 -56 End of Day 8

Solving Systems of Equations

Solve a System with a Quadratic Equation

Solve a System with a Quadratic Equation

Solve a System with a Quadratic Equation

Solve a System with a Quadratic Equation

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 [2 nd] [Y=], [ENTER], Highlight “On”, Press [ENTER] 2) Turn on diagnostic: Press [2 nd] [0] (for catalog), Scroll down to find Diagonstic. On. 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 L 1 Enter y-values (dependent) in L 2 Clear Lists (if needed): Press [STAT], [ENTER] (for EDIT), Highlight L 1 or L 2 (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: Quad. Reg 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 R 2 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 R 2 is between 0 and 1 (0 ≤ R 2 ≤ 1) • R 2 = 1 means all the data points “fit” the model (lie exactly on the graph with no scatter) – “knowing x lets you predict y perfectly!” • R 2 = 0 means none of the data points “fit” the model – “knowing x does not help predict y!” • An R 2 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 Wavelength Wave Speed the wavelength (in meters) (m/s) and the wave speed (in meters per second) of the deep water 3 6 ocean waves. Model the data 5 16 with a quadratic function then use the model to estimate: 7 31 a) the wave speed of a deep 8 40 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 R² value for each equation! End of Day 10