Algebra 2 Discuss: What does Algebra mean to you? When might you use Algebra in real life? What are some careers that may use Algebra?

Algebra 2 Warm-Up: In your journal, copy the following terms and definitions. Domain - The set of all possible input values of a relation or function. (x values) Function - A relation in which every input is paired with exactly one output. Inequality - A statement that compares two expressions by using one of the following signs: , ≤, ≥ or ≠. Quadratic function - A function that can be written in the form f (x) = a x 2 + bx + c, where a, b, and c are real numbers and a ≠ 0, or in the form f(x) = a (x - h) 2 + k, where a, h, and k are real numbers and a ≠ 0. Range - The set of output values of a function or relation. (y values) Transformation - A change in the position, size, or shape of a figure or graph. End behavior - The trends in the y-values of a function as the x-values approach positive and negative infinity Even function - A function in which f(-x) = f(x) for all x in the domain of the function. Inverse function - The function that results from exchanging the input and output values of a one-to-one function. The inverse of f(x) is denoted f -1(x) Odd function - A function in which f (-x) = -f (x) for all x in the domain of the function.

paste into your journal

Characteristics of a Function: Domain – Range – End Behavior – Intervals of increasing – Intervals of decreasing – Tuning points – Maximum value – Minimum value – Average rate of change – Zeros of the function –

Algebra 2 Practice: Draw a graph that follows the following qualities: Starts at (0,1) Has a rate of change of 1 Increases on the intervals 0<x<2 and 5<x<10 Decreases on the interval 2<x<5 Tweet: How many turning points are there? What is the maximum value and the minimum value? Homework: Pg.? (#1-3, 6-10, 15, 16, and pick one – 11, 12, or 13)

Algebra 2 Practice: Draw a graph that follows the following qualities: Starts at (-2,1) Has a rate of change of ½ on interval -2 < x < 2 Increases on the intervals -2<x<2 and 5<x<8 Has a rate of change of 1/3 on interval 5<x<8 Decreases on the interval 2<x<5 at a rate of -3 How many turning points are there? What is the maximum value and the minimum value?

Tweet: Algebra 2 Define in your own words the following terms: Domain – Range – End Behavior -

Algebra 2: Use the graph of the function to find the various characteristics of the function. Find each: Domain – Range – End Behavior – On what intervals is the function increasing? On what intervals is the function decreasing? How many tuning points? Maximum value? Minimum value? Average rate of change for -1< x < 0 and for 0 < x < 1 Zeros of the function –

Algebra 2: Use the graph of the function to find the various characteristics of the function. Find each: Domain – Range – End Behavior – On what intervals is the function increasing? On what intervals is the function decreasing? How many tuning points? Maximum value? Minimum value? Average rate of change for -1< x < 0 and for 0 < x < 1 Zeros of the function –

Algebra 2: Use the graph of the function to find the various characteristics of the function. Find each: Domain – Range – End Behavior – On what intervals is the function increasing? On what intervals is the function decreasing? How many tuning points? Maximum value? Minimum value? Average rate of change for Zeros of the function –

Algebra 2 Team Time Page 10 (#1-10) Page 18 (#1-10)

Algebra 2 Team Time Page 10 (#3-12) Page 18 (#4-13)

Algebra 2 In your journal, compare and contrast the two graphs: Graph A: Graph B: Compare Contrast

Algebra 2 Compare and contrast the two graphs: Graph A: Graph B: F(x) = x F(x) = x +7 F(x) = 𝑥 2 F(x) = (3𝑥) 2 F(x) = 𝑥 2 F(x) = −(𝑥 2 )

Algebra 2 Operations to be used Adding or subtracting Transformation Definition Operations to be used Change to x (inside parenthesis) Change to f(x) (outside parenthesis) Written as… Translations Sliding up, down, left, and/or right a given number of units Adding or subtracting Add to X moves the function right Subtract from X moves the function left Add to f(x) moves the function up Subtract from f(x) moves the function down Right g(x) = f(x-a) Left g(x) = f(x + a) Up g(x) = f(x) + a Down g(x) = f(x) -a Stretches Makes a function larger Multiplying by whole number Multiply X by a value gives a HORIZONTAL stretch Multiply f(x) by a value gives a VERTICAL stretch Horizontal g(x) = f(1/a x) Vertical g(x) = a f(x) Compression Makes a function smaller Multiplying by fraction (same as dividing) Multiply X by a fraction gives a HORIZONTAL compression Multiply f(x) by a fraction gives a VERTICAL compression g(x) = f(a x) g(x) = 1/a f(x) Reflections Flipped across an axis or line Multiplying by -1 / changing between positive and negative Change the sign of x values represents a reflection over the y-axis Change the sign of f(x) values represents a reflection over the x-axis Y-axis reflection g(x) = f(-x) X-axis reflection g(x) = -f(x)

Algebra 2 Practice: Create a function table of what you know. Solve for g(x) by performing the necessary operation. Plot your new points. Translate the graph of f(x) to the right 2 units Translate the graph of f(x) up 4 units Stretch the graph of f(x) horizontally by a factor of 2 Compress the graph of f(x) vertically by a factor of ½ Reflect the graph of f(x) over the x- axis Reflect the graph of f(x) over the y- axis

Algebra 2 Identifying transformations practice:

Algebra 2 Identifying more than 1 transformation: Example: Describe how to transform the graph of f (x) = x 2 to obtain the graph of the related function g (x). Then draw the graph of g (x). g (x) = -3ƒ(x - 2) -4 changes to x changes to f(x) x g(x) x f(x) -2 -1 1 2 x g(x)

Algebra 2 Tweet: Describe the transformations that have occurred to f(x) if g(x) = f(1/2(x+5))+2

Algebra 2 Functions can be considered even, odd, or neither. Function is … Symmetric… When plug –x in… Even Symmetric around y-axis Same f(x) Odd Symmetric around origin f(x) becomes –f(x), all signs switch Neither Symmetric around neither Some signs switch

Algebra 2 Even Function : ƒ(-x) = ƒ(x) for all x in the domain of the function Example: x f(x) = 2x -x f(-x) = 2(-x) -2 8 2 -1 1

Algebra 2 Odd Function : ƒ(-x) = -ƒ(x) for all x in the domain of the function Example: x f(x) = x -x f(-x) = (-x) -2 -8 2 8 -1 1

Algebra 2 Functions can be considered even, odd, or neither.

Algebra 2 Functions can be considered even, odd, or neither.

Algebra 2 Functions can be considered even, odd, or neither.

Algebra 2 Team Time Pg. 31 (#1-14)

Algebra 2 Function - A relation in which every input is paired with exactly one output. Tell if each is a function, yes or no? 1. Domain Range 2. Domain Range 3. 4.

Algebra 2 Inverse Relation – The inverse of the relation consisting of all ordered pairs (x, y) is the set of all ordered pairs (y, x). The graph of an inverse relation is the reflection of the graph of the relation across the line y = x. Inverse Function – The function that results from exchanging the input and output values of a one-to-one function. The inverse of f(x) is denoted f -1(x). Examples: Write the inverse function’s coordinates

Algebra 2 How to write an inverse function if given an equation? To check your work and verify that the functions are inverses, show that ƒ ( ƒ -1 (x)) = x and that ƒ -1 (ƒ (x)) = x. What does this mean??? Replace x in original equation with the inverse and it should simplify to x. Replace x in the inverse with the original equation and it should simplify to x.

Algebra 2 Examples: Find the inverse then use composition to verify the functions are inverses. F(x) = 3x + 4 F(x) = 2x – 2 F(x) = x+2 5

Algebra 2 Tweet: Give the inverse then tell if the inverse is a function: (-3, -2) (-1, 0) (0, 2) (2, -2)

Algebra 2 Team Time: Page 40 (#1-10)

Algebra 2 Team Time: Page 40 (#1-10, 13, 14)