Warm - up Evaluate the expression for x = -6. 1. |x| 2. -|x - 3|

TRANSFORMATIONS y = af(x – c) + d All transformations are in this form: y = af(bx – c) + d In the toolkit functions that we will study first, b has no meaning so we will concentrate on y = af(x – c) + d

Absolute Value Transformations

Graphing Absolute Value Functions Graph the Parent Function x y What would be the domain and range?

The absolute Value Equation The equation: Based on previous function characteristics, write down what do you think a, h and k do? Share your thoughts with your neighbor.

Let’s see if you were right… In the same coordinate plane graph y = a|x| for a = -2, -½, ½, and 2. What effect does a have on the graph of y = |x|? Were you right?

Try some more… In the same coordinate plane graph y = |x – h| for h = -2, 0, and 2. What effect does h have on the graph of y = |x|? Where you right about the effects of h?

Last One… In the same coordinate plane graph y = |x|+ k for k = -4, 0, and 4. What effects does k have on the graph? Were you right about k?

Like with quadratic, logarithmic and exponential equations you can have more than one transformation at a time. Ex: y = 2|x – 4| + 3 What would be the domain and range?

General transformation Rules For Absolute Value Graphs Parent Function: ALWAYS y = |x| The graph of y = a|x - h|+ k has the following characteristics: -The graph has vertex (h, k) and is symmetric in the line x = h (the axis of symmetry). -The graph is V-shaped. It opens up is a>0 and down if a<0. -The graph is wider than the parent function if a<1. -The graph is narrower that the parent function if a>1. y = a|x – h|+ k

Graphing an Absolute Function Graph y = -|x + 2|+3 1. What would be the domain and range? 2. End Behavior? 3. Where is the graph Increasing? 4. Decreasing?

Graphing an Absolute Function Graph y = 2|x - 3|- 4 1. What would be the domain and range? 2. End Behavior? 3. Where is the graph Increasing? 4. Decreasing?

Write an absolute value function Write the equation of the function below. 1. What would be the domain and range? 2. End Behavior? 3. Where is the graph Increasing? 4. Decreasing?

Write an absolute value function Write the equation of the function below. 1. What would be the domain and range? 2. End Behavior? 3. Where is the graph Increasing? 4. Decreasing?

Quadratic Transformations

The Graph of a Quadratic Function (cont) We will simplify your life by always using the form:

The Graph of a Quadratic Function The graph of a quadratic function is called a Parabola The parabola opens up when a > 0 The parabola opens down when a < 0 The vertex of the parabola is at (h, k) The value of ‘a’ determines the slope/steepness of the parabola

Graph of a Quadratic Function f (x)=(x -1) – 4 Moves right 1 and down 4. No vertical stretching. (1,-4) vertex

Graph of a Quadratic Function y = 3(x + 1) -1 Moves to the left 1, down 1, vertical stretch is a factor of 3. Vertex(-1, -1)

y = x2 + 3 y = (x-2)2 + 3 Domain: ______ Domain: _________ Range: _________ Range: _________ Increase: ______ Increase: __________ Decrease:______ Decrease: __________ End Behavior: ______ End Behavior: ________

Power Function Transformations

Power Functions: Any function where x is being raised to a power. Even powered functions have an even exponent, minimum or maximum and line symmetry. Odd powered functions have an odd exponent and rotational symmetry

Odd Powered Functions y = x3 + 2 y = (x-2)3 + 3 Domain: ______ Domain: _________ Range: _________ Range: _________ Increase: ______ Increase: __________ Decrease:______ Decrease: __________ End Behavior: ______ End Behavior: ________

Even Powered Functions y = x2 -3 y = (x+ 2)4 + 1 Domain: ______ Domain: _________ Range: _________ Range: _________ Increase: ______ Increase: __________ Decrease:______ Decrease: __________ End Behavior: ______ End Behavior: ________

Logarithms and Exponential Transformations

Bellwork State the transformation, Domian, Range and End Behavior. 1. f(x) = 2(x+4)2 2. f(x) = ½ |x – 2| + 3 3. f(x) = -3x3 - 5

Exponential Transformations Use what you know about transformation to make conjectures about the following equations with parent function y = 3(2)x. Check your conjectures with your groups and your calculator. y = 3(2)x-5 y = 3(2)x + 5 y = 3(2)x + 6 y = 3(2)x - 5 y = 3(2)x – 8 - 4 y = 3(2)x + 3 + 3

Apply to Logarithms Use what you know about transformation to make conjectures about the following equations with parent function y = log(x). Check your conjectures with your groups and your calculator. a. log (x + 4) b. log (x – 5) c. log (x) – 6 d. log (x) + 1 e. log (x + 3) – 2 f. log (x - 7) + 5

A few more… Make conjectures about the following transformations with parent function y = log(x). Check your conjectures with your groups and your calculator. a. y = 3 log(x) b. y = ½ log (x) c. y = ¼ log(x - 5) d. y = 2log(x) -4 Do the graphs look like you expected? Explain why or why not.

Domain, Range and End Behavior of Logarithms Graph the function y = log x in your calculator. Using the graph and your knowledge of asymptotes, find the following: End Behavior: Increase right and Decrease left Domain: (0, ∞) Range: (- ∞, ∞)

Transformed Domain and Range: Find the domain and range for the following: a. log (x + 4) b. log (x – 5) D: {x > -4), R: {all real} c. log (x) – 6 d. log (x) + 1 D: {x > 0), R: {all real} e. log (x + 3) – 2 f. log (x - 7) + 5 D: {x > -3), R: {all real} D: {x > 5), R: {all real} D: {x > 0), R: {all real} D: {x > 7), R: {all real}

Domain, Range and End Behavior of Exponentials (-∞, ∞) Range: (0, ∞) End Behavior: Increase right Left it decreases, but does it go all the way down to -∞? How would we phrase it?

Domain and Range of Exponentials Find the domain and range for the following exponential equations. y = 3(2)x y = 1(4)x +3 -5 y = 2(3)x + 4 y = 4(2)x y = -2(½)x - 2 y = 1(.75)x-4 +3