Absolute Value Functions and Graphs Lesson 2-7 Algebra 2 Absolute Value Functions and Graphs Lesson 2-7

Goals Goal Rubric To graph absolute value functions. Level 1 – Know the goals. Level 2 – Fully understand the goals. Level 3 – Use the goals to solve simple problems. Level 4 – Use the goals to solve more advanced problems. Level 5 – Adapts and applies the goals to different and more complex problems.

Vocabulary Absolute Value Function Axis of Symmetry Vertex

Essential Question Big Idea: Function What is an absolute value function?

Definitions Absolute-Value Function - is a function whose rule contains an absolute-value expression. The graph of the parent absolute-value function f(x) = |x| has a V shape and is symmetric about a vertical line called the axis of symmetry. The graph has either a single maximum point or a single minimum point, called the vertex. The parent absolute-value function has a vertex at (0, 0).

Absolute-Value Function

Absolute-Value Function The absolute-value parent function is composed of two linear pieces, one with a slope of –1 and one with a slope of 1. You can transform absolute-value functions.

The general forms for translations are Vertical: g(x) = f(x) + k Horizontal: g(x) = f(x – h) Remember!

Example: Perform the transformation on f(x) = |x|. Then graph the transformed function g(x). 5 units down f(x) = |x| g(x) = f(x) + k g(x) = |x| – 5 Substitute. The graph of g(x) = |x| – 5 is the graph of f(x) = |x| after a vertical shift of 5 units down. The vertex of g(x) is (0, –5).

Example: continued The graph of g(x) = |x|– 5 is the graph of f(x) = |x| after a vertical shift of 5 units down. The vertex of g(x) is (0, –5). f(x) g(x)

Example: Perform the transformation on f(x) = |x|. Then graph the transformed function g(x). 1 unit left f(x) = |x| g(x) = f(x – h ) g(x) = |x – (–1)| = |x + 1| Substitute.

Example: continued The graph of g(x) = |x + 1| is the graph of f(x) = |x| after a horizontal shift of 1 unit left. The vertex of g(x) is (–1, 0). f(x) g(x)

Your Turn: Let g(x) be the indicated transformation of f(x) = |x|. Write the rule for g(x) and graph the function. 4 units down f(x) = |x| g(x) = f(x) + k g(x) = |x| – 4 Substitute.

Your Turn: continued The graph of g(x) = |x| – 4 is the graph of f(x) = |x| after a vertical shift of 4 units down. The vertex of g(x) is (0, –4). f(x) g(x)

Your Turn: Perform the transformation on f(x) = |x|. Then graph the transformed function g(x). 2 units right f(x) = |x| g(x) = f(x – h) g(x) = |x – 2| = |x – 2| Substitute.

Your Turn: continued The graph of g(x) = |x – 2| is the graph of f(x) = |x| after a horizontal shift of 2 units right. The vertex of g(x) is (2, 0). f(x) g(x)

Absolute-Value Function Vertex Because the entire graph moves when shifted, the shift from f(x) = |x| determines the vertex of an absolute-value graph.

Example: Translate f(x) = |x| so that the vertex is at (–1, –3). Then graph. g(x) = |x – h| + k g(x) = |x – (–1)| + (–3) Substitute. g(x) = |x + 1| – 3

Example: continued The graph of g(x) = |x + 1| – 3 is the graph of f(x) = |x| after a vertical shift down 3 units and a horizontal shift left 1 unit. f(x) The graph confirms that the vertex is (–1, –3). g(x)

Your Turn: Translate f(x) = |x| so that the vertex is at (4, –2). Then graph. g(x) = |x – h| + k g(x) = |x – 4| + (–2) Substitute. g(x) = |x – 4| – 2

Your Turn: continued The graph of g(x) = |x – 4| – 2 is the graph of f(x) = |x| after a vertical down shift 2 units and a horizontal shift right 4 units. g(x) f(x) The graph confirms that the vertex is (4, –2).

Absolute-value functions can also be stretched, compressed, and reflected. Reflection across x-axis: g(x) = –f(x) Reflection across y-axis: g(x) = f(–x) Remember! Vertical stretch and compression : g(x) = af(x) Remember!

Example: Perform the transformation. Then graph. Reflect the graph. f(x) =|x – 2| + 3 across the y-axis. g(x) = f(–x) Take the opposite of the input value. g(x) = |(–x) – 2| + 3

Example: continued The vertex of the graph g(x) = |–x – 2| + 3 is (–2, 3). f g

Example: Stretch the graph. f(x) = |x| – 1 vertically by a factor of 2. g(x) = af(x) g(x) = 2(|x| – 1) Multiply the entire function by 2. g(x) = 2|x| – 2

Example: continued The graph of g(x) = 2|x| – 2 is the graph of f(x) = |x| – 1 after a vertical stretch by a factor of 2. The vertex of g is at (0, –2). g(x) f(x)

Your Turn: Perform the transformation. Then graph. Reflect the graph. f(x) = –|x – 4| + 3 across the y-axis. g(x) = f(–x) Take the opposite of the input value. g(x) = –|(–x) – 4| + 3 g(x) = –|–x – 4| + 3

Your Turn: continued The vertex of the graph g(x) = –|–x – 4| + 3 is (–4, 3). g f

Your Turn: Compress the graph of f(x) = |x| + 1 vertically by a factor of . g(x) = a(|x| + 1) g(x) = (|x| + 1) Multiply the entire function by . g(x) = ( |x| + ) Simplify.

Your Turn: continued The graph of g(x) = |x| + is the graph of g(x) = |x| + 1 after a vertical compression by a factor of . The vertex of g is at ( 0, ). f(x) g(x)

Absolute-Value Function Transformations

Your Turn: How does this transform the graph? Then graph. Reflect graph across the x axis:

Your Turn: How does this transform the graph? Then graph. Vertically stretches by a factor of 2:

Your Turn: How does this transform the graph? Then graph. Vertically compression by a factor of ½ :

Your Turn: How does this transform the graph? Then Graph. Moves it horizontally one unit to the right:

Your Turn: How does this transform the graph? Then graph. Translates it vertically down 4 units:

Your Turn: Perform the transformation. Then graph. Translate f(x) = |x| by 3 units right. f g g(x)=|x – 3|

Your Turn: Perform each transformation. Then graph. Translate f(x) = |x| so the vertex is at (2, –1). Then graph. f g g(x)=|x – 2| – 1

Your Turn: Graph y = |x+1| + 2 2 1

y = 2 |x - 3| + 1 Your Turn: Graph 1 3 stretch stretch Slope: 2

Your Turn: How does this transform the graph? Then graph. Horizontally translates 1 unit left Vertically translates 3 units down Vertically stretches by 2:

Absolute-Value Function Transformation Summary: f(x) = ± a|x-b|+c Vertex (b,c) b horizontal translation c vertical translation Direction: + opens up - Opens down Slope of sides ±

Essential Question Big Idea: Function What is an absolute value function? An absolute-value function gives the distance from the line y = 0 for each value of f(x). Write the equation of the function in the general form y = a|x – h| + k to identify the transformations of the parent function f(x) = |x|.

Assignment Section 2-7, Pg 125 – 127; #1 – 7 all, 8 – 42 even.