2.7: Absolute Value Functions and Graphs Objective: SWBAT graph an absolute value function by performing transformations on the parent function f (x) = |x|.

Vocabulary The function f(x) = |x| is an absolute value function. The highest of lowest point on the graph of an absolute value function is called the vertex. An axis of symmetry of the graph of a function is a vertical line that divides the graph into mirror images. An absolute value graph has one axis of symmetry that passes through the vertex.

Absolute Value Function Vertex Axis of Symmetry

Building the Absolute Value Function The absolute value function is defined by f (x) = |x|. The graph of the absolute value function is similar to the linear parent function, except it must always be positive.

Building the Absolute Value Function The absolute value function is defined by f (x) = |x|. So we just take the negative portion of the graph and reflect it across the x-axis making that part positive.

Building the Absolute Value Function The absolute value function is defined by f (x) = |x|. This is the absolute value parent function.

Parent Function V-shape It is symmetric about the y-axis (Axis of Symmetry) The vertex is the minimum point on the graph

Translation A translation is a transformation that shifts a graph horizontally or vertically, but doesn’t change the overall shape or orientation.

Translation The graph of y = |x – h| + k is the graph of y = |x| translated h horizontal units and y vertical units. The new vertex is at (h, k)

Stretching and Compression The graph of y = a|x| is graph of y = |x| vertically stretched or compressed depending on the |a|. The value of a acts like the slope.

Reflection The graph of y = a|x| is graph of y = |x| reflected across the x-axis.

y = -a |x – h| + k Transformations *Remember that (h, k) is your vertex* Transformations Reflection across the x-axis Vertical Translation Vertical Stretch a > 1 (makes it narrower) OR Vertical Compression 0 < a < 1 (makes it wider) Horizontal Translation (opposite of h)

Multiple Transformations In general, the graph of an absolute value function of the form y = a|x – h| + k can involve translations, reflections, stretches or compressions. To graph an absolute value function, start by identifying the vertex.

Graphing Absolute Value Functions Graphing y = a|x – h| + k is straight forward: Plot the vertex (h, k). (note…if +h inside that means h is negative(to the left); if – h inside that means h is positive (to the right) Use the a value as slope to plot more points. Remember you have to do positive and negative slope to get points on both sides of the V Connect the dots in a V-shape.

Example 1 Graph the following functions without making a table. y = |x – 2| + 3 This graph will go right 2 and up 3 so from the origin go right 2 and up 3. This is the vertex (2, 3). Now from that point use the positive and negative slope (a = 1 here) to get more points.

Your turn: Graph the following functions without making a table. Text book page 111 #14 y = |x - 1| + 3 Identify the vertex Vertex ( , ) Slope =

Example 2 Graph the following functions without making a table. y = (1/2)|x| This function does not have an “h” or “k” so the vertex is (0, 0). Since a = ½ the slope is ½. Go up 1 and right 2 then up one and left 2.

Your turn:  

Example 3 Graph the following functions without making a table. f (x) = -3|x + 1| – 2 This graph will go left 1 and down two so the vertex will be (-1, -2). Since “a” is negative the graph will open down. Since the value of “a” is 3 the slope will be 3 and -3 (just remember to go down.)

Your turn: Graph the following functions without making a table. Text book page 112 #42 y = 2|x + 2| - 3 Identify the vertex Vertex ( , ) Slope =

HOMEWORK Don’t forget to identify VERTEX first before graphing Homework: page 111 #17-27 (odds) Extra Challenge question Text book page 111 #26