Algebra II Piecewise Functions Edited by Mrs. Harlow

Up to now, we’ve been looking at functions represented by a single equation. In real life, however, functions are represented by a combination of equations, each corresponding to a part of the domain. These are called piecewise functions.

WISE FUNCTIONS These are functions that are defined differently on different parts of the domain.

This means for x’s less than 0, put them in f(x) = -x but for x’s greater than or equal to 0, put them in f(x) = x 2 What does the graph of f(x) = -x look like? Remember y = f(x) so let’s graph y = - x which is a line of slope –1 and y-intercept 0. Since we are only supposed to graph this for x< 0, we’ll stop the graph at x = 0. What does the graph of f(x) = x 2 look like? Since we are only supposed to graph this for x  0, we’ll only keep the right half of the graph. Remember y = f(x) so lets graph y = x 2 which is a square function (parabola) This then is the graph for the piecewise function given above.

For x values between –3 and 0 graph the line y = 2x + 5. Since you know the graph is a piece of a line, you can just plug in each end value to get the endpoints. f(-3) = -1 and f(0) = 5 For x = 0 the function value is supposed to be –3 so plot the point (0, -3) For x > 0 the function is supposed to be along the line y = - 5x. Since you know this graph is a piece of a line, you can just plug in 0 to see where to start the line and then count a – 5 slope. So this the graph of the piecewise function solid dot for "or equal to" open dot since not "or equal to"

One equation gives the value of f(x) when x ≤ 1 And the other when x>1

Evaluate f(x) when x=0, x=2, x=4 First you have to figure out which equation to use You NEVER use both X=0 This one fits into the top equation So: 0+2=2 f(0)=2 X=2 This one fits here So: 2(2) + 1 = 5 f(2) = 5 X=4 This one fits here So: 2(4) + 1 = 9 f(4) = 9

Graph: For all x’s < 1, use the top graph (to the left of 1) For all x’s ≥ 1, use the bottom graph (to the right of 1)

Graphing Piecewise Functions Domain - Range -

Domain - Range - [-6, 7] [-4, 2], (4, 7)

x=1 is the breaking point of the graph. To the left is the top equation. To the right is the bottom equation.

Graph:

Step Functions

Graph :

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

Parent Function V-shape It is symmetric about the y -axis vertexThe vertex is the minimum point on the graph

Translation 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 Shrinking The graph of y = a | x | is graph of y = | x | vertically stretched or shrunk 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 when a < 0.

Graphing Absolute Value Functions Graphing y = a | x – h | + k is easy: 1.Plot the vertex ( h, k ). (note…if +h inside that means h is negtive, if – h inside that means h is positive) 2.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 3.Connect the dots in a V-shape.

Transformations in General You can perform transformations on the graph of any function in manner similar to transformations on the absolute value function.

Characteristics of Absolute Value Functions If the function is ______________, the V opens upward. If the function is ___________, the V opens downward. If there is a number added or subtracted inside the absolute value bars, the V is shifted ________________. If there is a number added or subtracted outside the absolute value bars, the V is shifted ________________. If the function is multiplied by a coefficient, the V gets narrower or wider.

Graph the following functions without making a table. 1. 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.

Exercise 2 Graph the following functions without making a table. 1.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.

Graph the following functions without making a table. 1.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.)