Use Absolute Value Functions & Transformations Part II Chapter 2.7

Changing Parameter a The general form of an absolute value function is 𝑦=𝑎 𝑥−ℎ +𝑘 As we have previously seen, changing parameter h causes the graph of the parent function to shift either right or left Changing parameter k causes the graph of the parent function to shift either up or down What effect does changing a have? Before we investigate graphically, let’s try to make some sense out of what we already know

Changing Parameter a Recall the definition of absolute value 𝑥 = 𝑥, if 𝑥>0 0, if 𝑥=0 −𝑥, if 𝑥<0 Applying the definition to the general form gives us the following

Changing Parameter a 𝑎 𝑥−ℎ +𝑘= 𝑎 𝑥−ℎ +𝑘, if 𝑥>ℎ 𝑘, if 𝑥=ℎ 𝑎 −𝑥+ℎ +𝑘, if 𝑥<ℎ These are parts of two lines, one with positive slope and the other with negative slope: 𝑦=𝑎𝑥−𝑎ℎ+𝑘 and 𝑦=−𝑎𝑥+𝑎ℎ+𝑘 The slope of the first line is the same as the value of a, and the slope of the second line is the value of −𝑎. We can use these to graph the absolute value function after plotting the vertex point.

Changing Parameter a For example, suppose we want to graph the absolute value function 𝑓 𝑥 =2 𝑥−1 −3 As before, the vertex point is at (1,−3), and this divides the graph into two parts, each part of a line The two lines are 𝑦=2 𝑥−1 −3=2𝑥−5, if 𝑥>1 and 𝑦=2(−𝑥+1)−3=−2𝑥−1, for 𝑥<1 The graphs of each line are shown on the next slide

Changing Parameter a

Changing Parameter a The graph shows us that we can graph absolute value functions by using the parameter a as a slope How does changing parameter a affect the shape of the graph of the parent function? Go to the applet at http://ggbtu.be/m225623 and follow the directions Be sure to complete the statements at the bottom, copied into your notes

Graphing an Absolute Value Function Graph the absolute value function 𝑦=3 𝑥−2 −4 and compare to the parent function, 𝑦=|𝑥|.

Graphing an Absolute Value Function Graph the absolute value function 𝑦=3 𝑥−2 −4 and compare to the parent function, 𝑦=|𝑥|. Note that the vertex point is at 2,−4 and we use slope 3 1 to find a second point The line of symmetry is the vertical line 𝑥=2

Graphing an Absolute Value Function

Guided Practice Graph the absolute value function and compare to the parent function, 𝑦=|𝑥|. 𝑦=2 𝑥−1 −3 𝑦=− 𝑥+2 𝑦= 1 2 𝑥 +1 𝑦= −2 3 𝑥+1 +2 𝑓 𝑥 =−2 𝑥−3 +1

Guided Practice Graph the absolute value function and compare to the parent function, 𝑦=|𝑥|. 𝑦=2 𝑥−1 −3

Guided Practice Graph the absolute value function and compare to the parent function, 𝑦=|𝑥|. 𝑦=− 𝑥+2

Guided Practice Graph the absolute value function and compare to the parent function, 𝑦=|𝑥|. 𝑦= 1 2 𝑥 +1

Guided Practice Graph the absolute value function and compare to the parent function, 𝑦=|𝑥|. 𝑦= −2 3 𝑥+1 +2

Guided Practice Graph the absolute value function and compare to the parent function, 𝑦=|𝑥|. 𝑦=−2 𝑥−3 +1

Exercise 2.7b Page 127, #3-20