In order to be successful in math, students must learn to generalize concepts. Doing this requires a student to understand the principles of a concept in a given example and then connect the principles to broader applications. Function families exemplify this idea in that a student must first see the properties of the parent function before learning how various parameters change the graph of that parent function. While a teacher could hand his or her students a list of how a given parameter affects a given parent function, students are more likely to relationally understand the effects of the parameters if they discover the properties themselves. Using a graphing tool such as Desmos, students can graph generic functions in given families, adjusting the parameters to uncover the relationships among them.
To walk through this process, let’s examine the absolute value function family:
We will start by looking at the parent function, y=|x|, so a=1, b=1, c=0, and d=0. Noticing that the vertex of the graph is located at the origin, we also see that the graph has only positive y-values.
Leaving a=1, b=1, and c=0, we can adjust the value of d and see that increasing the value of d results in a vertical translation up, while decreasing the value of d results in a downward vertical translation. Next, we will let d=0 and adjust the c value. Doing this, we notice that increasing the value of c causes a horizontal translation to the right, while decreasing c results in a horizontal translation to the left. Putting these two findings together, we deduce that the vertex of our graph is the point (c, d), as shown below:
As seen in the above graph when d=-1, the absolute value function may produce negative y-values when d has a negative value. Thinking that absolute value is always positive, most students may be initially confused by these results, but this observation can lead students to an important understanding of the absolute value function. The key that students must understand is how the absolute value function operates. It operates such that the graph of the absolute value function is actually the graph of the linear equation y=a(b(x-c))+d where x<c is reflected over the line y=d. In this way, x=c serves as the line of symmetry, and students will observe that the function never produces values more negative than y=d when a>0 and never produces y-values more positive than y=d when a<0.
This leads us to examine the effect that a and b have on the graph. Allowing d=0, c=0, and b=1, we can manipulate the value of a to determine its effects:
- When a>0, the graph opens up and when a<0, the graph is reflected over the line y=d so that the graph opens down.
- When |a|>1, the graph is vertically stretched. For example, when a=1, the slope of the graph for x>0 is 1. When a=2, the slope for x>0 is 2, meaning that for every one unit travelled horizontally, two units are gained vertically, pulling the graph away from the x-axis.
- When |a|<1, the graph is vertically compressed. So, when a=1/2, when one unit is travelled horizontally, only 1/2 unit is gained vertically, pushing the graph towards the x-axis.
- When a=0, the graph is the horizontal line y=d, because the absolute value part of the function is zeroed out.
Now letting d=o, c=0, and a=1, it is evident that b has similar effects on the graph:
- Only the magnitude of b, not whether b is positive or negative, affects the graph, because the absolute value part of the equation causes b(x-c) to always be positive.
- When |b|>0, the graph is horizontally compressed. For example, when b=1, the graph travels horizontally one unit for every unit it travels up. When b=2, the graph travels horizontally one unit fore every 2 units travelled up.
- When |b|<0, the graph is horizontally stretched. When b=1/2, the graph travels horizontally 2 units for every unit travelled up.
- When b=0, the graph is the horizontal line y=d, because the absolute value part of the function is zeroed out.
This graphing activity is presented in Focus in High School Mathematics: Technology to Support Reasoning and Sense Making. In the book, Dick and Hollebrands (2011) recommend this as a group activity so that kids can individually interact with the graph while feeding off of each other’s ideas. Additionally, a teacher might have groups organize their conclusions in a diagram or table (Dick and Hollebrands, 2011, p. 29). As students investigate function families, it is crucial that a teacher facilitates student discussion, in this way ensuring students draw accurate conclusions.
Dick, T. P., & Hollebrands, K. F. (2011). Focus in high school mathematics: Technology to
support reasoning and sense making. Reston, VA: National Council of Teachers of