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Student Page 4.5.2 Transformation of Functions

In this activity, we examine the absolute value function family whose parent function is \(f(x) = |x|\) and the quadratic function family whose parent function is \(f(x) = x^2\text{.}\) To investigate these function families, we need three different parameters, \(a\text{,}\) \(h\text{,}\) and \(k\text{.}\) By changing each parameter, we will see how changes in \(a\text{,}\) \(h\text{,}\) and \(k\) affect the graphs of the parent functions.

To begin, we create a large coordinate plane on the floor with both \(x\)- and \(y\)-axes labeled from \(-10\) to 10. Volunteers use string to mark the \(x\)- and \(y\)-axes and sticky notes to label the axes. (Use 3 √ó 3 inch sticky notes or larger to label the axes). Make the grid large enough so that volunteers can stand next to each other on the \(x\)-axis. A tiled floor is very helpful!

We also need to prepare a set of hang tags that volunteers can wear. Tags can be made from light color cardstock. Punch a hole in the center of a short side, thread string, yarn, or ribbon through the hole so the tag can hang round a volunteer's neck. Label each tag with values of \(x\) between \(-10\) and 10. It is helpful if numbers are not all the same distance apart and if the positive numbers chosen are not all the same as the absolute value of the negative numbers chosen. A possible set of numbered tags might include: \(-10\text{,}\) \(-7\text{,}\) \(-4\text{,}\) \(-1\text{,}\) 0, 2, 5, 8, and 10.

Let's begin:

1.

Each of 9 volunteers wears a tag with a number on it. The number on the tag represents your assigned value of \(x\text{.}\)

2.

Stand on the \(x\)-axis at the point, \((x, 0)\text{,}\) where \(x\) is the number on your tag.

3.

Move to the ordered pair \((x, |x|)\text{.}\) Tape a colored cord to the floor to mark the graph you're standing on. You have just created the graph of \(y = |x|\text{.}\) Discuss the appearance of the graph.

4.

At your teacher's request, transform \(y = |x|\) in various ways. Consider the equation,

\begin{equation*} y = a \cdot | x - h | + k \end{equation*}
(b)

What are the values of \(a\text{,}\) \(h\text{,}\) and \(k\) for each graph? Write them in Table 4.5.2.1.

Table 4.5.2.1.
Move to: \(a\) \(h\) \(k\) Equation
\((x, |x|)\)        
\((x, |x| + 1)\)        
\((x, |x| - 2)\)        
\((x, -|x|)\)        
\((x, 0.5 \cdot |x|)\)        
\((x, |x - 2|)\)        
\((x, |x + 1|)\)        
\((x, 0.5 \cdot |x - 2|)\)        
\((x, 0.5 \cdot |x| - 2)\)        
(c)

Explain why the new graph is located where it is. Use the equation and the graph in your explanation. Compare each new graph to \(y = |x|\) and other related graphs.

(d)

In the final column, write the equation of the function defined by each ordered pair.

Return the room to its original condition. Use Desmos to explore transformations of the parent quadratic function, \(y = x^2\text{.}\)

5.

Continue the investigation using Desmos.

(a)

Enter \(y = a(x - h)2 + k\) into the entry line of Desmos. This is another important form of a quadratic function.

(b)

Set up sliders for \(a\text{,}\) \(h\text{,}\) and \(k\text{.}\)

(c)

Play with \(k\) first. What does \(k\) do? Why does the graph change as it does? Explain from the equation.

(d)

Play with \(a\) next. What does \(a\) do? Why does the graph change as it does? Explain from the equation.

(e)

Play with \(h\text{.}\) What does \(h\) do? Why does the graph change as it does? Explain from the equation.

6.

(a)

What is significant about \(h\) and \(k\text{?}\) What is significant about \(a\text{?}\)

(b)

This form of the quadratic equation, \(y = a(x - h)^2 + k\text{,}\) is called the Vertex Form. Why do you think that is?

7.

Your teacher will provide you a Class Code to complete the Desmos activity, Card Sort: Parabolas in the Teacher Desmos Quadratic Bundle. Complete the activity. The fourth slide is challenging because it does not include scales for any of the graphs. Describe how you knew how to sort each of the equations to fit the graphs provided. Provide illustrations in your description.

8.

Quadratic functions can be written in the following forms*:

Standard Form

\(\displaystyle y=ax^2 +bx+c\)

Vertex Form

\(\displaystyle y=a(x-h)^2 +k\)

Factored Form

\(y = a(x - p)(x - q)\) (*Some quadratic functions cannot be written in this form. Do you know why?)

(a)

What information about a quadratic function is most evident when it is given in:

(i)

Standard Form?

(ii)

Vertex Form?

(iii)

Factored Form?

(b)

What information about a quadratic function is the same for each form of the equation?