Lab: PLACEHOLDER Exploring Function Transformations

Objective

Discover how different parameters transform the graphs of functions through systematic exploration and pattern recognition.

Materials Needed

• Graphing calculator or computer with graphing software
• Graph paper
• Colored pencils or pens

Part 1: The Parent Function

Time: 10 minutes

Start with the parent function $f(x) = x^2$.

1. Graph $f(x) = x^2$ and record key features:

   • Vertex: ______
   • Axis of symmetry: ______
   • Direction of opening: ______

2. Create a data table with at least 7 points:

Part 2: Vertical Transformations

Time: 15 minutes

Investigation A: Adding Constants

Graph these functions on the same coordinate system:

• $g_1(x) = x^2 + 3$
• $g_2(x) = x^2 - 2$
• $g_3(x) = x^2 + 5$

Observations:

1. How do these graphs compare to $f(x) = x^2$?

2. What happens to the vertex of each function?

3. Make a conjecture: If $h(x) = x^2 + k$, what happens when:

   • $k > 0$? ________________
   • $k < 0$? ________________

Investigation B: Multiplying by Constants

Graph these functions:

• $p_1(x) = 2x^2$
• $p_2(x) = \frac{1}{2}x^2$
• $p_3(x) = -x^2$

Observations:

1. How does the shape change when you multiply by a number greater than 1?

2. What happens when you multiply by a number between 0 and 1?

3. What happens when you multiply by a negative number?

4. Make a conjecture: For $j(x) = ax^2$, what effect does $a$ have?

Part 3: Horizontal Transformations

Time: 15 minutes

Investigation C: Inside the Function

Graph these functions:

• $r_1(x) = (x - 2)^2$
• $r_2(x) = (x + 3)^2$
• $r_3(x) = (x - 1)^2$

Observations:

1. Where is the vertex of each function?

2. Counterintuitive discovery: When we have $(x - 2)^2$, which direction does the graph move?

3. Make a conjecture: For $s(x) = (x - h)^2$, what happens when:

   • $h > 0$? ________________
   • $h < 0$? ________________

Part 4: Combining Transformations

Time: 10-15 minutes

Now let's combine what we've learned!

Challenge Problems

For each function, predict the transformations before graphing:

4.1 $f(x) = 2(x - 1)^2 + 3$

• Prediction: ________________________________
• After graphing: ____________________________

4.2 $g(x) = -\frac{1}{2}(x + 2)^2 - 1$

• Prediction: ________________________________
• After graphing: ____________________________

4.3 Design your own function with these transformations:

• Reflected over x-axis
• Moved right 4 units
• Moved up 2 units
• Vertically compressed by factor of $\frac{1}{3}$

Your function: $h(x) = $ _______________________

Part 5: Real-World Application

Time: 5 minutes

A basketball follows the path $h(t) = -16(t - 1.5)^2 + 36$, where $h$ is height in feet and $t$ is time in seconds.

1. Without graphing, determine:

   • Maximum height: _____ feet
   • Time of maximum height: _____ seconds
   • How long until the ball hits the ground: _____ seconds

2. Verify by graphing and explain how the transformations help you understand the real situation.

Reflection Questions

1. Pattern Recognition: Write the general form for a transformed quadratic:

   $$f(x) = a(x - h)^2 + k$$

   Explain what each parameter does:

   • $a$: ________________________________
   • $h$: ________________________________
   • $k$: ________________________________

2. Counterintuitive Insight: Why does $(x - 3)^2$ move the graph right instead of left?

3. Extension: How might these transformation rules apply to other parent functions like $\sqrt{x}$, $|x|$, or $\frac{1}{x}$?

Lab Report

Write a brief summary (2-3 paragraphs) explaining:

• The most surprising discovery you made
• How transformations can help solve real-world problems
• One question you still have about function transformations

Take-Home Challenge

Explore transformations with a different parent function of your choice. Create your own mini-investigation and present your findings to the class next session.