Certainly! Below is a guide to answering each question for your portfolio worksheet. You'll need to create sketches on graph paper for several of these questions. I've provided explanations and guidelines for each question.
Question 1: Parent Functions and Families
Parent Functions:
- Linear Function: \( f(x) = x \)
- Quadratic Function: \( f(x) = x^2 \)
- Exponential Function: \( f(x) = 2^x \)
- Square Root Function: \( f(x) = \sqrt{x} \)
- Cube Root Function: \( f(x) = \sqrt[3]{x} \)
Definitions:
- Parent Function: The simplest function of a particular type that retains the essential characteristics of that family of functions (e.g., \( f(x) = x^2 \) is the parent function of quadratic functions).
- Function Family: A group of functions that share common characteristics and can be derived from a parent function by transformations (e.g., linear functions include \( f(x) = x + b \), where \( b \) is a constant).
Question 2: Transformation \( y = -f(x) \)
- Transformation Name: Reflection across the x-axis.
- Graph: Sketch both \( y = f(x) \) and \( y = -f(x) \) for one chosen parent function, e.g., \( y = x^2 \): The original graph is a parabola opening upwards, and the reflection opens downwards.
Question 3: Transformation \( y = f(-x) \)
- Transformation Name: Reflection across the y-axis.
- Graph: Sketch both \( y = f(x) \) and \( y = f(-x) \) for your chosen parent function, e.g., \( y = x^2 \): Both graphs will be symmetrical about the y-axis since \( x^2 \) is an even function.
Question 4: Even and Odd Functions
- Even Function: A function where \( f(x) = f(-x) \). Example: \( f(x) = x^2 \). Graph will be symmetrical about the y-axis.
- Odd Function: A function where \( f(-x) = -f(x) \). Example: \( f(x) = x^3 \). Graph will be symmetrical about the origin.
Answer: Not all functions are either even or odd; some functions may not exhibit these symmetries, like \( f(x) = x + 1 \).
Question 5: Transformation \( y = f(x) \pm k \)
- Transformation Name: Vertical shift.
- Graph: Choose a parent function and a \( k \) value (e.g., for \( f(x) = x^2 \) and \( k = 3 \), sketch \( y = f(x) \) and \( y = f(x) + 3 \), which shifts the graph up by 3 units).
Question 6: Transformation \( y = f(x \pm k) \)
- Transformation Name: Horizontal shift.
- Graph: Choose a parent function and a \( k \) value (e.g., for \( f(x) = x^2 \), sketch \( y = f(x) \) and \( y = f(x - 2) \), which shifts the graph right by 2 units).
Question 7: Transformation \( y = kf(x) \)
- Transformation Name: Vertical stretch or compression.
- Graph: Choose a parent function and a \( k \) value (e.g., for \( f(x) = x^2 \) and \( k = 2 \), sketch both \( y = x^2 \) and \( y = 2x^2 \), which will stretch the graph vertically).
Question 8: Transformation \( y = f(kx) \)
- Transformation Name: Horizontal stretch or compression.
- Graph: Choose a parent function and a \( k \) value (e.g., for \( f(x) = x^2 \) and \( k = 0.5 \), sketch both \( y = x^2 \) and \( y = f(0.5x) \), which compresses the graph horizontally).
Question 9: Comparing Transformations
-
Transformations of \( y = \sqrt{x} \) to \( y = -\sqrt{x} + 1 + 5 \):
- Reflection across the x-axis, shift up by 6 units.
-
Transformations of \( y = \sqrt{x} \) to \( y = -(\sqrt{x} + 1) + 5 \):
- Shift left by 1 unit, reflection across the x-axis and then shift up by 5 units.
-
Comparison:
- Same: Both transformations involve a reflection over the x-axis.
- Different: The first is a simple up shift of the reflected function, while the second includes both a left shift and a different vertical adjustment.
Final Steps:
- Draw all required sketches on graph paper and label them clearly.
- Save your worksheet and sketches and submit as instructed.
Feel free to ask for further clarification or help with any specific sketches or concepts!