Unit 3 Parent Functions And Transformations Homework 6 Answer Key

Unit 3 Parent Functions and Transformations Homework 6 Answer Key: A Comprehensive Guide

This comprehensive guide provides detailed explanations and solutions for common problems found in Unit 3, Homework 6, focusing on parent functions and their transformations. Understanding parent functions and how transformations affect their graphs is crucial for success in algebra and beyond. This guide aims to solidify your understanding and provide a strong foundation for future mathematical endeavors.

Understanding Parent Functions

Before delving into transformations, let's refresh our understanding of parent functions. These are the most basic forms of functions, serving as building blocks for more complex functions. Some key parent functions include:

• Linear Function: f(x) = x This is a straight line passing through the origin with a slope of 1.

• Quadratic Function: f(x) = x² This is a parabola opening upwards with its vertex at the origin.

• Cubic Function: f(x) = x³ This is a cubic curve passing through the origin.

• Square Root Function: f(x) = √x This is a curve starting at the origin and extending to the right.

• Absolute Value Function: f(x) = |x| This is a V-shaped graph with its vertex at the origin.

• Reciprocal Function: f(x) = 1/x This is a hyperbola with asymptotes at x=0 and y=0.

• Exponential Function: f(x) = aˣ (where 'a' is a constant greater than 0 and not equal to 1) This shows exponential growth or decay.

• Logarithmic Function: f(x) = logₐ(x) (where 'a' is a constant greater than 0 and not equal to 1) This is the inverse of the exponential function.

Transformations of Parent Functions

Transformations alter the graph of a parent function, shifting, stretching, compressing, or reflecting it. These transformations are typically represented in the function's equation:

• Vertical Shifts: Adding a constant 'k' to the function shifts it vertically: f(x) + k. A positive 'k' shifts it upwards, and a negative 'k' shifts it downwards.

• Horizontal Shifts: Adding a constant 'h' inside the function's parentheses shifts it horizontally: f(x - h). A positive 'h' shifts it to the right, and a negative 'h' shifts it to the left.

• Vertical Stretches/Compressions: Multiplying the function by a constant 'a' stretches it vertically if |a| > 1 and compresses it if 0 < |a| < 1. A negative 'a' reflects the graph across the x-axis.

• Horizontal Stretches/Compressions: Multiplying 'x' by a constant 'b' inside the function's parentheses stretches it horizontally if 0 < |b| < 1 and compresses it if |b| > 1. A negative 'b' reflects the graph across the y-axis.

General Transformation Form: A general form incorporating all these transformations is: a * f(b(x - h)) + k

Example Problems and Solutions:

Let's work through some example problems that might appear in your Homework 6. Remember, the key is to identify the parent function and then analyze the transformations applied.

Problem 1: Describe the transformations applied to the parent function f(x) = x² to obtain g(x) = -2(x + 3)² - 1.

Solution:

The parent function is f(x) = x². The transformations applied to obtain g(x) are:

1. Horizontal Shift: The (x + 3) term indicates a horizontal shift of 3 units to the left.

2. Vertical Stretch: The 2 multiplier stretches the graph vertically by a factor of 2.

3. Reflection: The negative sign (-) reflects the graph across the x-axis.

4. Vertical Shift: The -1 term indicates a vertical shift of 1 unit down.

Problem 2: Graph the function h(x) = √(x - 2) + 1. Identify the domain and range.

Solution:

The parent function is f(x) = √x. The transformations are:

1. Horizontal Shift: The (x - 2) term shifts the graph 2 units to the right.

2. Vertical Shift: The +1 term shifts the graph 1 unit up.

The graph will start at the point (2, 1) and extend to the right.

• Domain: [2, ∞) (The function is only defined for x ≥ 2)

• Range: [1, ∞) (The y-values start at 1 and extend upwards)

Problem 3: Write the equation of the function whose graph is a parabola that opens downwards, is vertically compressed by a factor of 1/3, shifted 4 units to the right, and shifted 2 units up, with a parent function of f(x) = x².

Solution:

1. Reflection: Since it opens downwards, we'll use a negative sign: -

2. Vertical Compression: We multiply by 1/3: -(1/3)

3. Horizontal Shift: We subtract 4 from x: -(1/3)(x - 4)²

4. Vertical Shift: We add 2: -(1/3)(x - 4)² + 2

Therefore, the equation is: g(x) = -(1/3)(x - 4)² + 2

Problem 4: Given the function k(x) = |x - 1| - 3, describe the transformation and find the x- and y-intercepts.

Solution:

The parent function is f(x) = |x|.

1. Horizontal Shift: The (x-1) term shifts the graph 1 unit to the right.

2. Vertical Shift: The -3 term shifts the graph 3 units down.

• x-intercepts: To find the x-intercepts, set k(x) = 0 and solve for x: |x - 1| - 3 = 0 => |x - 1| = 3. This gives us two solutions: x = 4 and x = -2.

• y-intercept: To find the y-intercept, set x = 0: k(0) = |0 - 1| - 3 = -2. The y-intercept is (0, -2).

Problem 5 (More Challenging): Analyze the function p(x) = 2^(x+1) - 4. Describe the transformations, find the horizontal asymptote, and state whether it represents exponential growth or decay.

Solution:

The parent function is f(x) = 2ˣ.

1. Horizontal Shift: The (x + 1) term shifts the graph 1 unit to the left.

2. Vertical Shift: The -4 term shifts the graph 4 units down.

The horizontal asymptote is y = -4 (because the graph approaches this line but doesn't intersect it). Since the base is 2 (greater than 1), this represents exponential growth.

Further Practice and Resources

This guide provides a foundation for understanding parent functions and transformations. To solidify your understanding, practice more problems from your textbook or online resources. Focus on visualizing the transformations and how they affect the graph. Remember to always identify the parent function first before analyzing the transformations. By mastering these concepts, you'll be well-prepared for more advanced topics in mathematics. Good luck!