Unit 3 Parent Functions And Transformations Homework 3 Answer Key

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

This comprehensive guide delves into the complexities of Unit 3: Parent Functions and Transformations, providing detailed explanations and solutions for Homework 3. Understanding parent functions and their transformations is fundamental to mastering algebra and precalculus. This article aims to not only provide the answers but also foster a deeper understanding of the underlying concepts. We will explore various types of parent functions, common transformations, and how to apply them effectively.

Understanding Parent Functions

Parent functions are the most basic forms of functions, serving as building blocks for more complex functions. Understanding their characteristics is crucial before exploring transformations. Some common parent functions include:

• Linear Function: f(x) = x. This function represents a straight line with a slope of 1 and a y-intercept of 0.

• Quadratic Function: f(x) = x². This function represents a parabola opening upwards with its vertex at the origin (0,0).

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

• Square Root Function: f(x) = √x. This function represents a curve starting at the origin and extending to positive x-values.

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

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

• Exponential Function: f(x) = aˣ (where a > 0 and a ≠ 1). This function represents exponential growth or decay.

• Logarithmic Function: f(x) = logₐ(x) (where a > 0 and a ≠ 1). This function is the inverse of the exponential function.

Understanding the graph and key features (domain, range, intercepts, asymptotes) of each parent function is essential.

Transformations of Parent Functions

Transformations alter the parent function's graph, creating variations while retaining its core characteristics. Common transformations include:

• Vertical Shifts: Adding a constant 'k' to the function shifts it vertically. f(x) + k shifts the graph up 'k' units, while f(x) - k shifts it down 'k' units.

• Horizontal Shifts: Adding a constant 'h' inside the function's parentheses shifts it horizontally. f(x - h) shifts the graph to the right 'h' units, while f(x + h) shifts it to the left 'h' units.

• Vertical Stretches and Compressions: Multiplying the function by a constant 'a' stretches or compresses it vertically. If |a| > 1, the graph is stretched; if 0 < |a| < 1, it's compressed. A negative 'a' reflects the graph across the x-axis.

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

Homework 3 Problem Breakdown and Solutions (Illustrative Examples)

Let's assume Homework 3 involves several problems requiring identification of parent functions and their transformations. We'll work through illustrative examples to clarify the process. Note: Specific problems will vary depending on your curriculum.

Problem 1: Identify the parent function and describe the transformations.

Given the function: g(x) = (x + 2)² - 3

Solution:

• Parent Function: The parent function is f(x) = x². This is a quadratic function.

• Transformations:

  • Horizontal Shift: The "+2" inside the parentheses indicates a horizontal shift 2 units to the left.
  • Vertical Shift: The "-3" outside the parentheses indicates a vertical shift 3 units down.

Therefore, g(x) is a parabola obtained by shifting the parent function f(x) = x² two units to the left and three units down.

Problem 2: Write the equation of the function shown in the graph.

(Assume a graph is provided showing a parabola that opens upwards, with a vertex at (1, -4) and passing through the point (2, -3))

Solution:

The graph shows a parabola, indicating a quadratic parent function, f(x) = x². The vertex is shifted from (0,0) to (1,-4), and we can deduce the equation using the vertex form of a quadratic: f(x) = a(x - h)² + k, where (h, k) is the vertex.

• Vertex: (h, k) = (1, -4)

• Point on the graph: (2, -3)

Substituting these values into the vertex form:

-3 = a(2 - 1)² + (-4) -3 = a + (-4) a = 1

Therefore, the equation of the function is: f(x) = (x - 1)² - 4

Problem 3: Graph the function and describe its transformations.

Given the function: h(x) = -√(x - 1) + 2

Solution:

• Parent Function: f(x) = √x, the square root function.

• Transformations:

  • Reflection across the x-axis: The negative sign in front of the square root reflects the graph across the x-axis.
  • Horizontal Shift: The "-1" inside the parentheses indicates a horizontal shift 1 unit to the right.
  • Vertical Shift: The "+2" outside the parentheses indicates a vertical shift 2 units up.

To graph this, start with the basic square root function, then apply the transformations sequentially. First, reflect across the x-axis, then shift 1 unit to the right, and finally, shift 2 units up.

Problem 4: Find the domain and range of the transformed function.

Given the function: j(x) = 2|x + 3| - 1

Solution:

• Parent Function: f(x) = |x|, the absolute value function.

• Transformations:

  • Vertical Stretch: The "2" in front stretches the graph vertically by a factor of 2.
  • Horizontal Shift: The "+3" inside the parentheses shifts the graph 3 units to the left.
  • Vertical Shift: The "-1" outside the parentheses shifts the graph 1 unit down.

• Domain: The domain of the absolute value function is all real numbers (-∞, ∞). The transformations do not affect the domain. Therefore, the domain of j(x) is (-∞, ∞).

• Range: The range of the parent absolute value function is [0, ∞). The vertical stretch and shift down change the range. The minimum value is now -1. Therefore, the range of j(x) is [-1, ∞).

Advanced Concepts and Further Exploration

This guide provides a foundation for understanding parent functions and transformations. However, more advanced concepts exist, such as:

• Combining Transformations: Problems may involve multiple transformations applied simultaneously. Understanding the order of operations is crucial. Generally, horizontal transformations are applied before vertical transformations.

• Piecewise Functions: These functions are defined by multiple sub-functions over different intervals. Transformations can be applied to each sub-function individually.

• Inverse Functions: Understanding how transformations affect the inverse of a function is also important.

• Trigonometric Functions: The same principles of transformations apply to trigonometric functions (sine, cosine, tangent, etc.), although their graphs and properties are more complex.

Conclusion

Mastering parent functions and their transformations is a cornerstone of mathematical understanding. This detailed guide, through illustrative examples and explanations, aims to equip you with the necessary knowledge and problem-solving skills to tackle even the most challenging problems. Remember to practice regularly and consult additional resources if needed. By diligently working through problems and understanding the underlying principles, you can build a strong foundation in algebra and prepare yourself for more advanced mathematical concepts. Remember to always check your work and utilize available resources to solidify your understanding. Consistent practice and a thorough grasp of the concepts will lead to success in your studies.