Unit 3 Parent Functions And Transformations Homework 3 Answer Key

unit 3 parent functions and transformations homework 3 answer key provides a concise guide that walks students through each problem, clarifies the underlying concepts, and reinforces the correct application of function shifts, stretches, and reflections. This article serves as both a solution manual and a study aid, ensuring that learners can verify their work, understand common pitfalls, and build confidence in manipulating parent functions.

Overview of Parent Functions

Before diving into the specific problems of homework 3, it is essential to review the seven basic parent functions that form the foundation of this unit:

1. Linear – f(x) = x
2. Quadratic – f(x) = x²
3. Cubic – f(x) = x³
4. Absolute Value – f(x) = |x|
5. Reciprocal – f(x) = 1/x 6. Exponential – f(x) = a^x (usually a = e)
6. Logarithmic – f(x) = logₐ(x) (usually a = e)

Each of these functions serves as a reference point. Transformations—such as translations, dilations, and reflections—modify the graph of a parent function while preserving its core shape. Recognizing how each parameter in the transformed equation influences the graph is the key to solving homework problems accurately.

Understanding Transformations

Transformations can be categorized into shifts, stretches/compressions, and reflections. The general form of a transformed function is:

[ g(x) = a , f(b(x - h)) + k ]

where:

• a controls vertical stretch/compression and reflection across the x‑axis.
• b controls horizontal stretch/compression and reflection across the y‑axis.
• h translates the graph horizontally.
• k translates the graph vertically.

Italic emphasis is used for foreign terms like dilation to highlight their meaning without breaking the flow. When a is negative, the graph flips over the x‑axis; when b is negative, the flip occurs over the y‑axis. Positive values of h shift the graph to the right, while negative h shifts it left. Similarly, positive k moves the graph upward, and negative k moves it downward.

Homework 3: Problem Set and Solutions

Below is a detailed answer key for the three primary problems typically found in unit 3 parent functions and transformations homework 3. Each solution includes the original parent function, the transformation applied, the resulting equation, and a brief explanation of the graph’s key features.

Problem 1: Translating a Quadratic

Original Problem:
Given the parent function f(x) = x², write the equation of the function that is shifted 3 units to the right, 2 units up, and vertically stretched by a factor of 4.

Solution:

1. Start with the parent: f(x) = x².
2. Apply a horizontal shift right by 3: replace x with (x‑3). → f₁(x) = (x‑3)².
3. Apply a vertical shift up by 2: add 2 to the entire function. → f₂(x) = (x‑3)² + 2.
4. Apply a vertical stretch by factor 4: multiply the whole expression by 4. → g(x) = 4[(x‑3)² + 2].

Final Equation: [ \boxed{g(x) = 4(x-3)^2 + 8} ]

Explanation:
The vertex of the parabola moves from (0,0) to (3,2), and the coefficient 4 amplifies the opening, making the parabola narrower. The y‑intercept can be found by substituting x = 0: g(0) = 4(‑3)² + 8 = 4·9 + 8 = 44.

Problem 2: Reflecting and Compressing an Exponential

Original Problem:
Transform the parent exponential function f(x) = e^x by reflecting it across the x‑axis, shifting it 1 unit left, and compressing it horizontally by a factor of ½.

Solution:

1. Reflection across the x‑axis: multiply by –1 → –e^x.
2. Horizontal shift left by 1: replace x with (x+1). → –e^{x+1}.
3. Horizontal compression by ½: replace x with 2x (since b = 2 compresses). → –e^{2(x+1)}.

Final Equation:
[ \boxed{h(x) = -e^{2(x+1)}} ]

Explanation:
The negative sign flips the graph downward, the +1 inside the exponent moves every point one unit left, and the factor 2 inside the exponent squeezes the graph toward the y‑axis, making it steeper. The horizontal asymptote remains at y = 0, but the y‑intercept becomes h(0) = -e^{2} ≈ -7.39.

Problem 3: Combining Absolute Value Stretch and Shift

Original Problem:
Create the equation for the absolute value function that is vertically stretched by a factor of ½, reflected over the x‑axis, and shifted 4 units down from the parent f(x) = |x|.

Solution:

1. Vertical stretch by ½: multiply the whole function by ½ → ½|x|. 2. Reflection over the x‑axis: multiply by –1 → ‑½|x|.
2. Shift down 4 units: subtract 4 → ‑½|x| - 4.

Final Equation:
[ \boxed{p(x) = -\frac{1}{2}|x| - 4} ]

Explanation:
The graph opens downward because of

Continuing the explanation for Problem 3:

Explanation:
The negative sign reflects the graph over the x-axis, causing the V-shape to open downward. The vertical stretch by a factor of ½ makes the V narrower than the parent function. The downward shift of 4 units moves the vertex from (0,0) to (0,-4). The graph has a vertex at (0, -4), a y-intercept at (0, -4), and a horizontal asymptote does not exist for this absolute value function. The slope becomes steeper near the vertex due to the compression.

Problem 4: Shifting and Stretching a Logarithmic Function

Original Problem:
Transform the parent logarithmic function f(x) = log(x) by shifting it 2 units to the left, reflecting it over the y-axis, and vertically compressing it by a factor of ⅓.

Solution:

1. Horizontal shift left by 2: replace x with (x+2). → log(x+2).
2. Reflection over the y-axis: replace x with (-x). → log(-x+2).
3. Vertical compression by ⅓: multiply the entire function by ⅓. → ⅓ log(-x+2).

Final Equation:
[ \boxed{q(x) = \frac{1}{3}\log(-x+2)} ]

Explanation:
The shift left moves the vertical asymptote from x=0 to x=-2. The reflection over the y-axis flips the graph horizontally, changing the domain from (0,∞) to (-2,0). The vertical compression by ⅓ makes the curve rise more gradually. The vertical asymptote is at x = -2, the domain is (-2, 0), and the range remains (−∞, ∞).

Conclusion:
This exploration of function transformations—horizontal and vertical shifts, reflections, stretches, and compressions—demonstrates the profound impact of algebraic modifications on a function's graph. Each transformation systematically alters the position, orientation, and shape of the parent function, revealing the deep connection between algebraic expressions and their geometric representations. Understanding these principles is fundamental for analyzing and sketching complex functions efficiently, providing a powerful toolkit for modeling real-world phenomena across mathematics, science, and engineering. Mastery of these transformations enables precise prediction of a function's behavior and key characteristics, such as vertices, asymptotes, intercepts, and domains.

Continuing theexploration of function transformations:

Problem 5: Combining Transformations for a Rational Function Original Problem:
Transform the parent rational function f(x) = 1/x by shifting it 3 units to the right, reflecting it over the x-axis, and vertically stretching it by a factor of 4.

Solution:

1. Horizontal shift right by 3: replace x with (x-3). → 1/(x-3).
2. Reflection over the x-axis: multiply by -1. → -1/(x-3).
3. Vertical stretch by factor of 4: multiply the entire function by 4. → 4 * (-1/(x-3)) = -4/(x-3).

Final Equation:
[ \boxed{r(x) = -\frac{4}{x-3}} ]

Explanation:
The shift right moves the vertical asymptote from x=0 to x=3. The negative sign reflects the hyperbola over the x-axis, flipping it into the third and fourth quadrants. The vertical stretch by 4 makes the branches steeper. The vertical asymptote is at x = 3, the horizontal asymptote is y = 0, the domain is (-∞, 3) ∪ (3, ∞), and the range is (-∞, 0) ∪ (0, ∞).

Conclusion:
This exploration of function transformations—horizontal and vertical shifts, reflections, stretches, and compressions—demonstrates the profound impact of algebraic modifications on a function's graph. Each transformation systematically alters the position, orientation, and shape of the parent function, revealing the deep connection between algebraic expressions and their geometric representations. Understanding these principles is fundamental for analyzing and sketching complex functions efficiently, providing a powerful toolkit for modeling real-world phenomena across mathematics, science, and engineering. Mastery of these transformations enables precise prediction of a function's behavior and key characteristics, such as vertices, asymptotes, intercepts, and domains. This foundational knowledge is indispensable for advancing into higher-level mathematics and its myriad practical applications.