1.2 Graphs Of Functions Homework Answers

1.2 Graphs of Functions: Homework Answers and Deep Dive into Concepts

This comprehensive guide tackles the complexities of graphing functions, specifically addressing common homework problems encountered in the 1.2 section of many introductory algebra or precalculus courses. We'll move beyond simply providing answers; instead, we'll delve into the underlying concepts, providing you with a solid understanding that will empower you to tackle any graphing problem with confidence. We will cover key aspects like identifying domain and range, interpreting intercepts, understanding function behavior, and mastering various graphing techniques.

Understanding the Fundamentals: What Makes a Function a Function?

Before we jump into graphing, let's solidify our understanding of functions themselves. A function is a special type of relation where each input (x-value) corresponds to exactly one output (y-value). This "one-to-one" relationship is crucial. Think of a function as a machine: you put in an input, and it spits out a unique output. If you put in the same input twice, you must get the same output. A graph that fails the vertical line test (meaning a vertical line intersects the graph at more than one point) is not a function.

Key Concepts for Graphing Functions

Several key concepts underpin our ability to effectively graph functions. Let's explore them:

1. Domain and Range: The Foundation of Graphing

The domain of a function is the set of all possible input values (x-values). The range is the set of all possible output values (y-values). Understanding the domain and range is essential for accurately sketching a graph, as it defines the boundaries within which the function exists.

• Finding the Domain: Often, the domain is all real numbers except for values that make the denominator zero (in rational functions) or those that result in an even root of a negative number (in radical functions).
• Finding the Range: This can be more challenging. Analyzing the function's behavior (increasing, decreasing, asymptotes, etc.) and the domain helps determine the range. Looking at the graph after plotting points often reveals the range visually.

Example: Consider the function f(x) = √(x - 2). The domain is x ≥ 2 (because we can't take the square root of a negative number), and the range is y ≥ 0 (since the square root of a non-negative number is always non-negative).

2. Intercepts: Where the Graph Meets the Axes

• x-intercepts: These are the points where the graph crosses the x-axis (where y = 0). To find them, set y (or f(x)) to 0 and solve for x. These are also known as the roots or zeros of the function.
• y-intercepts: These are the points where the graph crosses the y-axis (where x = 0). To find them, set x = 0 and solve for y (or f(x)).

Example: For the function f(x) = x² - 4, the x-intercepts are found by setting x² - 4 = 0, which gives x = 2 and x = -2. The y-intercept is found by setting x = 0, which gives y = -4.

3. Asymptotes: Lines the Graph Approaches but Never Touches

Asymptotes represent lines that the graph approaches infinitely closely but never actually reaches. There are three main types:

• Vertical Asymptotes: Occur when the denominator of a rational function equals zero, and the numerator is non-zero at that point.
• Horizontal Asymptotes: Describe the end behavior of the graph as x approaches positive or negative infinity. Their existence and location depend on the degrees of the numerator and denominator of rational functions.
• Oblique (Slant) Asymptotes: Occur in rational functions where the degree of the numerator is exactly one greater than the degree of the denominator.

Example: The function f(x) = 1/(x - 2) has a vertical asymptote at x = 2 (the denominator is zero). It has a horizontal asymptote at y = 0 (as x approaches infinity, the function approaches zero).

4. Increasing and Decreasing Intervals: Analyzing Function Behavior

A function is increasing on an interval if its values increase as x increases within that interval. It's decreasing if its values decrease as x increases. Identifying these intervals helps understand the overall shape of the graph.

Example: A quadratic function like f(x) = x² is decreasing for x < 0 and increasing for x > 0.

5. Symmetry: Recognizing Patterns in the Graph

Understanding symmetry can significantly simplify the graphing process. Two common types of symmetry are:

• Even Functions: f(-x) = f(x). These functions are symmetric about the y-axis. Example: f(x) = x².
• Odd Functions: f(-x) = -f(x). These functions are symmetric about the origin. Example: f(x) = x³.

6. Transformations: Shifting, Stretching, and Reflecting

Transformations allow us to create new graphs from existing ones by performing operations like shifting, stretching, or reflecting. Understanding these transformations is crucial for quickly sketching graphs.

• Vertical Shift: Adding a constant to the function shifts it vertically. f(x) + c shifts upwards by c units, f(x) - c shifts downwards by c units.
• Horizontal Shift: Adding a constant to x shifts the graph horizontally. f(x + c) shifts left by c units, f(x - c) shifts right by c units.
• Vertical Stretch/Compression: Multiplying the function by a constant stretches or compresses it vertically. cf(x) (c > 1) stretches, cf(x) (0 < c < 1) compresses.
• Horizontal Stretch/Compression: Multiplying x by a constant stretches or compresses the graph horizontally. f(cx) (0 < c < 1) stretches, f(cx) (c > 1) compresses.
• Reflection: Multiplying the function by -1 reflects it across the x-axis. Multiplying x by -1 reflects it across the y-axis.

Example: The graph of f(x) = (x - 2)² + 1 is a parabola shifted 2 units to the right and 1 unit upward from the basic parabola f(x) = x².

Tackling Homework Problems: A Step-by-Step Approach

Let's consider a few sample problems and break down how to solve them step-by-step:

Problem 1: Graph the function f(x) = |x - 3| + 2.

1. Identify the parent function: The parent function is f(x) = |x|, which is a V-shaped graph with its vertex at the origin.
2. Apply transformations: The function is shifted 3 units to the right (because of the "-3" inside the absolute value) and 2 units upward (because of the "+2").
3. Find intercepts: The y-intercept is f(0) = |0 - 3| + 2 = 5. The x-intercept is found by setting |x - 3| + 2 = 0, which has no solution (the graph never crosses the x-axis because it's always above the x-axis).
4. Sketch the graph: Start with the parent function and apply the transformations, noting the vertex, intercepts, and overall shape.

Problem 2: Graph the rational function f(x) = (x + 1) / (x - 2).

1. Find the vertical asymptote: The denominator is zero when x = 2, so there's a vertical asymptote at x = 2.
2. Find the horizontal asymptote: The degrees of the numerator and denominator are equal, so the horizontal asymptote is the ratio of the leading coefficients: y = 1.
3. Find the x-intercept: Set f(x) = 0, which gives x = -1.
4. Find the y-intercept: Set x = 0, which gives f(0) = -1/2.
5. Sketch the graph: Draw the asymptotes, plot the intercepts, and sketch the branches of the graph approaching the asymptotes. Note that the graph will be in two separate sections because of the vertical asymptote.

Problem 3: Determine the domain and range of the function f(x) = √(4 - x²).

1. Domain: The expression under the square root must be non-negative: 4 - x² ≥ 0. This inequality solves to -2 ≤ x ≤ 2. Thus, the domain is [-2, 2].
2. Range: Since the square root is always non-negative, the range is [0, 2] (the maximum value occurs when x = 0, giving √4 = 2).

Beyond the Basics: Advanced Graphing Techniques

This guide has provided a foundation for graphing functions. As you progress, you'll encounter more complex functions requiring advanced techniques, such as:

• Using calculus: Derivatives help determine increasing/decreasing intervals and concavity.
• Piecewise functions: Functions defined differently on different intervals.
• Polar coordinates: Graphing functions in a polar coordinate system.
• Parametric equations: Representing curves using two separate equations, one for x and one for y, both in terms of a parameter (often 't').

Mastering these concepts will equip you to handle virtually any graphing problem you might encounter. Remember, practice is key! Work through numerous examples, and don't hesitate to seek additional help when needed. By understanding the fundamental principles and using a methodical approach, graphing functions will transition from a challenging task to a skill you confidently apply.