Which Statement Best Describes The Function Represented By The Graph

Which Statement Best Describes the Function Represented by the Graph? A Comprehensive Guide

Analyzing graphs to determine the function they represent is a fundamental skill in mathematics and numerous related fields. This comprehensive guide will delve into various techniques for interpreting graphs and selecting the statement that best describes the underlying function. We'll cover linear, quadratic, exponential, and other function types, illustrating each with examples and clear explanations. By the end, you'll be equipped to confidently analyze graphs and accurately represent their function.

Understanding Function Representation

Before we dive into specific graph types, let's clarify what it means to "represent a function." A function, in its simplest form, is a relationship between an input (often denoted as 'x') and an output (often denoted as 'y') where each input has only one output. A graph visually represents this relationship by plotting points (x, y) on a coordinate plane. The function's representation describes the mathematical equation that governs this relationship.

Key Features to Analyze

Analyzing a graph to determine its function involves scrutinizing several key characteristics:

• Shape: The overall shape of the graph (e.g., straight line, parabola, curve) provides significant clues about the type of function.
• Intercepts: The points where the graph intersects the x-axis (x-intercepts or roots) and the y-axis (y-intercept) offer valuable information.
• Slope (for linear functions): The steepness of a straight line indicates the rate of change of the function.
• Vertex (for quadratic functions): The highest or lowest point of a parabola reveals the function's maximum or minimum value.
• Asymptotes: Lines that the graph approaches but never touches provide insight into the function's behavior at extreme values.
• Symmetry: Symmetry around the y-axis or the origin reveals properties of even and odd functions respectively.
• Increasing/Decreasing Intervals: Determining the intervals where the function's value increases or decreases is crucial for understanding its behavior.
• Continuity: Is the graph a continuous line, or are there breaks or discontinuities?

Types of Functions and Their Graphical Representations

Let's explore common function types and how their graphs appear:

1. Linear Functions

Linear functions have the form y = mx + c, where 'm' is the slope and 'c' is the y-intercept. Their graphs are always straight lines.

• Positive Slope (m > 0): The line slopes upward from left to right.
• Negative Slope (m < 0): The line slopes downward from left to right.
• Zero Slope (m = 0): The line is horizontal.
• Undefined Slope: The line is vertical (represented by x = a constant).

Example: A graph showing a straight line passing through (0, 2) and (1, 5) represents the linear function y = 3x + 2.

2. Quadratic Functions

Quadratic functions have the form y = ax² + bx + c, where 'a', 'b', and 'c' are constants. Their graphs are parabolas.

• a > 0: The parabola opens upwards (U-shaped), having a minimum value at its vertex.
• a < 0: The parabola opens downwards (∩-shaped), having a maximum value at its vertex.
• Vertex: The turning point of the parabola. Its x-coordinate is given by -b/2a.

Example: A U-shaped parabola with a vertex at (1, -2) could represent a quadratic function such as y = (x - 1)² - 2.

3. Exponential Functions

Exponential functions have the form y = abˣ, where 'a' and 'b' are constants (b > 0 and b ≠ 1). Their graphs are curves that either increase or decrease rapidly.

• b > 1: The function represents exponential growth; the graph increases rapidly as x increases.
• 0 < b < 1: The function represents exponential decay; the graph decreases rapidly as x increases.

Example: A graph showing a curve that increases rapidly as x increases, passing through (0, 1) and (1, 2) could represent an exponential function like y = 2ˣ.

4. Polynomial Functions

Polynomial functions have the general form y = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₁x + a₀, where 'n' is a non-negative integer and aₙ, aₙ₋₁, ..., a₀ are constants. Their graphs can have various shapes depending on the degree ('n') of the polynomial.

• n = 1: Linear function
• n = 2: Quadratic function
• n = 3: Cubic function (S-shaped)
• Higher degrees: More complex curves with multiple turning points.

Example: An S-shaped curve could represent a cubic function such as y = x³ - 3x.

5. Trigonometric Functions

Trigonometric functions such as sine (sin x), cosine (cos x), and tangent (tan x) have periodic graphs that repeat themselves. Their graphs involve waves.

• Sine and Cosine: Oscillate between -1 and 1.
• Tangent: Has vertical asymptotes.

Example: A wave-like graph oscillating between -1 and 1 represents a sine or cosine function.

6. Logarithmic Functions

Logarithmic functions are the inverse of exponential functions. They have the general form y = logₓ(f(x)), where x is the base. Their graphs are curves that increase slowly.

• Base > 1: The graph increases slowly as x increases.
• 0 < Base < 1: The graph decreases slowly as x increases.

Example: A graph showing a slowly increasing curve passing through (1,0) could represent a logarithmic function such as y = log₁₀(x).

Strategies for Identifying the Function from a Graph

Here’s a step-by-step approach:

1. Determine the overall shape: Is it a straight line, parabola, curve, or wave? This immediately narrows down the possibilities.

2. Identify key points: Note the x-intercepts (where the graph crosses the x-axis), the y-intercept (where the graph crosses the y-axis), and any other significant points like the vertex of a parabola.

3. Analyze the slope (for lines): Determine if the slope is positive, negative, zero, or undefined.

4. Check for symmetry: Is the graph symmetrical about the y-axis (even function), the origin (odd function), or neither?

5. Consider asymptotes: Are there any vertical or horizontal asymptotes? This is particularly relevant for exponential and logarithmic functions.

6. Examine the behavior at extreme values: What happens to the function's value as x approaches positive or negative infinity?

7. Consider the context: If the graph represents a real-world scenario, the context can provide additional clues.

8. Eliminate incorrect options: Once you have a good understanding of the graph's characteristics, eliminate the statements that are inconsistent with these characteristics.

Examples and Solutions

Let's work through some examples to solidify your understanding:

Example 1:

A graph shows a straight line passing through points (0, 3) and (1, 5). Which statement best describes the function represented by this graph?

a) y = 2x + 3 b) y = x² + 3 c) y = 2ˣ + 3 d) y = 3ˣ + 2

Solution: The graph is a straight line, indicating a linear function. The y-intercept is 3, and the slope is (5 - 3)/(1 - 0) = 2. Therefore, the correct statement is a) y = 2x + 3.

Example 2:

A graph shows a U-shaped parabola with a vertex at (2, 1). Which statement best describes the function represented by this graph?

a) y = (x + 2)² + 1 b) y = (x - 2)² + 1 c) y = -(x - 2)² + 1 d) y = (x - 2)² - 1

Solution: The parabola opens upwards (a > 0), and its vertex is at (2, 1). This indicates a quadratic function of the form y = a(x - h)² + k, where (h, k) is the vertex. Therefore, the correct statement is b) y = (x - 2)² + 1.

By carefully considering the shape, key points, and behavior of the graph, you can confidently determine the function it represents and choose the statement that best describes it. Remember to practice analyzing different graph types to improve your skills and understanding. This guide provides a solid foundation for interpreting graphs and selecting the appropriate function representation. Remember to always carefully examine the given graph's features to ensure accurate function identification.