Which Rule Describes The Function Whose Graph Is Shown

Which Rule Describes the Function Whose Graph is Shown? A Comprehensive Guide

Determining the rule (or equation) that describes a function from its graph is a fundamental skill in algebra and pre-calculus. This process involves analyzing key features of the graph, such as intercepts, slope, asymptotes, and vertex, to deduce the underlying mathematical relationship. This guide will walk you through various techniques and examples, equipping you to confidently tackle such problems.

Understanding Function Representation

Before diving into specific methods, let's solidify our understanding of how functions are represented. A function, in essence, describes a relationship between an input (typically denoted by 'x') and an output (typically denoted by 'y'). This relationship can be expressed in several ways:

• Graphically: A visual representation showing the relationship between input and output values as points on a coordinate plane.
• Algebraically: An equation that defines the output (y) as a function of the input (x). This is what we aim to find when given a graph.
• Numerically: A table of values pairing inputs and their corresponding outputs.
• Verbally: A description in words that explains the relationship between the input and the output.

Analyzing Key Features of the Graph

The process of finding the function rule from its graph involves carefully observing and interpreting several key characteristics:

1. Type of Function

The overall shape of the graph provides a crucial first clue. Identifying the general type of function (linear, quadratic, exponential, logarithmic, trigonometric, etc.) significantly narrows down the possibilities.

• Linear Functions: These have straight-line graphs. Their rule is of the form y = mx + b, where 'm' is the slope and 'b' is the y-intercept.
• Quadratic Functions: These have parabolic (U-shaped) graphs. Their rule is generally of the form y = ax² + bx + c, where 'a', 'b', and 'c' are constants. The sign of 'a' determines whether the parabola opens upwards (a > 0) or downwards (a < 0). The vertex represents the minimum or maximum point.
• Exponential Functions: These graphs show rapid increase or decrease. Their rule is typically of the form y = abˣ, where 'a' is the initial value and 'b' is the base (growth or decay factor).
• Logarithmic Functions: These are the inverse of exponential functions, exhibiting slow growth or decay. Their rule is typically of the form y = logₐ(x), where 'a' is the base.
• Trigonometric Functions: These functions (sine, cosine, tangent, etc.) have periodic, wave-like graphs. Their rules involve trigonometric ratios.

2. Intercepts

The points where the graph intersects the x-axis (x-intercepts) and the y-axis (y-intercept) provide valuable information.

• Y-intercept: The y-coordinate of the point where the graph crosses the y-axis (where x = 0). This value often appears directly in the function rule (e.g., the 'b' in y = mx + b).
• X-intercepts: The x-coordinates of the points where the graph crosses the x-axis (where y = 0). These are also known as roots or zeros of the function. Finding x-intercepts often involves solving the equation y = 0.

3. Slope (for Linear Functions)

For linear functions, the slope (m) represents the rate of change. It is calculated as the change in y divided by the change in x between any two points on the line. A positive slope indicates an increasing function, while a negative slope indicates a decreasing function.

4. Vertex (for Quadratic Functions)

The vertex of a parabola is its lowest (minimum) or highest (maximum) point. The x-coordinate of the vertex can be found using the formula x = -b / 2a for a quadratic function y = ax² + bx + c. Substituting this x-value back into the equation gives the y-coordinate of the vertex.

5. Asymptotes

Asymptotes are lines that the graph approaches but never touches. They are common in exponential and logarithmic functions. Horizontal asymptotes describe the behavior of the function as x approaches positive or negative infinity. Vertical asymptotes indicate values of x where the function is undefined (e.g., division by zero).

Step-by-Step Process with Examples

Let's illustrate the process with a few examples:

Example 1: Linear Function

Imagine a graph showing a straight line passing through points (1, 3) and (3, 7).

1. Type: The graph is a straight line, indicating a linear function.

2. Slope: The slope (m) is (7 - 3) / (3 - 1) = 2.

3. Y-intercept: Using the point-slope form (y - y₁) = m(x - x₁), we can find the equation: y - 3 = 2(x - 1), which simplifies to y = 2x + 1. The y-intercept is 1.

Therefore, the function rule is y = 2x + 1.

Example 2: Quadratic Function

Consider a parabolic graph with vertex at (2, -1) and passing through the point (3, 2).

1. Type: The graph is a parabola, representing a quadratic function.

2. Vertex Form: The vertex form of a quadratic function is y = a(x - h)² + k, where (h, k) is the vertex. Substituting the vertex (2, -1), we get y = a(x - 2)² - 1.

3. Finding 'a': We use the point (3, 2) to solve for 'a': 2 = a(3 - 2)² - 1, which simplifies to a = 3.

Therefore, the function rule is y = 3(x - 2)² - 1.

Example 3: Exponential Function

Suppose a graph shows exponential growth, passing through points (0, 2) and (1, 6).

1. Type: The graph exhibits exponential growth.

2. Exponential Form: The general form is y = abˣ.

3. Finding 'a' and 'b': Using the point (0, 2), we get 2 = ab⁰, which means a = 2. Using the point (1, 6), we get 6 = 2b¹, which means b = 3.

Therefore, the function rule is y = 2(3)ˣ.

Handling More Complex Scenarios

Graphs can be more complex, featuring multiple x-intercepts, asymptotes, or other characteristics that require advanced techniques. These may involve:

• Polynomial Functions: Functions with higher degrees (e.g., cubic, quartic) require factoring techniques to find x-intercepts and determining the leading coefficient.
• Piecewise Functions: Functions defined differently over different intervals.
• Rational Functions: Functions that are ratios of polynomials. These often have vertical and horizontal asymptotes.

For these situations, a deeper understanding of algebraic manipulations, calculus concepts (like derivatives for finding extrema and concavity), and potentially graphing calculators or software is beneficial.

Conclusion

Determining the rule of a function from its graph is a skill that strengthens with practice. By systematically analyzing key features like the type of function, intercepts, slope, vertex, and asymptotes, you can effectively deduce the underlying mathematical relationship. Remember to always start by identifying the overall shape of the graph to narrow down the possibilities and apply the appropriate techniques based on the function type. While simple functions can be tackled with elementary algebra, more complex graphs require a broader mathematical toolkit. Consistent practice and a methodical approach are key to mastering this crucial skill.