Which Rule Describes The Function In The Graph Below

Which Rule Describes the Function in the Graph Below? A Comprehensive Guide

Determining the rule that describes a function from its graph involves careful observation and understanding of various mathematical concepts. This guide will walk you through a systematic approach to identifying the function, covering linear, quadratic, exponential, and other common function types. We'll explore how to extract key features from the graph, including intercepts, slopes, asymptotes, and turning points, to arrive at the precise function rule.

Understanding Function Types and Their Graphical Representations

Before diving into analyzing a specific graph, let's review the fundamental characteristics of some common function types:

1. Linear Functions

• Rule: f(x) = mx + c where 'm' is the slope and 'c' is the y-intercept.
• Graphical Representation: A straight line. The slope (m) represents the steepness of the line, and the y-intercept (c) is the point where the line crosses the y-axis. A positive slope indicates an upward trend, while a negative slope indicates a downward trend. A zero slope results in a horizontal line.

2. Quadratic Functions

• Rule: f(x) = ax² + bx + c where 'a', 'b', and 'c' are constants.
• Graphical Representation: A parabola (U-shaped curve). The value of 'a' determines the parabola's orientation (opens upwards if a > 0, downwards if a < 0). The vertex represents the minimum (if a > 0) or maximum (if a < 0) point of the function.

3. Exponential Functions

• Rule: f(x) = abˣ where 'a' is the initial value and 'b' is the base (b > 0 and b ≠ 1).
• Graphical Representation: A curve that increases or decreases rapidly. If b > 1, the function represents exponential growth; if 0 < b < 1, it represents exponential decay. The graph never touches the x-axis (asymptote at y = 0).

4. Polynomial Functions

• Rule: f(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₁x + a₀ where 'n' is a non-negative integer (degree of the polynomial), and 'aₙ', 'aₙ₋₁', ..., 'a₀' are constants.
• Graphical Representation: Can have multiple turning points (local maxima and minima) depending on the degree of the polynomial. The degree dictates the maximum number of x-intercepts (roots) the function can have.

5. Trigonometric Functions

• Rule: Examples include sine (sin(x)), cosine (cos(x)), and tangent (tan(x)).
• Graphical Representation: Periodic functions with characteristic wave-like patterns. The period determines the length of one complete cycle of the wave.

6. Logarithmic Functions

• Rule: f(x) = logₐ(x) where 'a' is the base (a > 0 and a ≠ 1).
• Graphical Representation: A curve that increases slowly and has a vertical asymptote at x = 0 (the y-axis). The function is the inverse of an exponential function.

Analyzing a Graph to Determine the Function Rule: A Step-by-Step Approach

To illustrate the process, let's assume we have a graph in front of us. We will systematically analyze its features to determine the underlying function. Note that a precise function can only be determined if the graph is accurate and provides sufficient data points.

Step 1: Identify the General Shape of the Graph

The first step is to observe the overall shape of the graph. Is it a straight line, a parabola, an exponential curve, a wave-like pattern, or something else? This initial observation narrows down the possible function types significantly.

Step 2: Determine Key Features

Once the general shape is identified, pinpoint key features:

• Intercepts: Where does the graph intersect the x-axis (x-intercepts or roots) and the y-axis (y-intercept)? These points provide crucial information about the function. For instance, the y-intercept is the value of f(0).

• Slope (for Linear Functions): If the graph is a straight line, calculate the slope using two distinct points on the line: m = (y₂ - y₁) / (x₂ - x₁).

• Vertex (for Quadratic Functions): If the graph is a parabola, locate the vertex (the highest or lowest point). The x-coordinate of the vertex is given by x = -b / 2a for the quadratic function f(x) = ax² + bx + c.

• Asymptotes: Does the graph approach a horizontal or vertical line without ever touching it? These are asymptotes and provide clues about the function's behavior. Horizontal asymptotes often indicate limitations in growth (e.g., exponential decay or logarithmic functions).

• Turning Points (for Polynomial Functions): Identify any points where the graph changes direction from increasing to decreasing or vice versa. The number of turning points can provide hints about the polynomial's degree.

Step 3: Use the Key Features to Formulate the Function Rule

Based on the general shape and key features, construct a potential function rule. For example:

• Linear Function: Use the slope (m) and y-intercept (c) to write the function as f(x) = mx + c.

• Quadratic Function: If you know the vertex (h, k) and another point (x, y) on the parabola, you can use the vertex form: f(x) = a(x - h)² + k. Solve for 'a' using the other point's coordinates.

• Exponential Function: If you know the initial value (a) and another point (x, y), you can use the general form f(x) = abˣ and solve for 'b'.

• Other Functions: For more complex functions, you may need to use curve fitting techniques or other advanced mathematical tools to find a precise representation.

Step 4: Verify the Function Rule

After formulating a potential function rule, verify it by checking if the rule accurately predicts the behavior of the graph for several other points. If the predictions match the graph, the rule is likely correct. If not, refine the rule or explore alternative function types.

Illustrative Examples

Let's illustrate this process with two examples:

Example 1: Linear Function

Suppose the graph is a straight line passing through points (1, 3) and (3, 7).

1. Shape: Linear
2. Key Features:
   • Slope: m = (7 - 3) / (3 - 1) = 2
   • Y-intercept: Using the point-slope form y - y₁ = m(x - x₁), we have y - 3 = 2(x - 1), which simplifies to y = 2x + 1. The y-intercept is 1.
3. Rule: f(x) = 2x + 1

Example 2: Quadratic Function

Suppose the graph is a parabola with vertex at (2, 1) and passing through the point (3, 4).

1. Shape: Quadratic
2. Key Features:
   • Vertex: (2, 1)
3. Rule: Using the vertex form f(x) = a(x - h)² + k, we have f(x) = a(x - 2)² + 1. Substituting the point (3, 4), we get 4 = a(3 - 2)² + 1, which simplifies to a = 3. Thus, the rule is f(x) = 3(x - 2)² + 1.

Conclusion

Identifying the function rule from its graph requires a methodical approach combining visual inspection with mathematical analysis. By systematically analyzing the graph's shape, key features, and using appropriate mathematical formulas, you can accurately determine the function that represents the data. Remember to always verify your findings by comparing the predicted values with the actual data points on the graph. This iterative process ensures accuracy and allows you to confidently determine which rule accurately describes the function depicted in the graph.