Write The Equation Of The Function Whose Graph Is Shown.

Write the Equation of the Function Whose Graph is Shown: A Comprehensive Guide

Determining the equation of a function from its graph is a fundamental skill in mathematics, crucial for understanding function behavior and applying mathematical concepts to real-world problems. This skill bridges the gap between visual representation and algebraic expression, allowing for a deeper understanding of the relationship between variables. This comprehensive guide will equip you with the strategies and techniques needed to confidently write the equation of a function given its graph. We'll explore various function types and provide step-by-step examples.

Identifying the Type of Function

The first step in writing the equation is correctly identifying the type of function depicted in the graph. Different function types have characteristic shapes and properties that can be readily identified. Common types include:

1. Linear Functions:

Linear functions are characterized by a straight line. Their equation is of the form y = mx + b, where 'm' represents the slope (the steepness of the line) and 'b' represents the y-intercept (the point where the line crosses the y-axis).

Identifying the slope (m): Choose two distinct points on the line (x₁, y₁) and (x₂, y₂). The slope is calculated as: **m = (y₂ - y₁) / (x₂ - x₁) **.

Identifying the y-intercept (b): This is the y-coordinate of the point where the line intersects the y-axis (where x = 0).

Example: If a line passes through points (1, 2) and (3, 6), the slope is (6-2)/(3-1) = 2. If the y-intercept is 0, the equation is y = 2x.

2. Quadratic Functions:

Quadratic functions are represented by parabolas (U-shaped curves). Their equation is of the form y = ax² + bx + c, where 'a', 'b', and 'c' are constants. The value of 'a' determines whether the parabola opens upwards (a > 0) or downwards (a < 0).

Identifying key features:

• Vertex: The lowest (or highest) point on the parabola. The x-coordinate of the vertex is given by x = -b / 2a.
• x-intercepts (roots): The points where the parabola intersects the x-axis (where y = 0).
• y-intercept: The point where the parabola intersects the y-axis (where x = 0). This is simply the value of 'c'.

Example: If the parabola has x-intercepts at x = 1 and x = 3, and passes through the point (2, -1), we can write the equation in factored form: y = a(x - 1)(x - 3). Substituting (2, -1), we get -1 = a(1)(-1), so a = 1. The equation is y = (x - 1)(x - 3) = x² - 4x + 3.

3. Polynomial Functions:

Polynomial functions are of the form y = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₁x + a₀, where 'n' is a non-negative integer (the degree of the polynomial) and 'aₙ, aₙ₋₁, ..., a₀' are constants. The graph's shape depends on the degree and the coefficients. Higher-degree polynomials can have multiple turning points and x-intercepts.

Identifying key features:

• x-intercepts (roots): These are the points where the graph intersects the x-axis.
• y-intercept: The point where the graph intersects the y-axis.
• Turning points: The points where the graph changes from increasing to decreasing or vice-versa. A polynomial of degree 'n' has at most (n-1) turning points.

Example: A cubic function might have three x-intercepts. If these are at x = -1, x = 0, and x = 2, the equation could be y = ax(x + 1)(x - 2). To find 'a', an additional point on the graph is needed.

4. Exponential Functions:

Exponential functions have the form y = abˣ, where 'a' is the initial value and 'b' is the base (b > 0, b ≠ 1). They exhibit rapid growth or decay. The graph approaches but never touches the x-axis (asymptote) if b > 1 and approaches but never touches the y-axis if b < 1.

Identifying key features:

• y-intercept: The value of 'a' (when x = 0).
• Growth or decay rate: The value of 'b' determines the rate of growth (b > 1) or decay (0 < b < 1).

Example: If the graph passes through (0, 2) and (1, 6), then a = 2. Substituting (1, 6) into the equation, we have 6 = 2b¹, which gives b = 3. The equation is y = 2(3ˣ).

5. Logarithmic Functions:

Logarithmic functions are the inverse of exponential functions. They have the form y = a logₓ(b). The graph approaches but never touches the y-axis (asymptote).

Identifying key features:

• x-intercept: The point where the graph intersects the x-axis.
• Asymptote: The vertical line that the graph approaches but never touches.

Example: A graph might show that the function passes through (1,0) and this point would be important in determining the constant of the logarithmic function.

6. Trigonometric Functions:

Trigonometric functions like sine, cosine, and tangent have characteristic periodic wave patterns. Their equations involve trigonometric functions (sin x, cos x, tan x) and may include amplitude, period, and phase shifts.

Identifying key features:

• Amplitude: The distance from the center line to the peak (or trough) of the wave.
• Period: The horizontal distance it takes for the wave to complete one full cycle.
• Phase shift: A horizontal shift of the graph.
• Vertical shift: A vertical shift of the graph.

Example: A sine wave with an amplitude of 2, a period of π, and a vertical shift of 1 could be represented as y = 2sin(2x) + 1.

Strategies for Finding the Equation

Once you've identified the function type, several strategies can help you determine the equation:

• Using key points: Identify key points on the graph, such as intercepts, vertices, or turning points. Substitute these points into the general equation for the function type to solve for the constants.

• Using the slope-intercept form (for linear functions): Determine the slope and y-intercept from the graph and substitute them into the equation y = mx + b.

• Using the vertex form (for quadratic functions): Use the vertex and another point on the parabola to find the equation in the form y = a(x - h)² + k, where (h, k) is the vertex.

• Using factored form (for quadratic and polynomial functions): If the x-intercepts are known, the equation can be written in factored form. An additional point is then needed to determine the leading coefficient.

• Using transformations: Recognize how the graph relates to a parent function (e.g., y = x², y = eˣ). Transformations such as vertical or horizontal shifts, stretches, or reflections can be reflected in the equation.

Working Through Examples

Let's work through some examples to solidify these concepts.

Example 1: A Linear Function

Imagine a graph showing a straight line passing through points (1, 3) and (4, 9).

1. Calculate the slope: m = (9 - 3) / (4 - 1) = 2
2. Use point-slope form: y - y₁ = m(x - x₁). Using (1, 3), we get y - 3 = 2(x - 1).
3. Simplify: y = 2x + 1

Example 2: A Quadratic Function

A parabola opens upwards, its vertex is at (2, -1), and it passes through the point (3, 0).

1. Use vertex form: y = a(x - h)² + k, where (h, k) = (2, -1).
2. Substitute the point (3, 0): 0 = a(3 - 2)² - 1.
3. Solve for 'a': a = 1.
4. The equation is: y = (x - 2)² - 1 = x² - 4x + 3.

Example 3: An Exponential Function

The graph passes through points (0, 1) and (1, 3).

1. Use the general form: y = abˣ.
2. Substitute (0, 1): 1 = ab⁰ = a. So a = 1.
3. Substitute (1, 3): 3 = 1 * b¹. So b = 3.
4. The equation is: y = 3ˣ.

Conclusion: Mastering Function Equations from Graphs

Determining the equation of a function from its graph is a powerful skill that enhances your understanding of mathematical relationships. By systematically identifying the function type, using key features, and applying appropriate strategies, you can confidently translate visual representations into precise algebraic expressions. This ability is invaluable across various mathematical disciplines and real-world applications. Remember to practice regularly, starting with simpler functions and gradually tackling more complex ones. The more you practice, the more intuitive this process will become. With consistent effort, you'll master the art of writing function equations from graphs!