Table of Contents

Graphs of functions Level 7

Introduction

Have you ever wondered how to visualize relationships between different quantities? Graphs of functions help us see these relationships clearly! Whether it’s tracking the growth of a plant or understanding how far you can travel over time, graphs are essential tools in mathematics. In this article, we will explore how to draw and interpret graphs of both linear and non-linear functions.

Definition and Concept

A function is a relationship between two sets of values, where each input (or x-value) is related to exactly one output (or y-value). Graphs are visual representations of these functions, showing how the output changes as the input varies.

Types of Functions:

  • Linear Functions: Functions that create a straight line when graphed. They can be expressed in the form y = mx + b, where m is the slope and b is the y-intercept.
  • Non-linear Functions: Functions that create curves when graphed, such as quadratics (y = ax² + bx + c) or exponential functions (y = a * b^x).

Historical Context or Origin​

The concept of graphing functions dates back to the early 17th century with the work of mathematicians like René Descartes, who introduced the Cartesian coordinate system. This system allowed for the graphical representation of algebraic equations, revolutionizing mathematics and paving the way for modern graphing techniques.

Understanding the Problem

To draw a graph of a function, you need to understand the relationship between the x and y values. Let’s take a simple linear function as an example:

Example Function: y = 2x + 1

  • Identify x values (inputs): Choose a range of x values, such as -2, -1, 0, 1, and 2.
  • Calculate corresponding y values (outputs): For each x, compute y using the function.

Methods to Solve the Problem with different types of problems​

Method 1: Creating a Table of Values

  • Choose several x-values.
  • Calculate the corresponding y-values.
  • Plot the points on a graph.

    • Example:
    For y = 2x + 1, choose x = -2, -1, 0, 1, 2. The points would be (-2, -3), (-1, -1), (0, 1), (1, 3), (2, 5).

    Method 2: Using the Slope-Intercept Form
    If the function is in the form y = mx + b, identify m (slope) and b (y-intercept).
    Example: For y = 2x + 1, the slope is 2 (rise over run) and the y-intercept is 1.

    Method 3: Graphing Technology
    Use graphing calculators or software to plot functions quickly and accurately.
    Example: Input the function into the software, and it will generate the graph for you.

    Exceptions and Special Cases​

  • Horizontal Lines: Functions like y = b are constant and will appear as horizontal lines on a graph.
  • Vertical Lines: An equation like x = a is not a function since it does not pass the vertical line test.

    • Step-by-Step Practice​

    Problem 1: Graph the function y = x².

    Solution:

  • Choose x values: -2, -1, 0, 1, 2.
  • Calculate y values: 4, 1, 0, 1, 4.
  • Plot points: (-2, 4), (-1, 1), (0, 0), (1, 1), (2, 4).

    • Problem 2: Graph the function y = -3x + 4.

    Solution:

  • Choose x values: -1, 0, 1.
  • Calculate y values: 7, 4, 1.
  • Plot points: (-1, 7), (0, 4), (1, 1).

    • Examples and Variations

    Linear Function Example:

    • Function: y = 3x – 2
    • Points: Choose x = -1, 0, 1, 2. Calculate y: 3, -2, 1, 4. Plot points: (-1, -5), (0, -2), (1, 1), (2, 4).

    Non-linear Function Example:

    • Function: y = x² – 4
    • Points: Choose x = -3, -2, -1, 0, 1, 2, 3. Calculate y: 5, 0, -3, -4, -3, 0, 5. Plot points: (-3, 5), (-2, 0), (-1, -3), (0, -4), (1, -3), (2, 0), (3, 5).

    Interactive Quiz with Feedback System​

    You do not have access to this page.

    If you are not a subscriber, please click here to subscribe.
    OR

    Common Mistakes and Pitfalls

    • Forgetting to label axes on the graph.
    • Not plotting points accurately.
    • Misinterpreting the slope of a line.

    Tips and Tricks for Efficiency

    • Use graph paper to keep points aligned and neat.
    • Check the scale of your axes for accurate representation.
    • Practice drawing different types of functions to build familiarity.

    Real life application

    • Economics: Graphing supply and demand curves to understand market behavior.
    • Science: Plotting data points to analyze trends in experiments.
    • Sports: Tracking performance metrics over time to improve training.

    FAQ's

    Linear graphs create straight lines, while non-linear graphs form curves or other shapes.
    The slope is calculated as the change in y divided by the change in x (rise/run).
    The y-intercept is the point where the graph crosses the y-axis (when x = 0).
    Yes, you can use the slope-intercept form to quickly sketch a linear function.
    Graphs provide a visual representation of data, making it easier to understand relationships and trends.

    Conclusion

    Understanding how to graph functions is an essential skill in mathematics that helps us visualize relationships between variables. By practicing with both linear and non-linear functions, you’ll become more confident in interpreting and creating graphs, which is invaluable in many real-world applications.

    References and Further Exploration

    • Khan Academy: Lessons on graphing functions.
    • Book: Pre-Algebra by Richard Rusczyk.

    Like? Share it with your friends

    Facebook
    Twitter
    LinkedIn

    Filter

    • Category

    • Levels

    • Subject