9th grade math - Algebra

Graphing Functions

Graphing functions is a foundational skill in algebra and beyond. Whether you’re analyzing real-world relationships, solving systems of equations, or preparing for calculus, understanding how to visualize functions gives you powerful insight into their behavior. In this article, we’ll explore the basics of graphing different types of functions, including linear, quadratic, absolute value, and exponential functions. We’ll also revisit key ideas from our previous articles on graphing linear equations and graphing linear inequalities to build a deeper understanding.

What Is a Function?

In mathematics, a function is a relationship between two variables — usually called x (input) and y (output) — where each input has exactly one output. In simpler terms, a function is a rule that assigns each x a single y.

Function Notation

Instead of writing equations as y = 2x + 1, we often use function notation:
f(x) = 2x + 1

Here, f(x) means “the value of the function at x.” It behaves exactly like y in a regular equation.

Types of Functions and Their Graphs

1. Linear Functions

These are the simplest type of function, and we’ve already explored them in our graphing linear equations article. A linear function is written as:
f(x) = mx + b

  • m is the slope (rate of change)
  • b is the y-intercept (value where the graph crosses the y-axis)

Example:
f(x) = 3x - 2
This line has a slope of 3 and a y-intercept of -2.

2. Quadratic Functions

Quadratic functions have the form:
f(x) = ax² + bx + c

These create parabolas, which are U-shaped graphs. If a > 0, the parabola opens upward; if a < 0, it opens downward.

Example:
f(x) = x² - 4x + 3

To graph it:

  • Find the vertex (turning point)
  • Plot intercepts (where the graph crosses the x- and y-axes)
  • Sketch the symmetrical curve

3. Absolute Value Functions

These look like:
f(x) = |x|

The graph is V-shaped, with its point (the vertex) at the origin unless it’s shifted. Absolute value graphs are symmetrical and increase in both directions from the vertex.

Example:
f(x) = |x - 2| + 1
The graph shifts right 2 units and up 1. The vertex is at (2, 1).

4. Exponential Functions

Exponential functions have the form:
f(x) = a · b^x

  • a is the starting value
  • b is the base (if b > 1, it’s growth; if 0 < b < 1, it’s decay)

Example:
f(x) = 2^x
Passes through (0, 1), increases quickly for positive x, and approaches 0 for negative x but never touches the x-axis.

How to Graph a Function

  1. Create a Table of Values: Choose x-values and calculate f(x).
  2. Identify Key Features: Intercepts, symmetry, asymptotes, vertex, etc.
  3. Plot and Connect Points: Use smooth curves or lines as needed.

Comparing with Linear Inequalities

In our article on graphing linear inequalities, we learned that instead of just a line, you shade part of the graph to show solutions. This idea can extend to quadratic and absolute value inequalities as well.

Example:
Graph y > |x - 1|

  • Graph the boundary y = |x - 1| using a dashed line
  • Shade above the V-shaped graph

Real-World Applications

  • Physics: Position vs. time graphs (often quadratic)
  • Economics: Revenue and cost functions
  • Biology: Population growth models
  • Engineering: Load curves and system response

Common Mistakes to Avoid

  • Forgetting to label axes and scale
  • Plotting too few points (especially for curves)
  • Not identifying domain and range
  • Incorrect graph transformations

Conclusion

Graphing functions gives you a powerful visual tool for understanding algebraic relationships. Whether it’s a simple line or a complex exponential curve, the graph tells a story — and now you have the tools to read and draw that story.

This article builds upon your earlier learning in: