Identify linear quadratic and exponential functions from graphs

• Melissa Bialowas

  Melissa Bialowas has taught preschool through high school for over 20 years. She specializes in math, science, gifted and talented, and special education. She has a Master's Degree in Education from Western Governor's University and a Bachelor's Degree in Sociology from Southern Methodist University. She is a certified teacher in Texas as well as a trainer and mentor throughout the United States.

  Table of Contents Show

  • Table of Contents
  • Linear Functions
  • Quadratic Functions
  • Exponential Functions
  • Linear, Quadratic, and Exponential Graphs
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  • Resources created by teachers for teachers
  • How do you determine if a graph is linear quadratic or exponential?
  • How do you tell if a function is exponential from a graph?
  • What is a linear quadratic and exponential function?
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• Instructor Laura Pennington

  Laura received her Master's degree in Pure Mathematics from Michigan State University, and her Bachelor's degree in Mathematics from Grand Valley State University. She has 20 years of experience teaching collegiate mathematics at various institutions.

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Learn to define linear, quadratic, and exponential functions and use their models for real-life applications. Discover their graphs and see examples. Updated: 02/28/2022

There are three main types of functions: linear, quadratic, and exponential. This lesson will discuss each in detail, so it is easy to understand their similarities and differences. First, a function is a mathematical relationship between inputs and outputs. Functions can be as simple as "Jenny offers a smile then Frank blushes," or as complex as the inputs needed to successfully land a rover on the moon.

Linear Functions

Linear functions are relationships between one variable and the associated outputs. When graphed, the function makes a line, thus it is called linear. A linear function is a polynomial function where 1 is the highest exponent. The standard form of a linear function is: y = mx + b. Where 'x' is the input and 'y' is the output. The coefficient 'm' represents the slope of the line, and the number 'b' is the intercept. Linear functions are found throughout daily life. When a person buys a stamp, 5 stamps, or 20 stamps, the relationship between the number of stamps purchased and their cost is a linear function.

The blue function is moving down while the red function moves up. They both form lines and are considered linear functions.

The constant slope is positive for lines that move up, and negative for lines that move down. Even if the lines are vertical or horizontal, they are still lines and therefore linear functions.

Quadratic Functions

Quadratic functions are also relationships between one variable and the associated outputs, but when graphed, they make parabolas. Quadratic functions are polynomial functions where 2 is the highest exponent. The general form of a quadratic function is: {eq}y = ax^{2} + bx + c {/eq}. Where 'x' is the input and 'y' is the output. The letters 'a', 'b', and 'c' represent numbers. Quadratic functions are also found throughout daily life. For example, when a person throws a ball up in the air, it first goes up and then down. The input in this case is time and the output is the height of the ball.

Quadratic functions can vary in intercepts, minimums, maximums, and steepness.

It is important to know that parabolas can open up, down, right, or left. No matter their orientation, they are all still quadratic equations.

Exponential Functions

Exponential functions are relationships between one variable and the associated outputs, but the variable in the function is in the exponent. The general form of an exponential function is: {eq}y = e^{x} {/eq}. Where 'x' is the input, 'y' is the output, and 'e' is a number. Exponential functions can be found in many unexpected situations in daily life. One example is the spread of a fire. because it grows exponentially without intervention. In this scenario, the input would be time and the output would be the area on fire.

This is a graph of exponential growth.

Exponential function graphs are typically either displaying exponential growth (curve going up) or exponential decay (curve going down).

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Linear, Quadratic, and Exponential Graphs

Linear graphs always look like a straight line with no curve. Quadratic graphs have a parabola shape. An exponential graph has a curve, but the curve will start out vertical and become more horizontal, or the curve will start out horizontal and grow to be more vertical.

This graph has a linear (red) and a quadratic (blue) function pictured.

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On the graph, two different functions have been drawn. The red graph is a straight line, so it is a linear function. The blue graph is a parabola, so it is a quadratic function.

This graph has a linear (blue) and an exponential (red) function displayed.

Here, a blue line displays a linear function and a red curve displays an exponential function.

There are many different methods for graphing functions. The method listed below is the easiest to learn because it works with all function types:

• Plug a variety of inputs (x) into the given function.
• Solve for the matching output (y) for each input.
• Draw a dot at each point determined by the inputs and outputs on the graph.
• Connect the dots.
• Observe the shape of the graph.

How can you tell if a function is quadratic or exponential?

Quadratic functions have a highest exponent of two, and they form a parabola when graphed. Exponential functions have a variable in the exponent.

What is the difference between a linear, quadratic, and exponential function?

A linear function forms a line. A quadratic function forms a parabola. An exponential function forms a curve that grows steeper over time.

How do you identify linear, quadratic, and exponential functions?

One could look at their shapes or their general equations. When using general equations, the exponents determine the identity of the function. A linear equation has no exponent. A quadratic equation has a highest exponent of two. An exponential equation has a variable in the exponent.

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