Comparing Graphs of Quadratic & Linear Functions

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  • 0:04 Bridges
  • 0:57 Parent Graphs
  • 1:35 Examples
  • 2:04 How to Graph a Linear Function
  • 3:00 How to Graph a…
  • 4:52 Lesson Summary
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Lesson Transcript
Instructor: Stephanie Matalone

Stephanie taught high school science and math and has a Master's Degree in Secondary Education.

In this lesson, we will discuss the basics of linear and quadratic functions and their graphs. We will see some examples and discuss how to graph each type when given an equation.


Have you ever seen a bridge before? They are perfect examples of real-world graphs of linear and quadratic functions. A bridge that goes straight across from one spot to another would represent a linear function. It has to be nice and flat to be linear. A bridge with a curved arch like a hill would represent a quadratic function (think of an upside-down U shape).

Linear functions are typically in the form of y = mx + b where m stands for the slope, or rate of change, and b is the y intercept. Linear functions are graphed as straight lines because the x variable is not raised to any exponent. They are like the flat bridge.

Linear Bridge

Quadratic functions are typically in the form y = ax2 + bx + c. Quadratic functions will always have the x variable to the second power. In other words, the x is squared. This makes for a parabola, which is a symmetrical, curved graph.

Quadratic Bridge

Parent Graphs

Parent functions are the most basic form of a function with coefficients of 1 or 0. All other graphs are just transformations of these parent graphs. The parent graph for a linear function is simply y = x. In this parent function, m is equal to 1 and b is equal to 0. This is graphed in red in the image.

The parent graph for the quadratic function is y = x2. Here, a is equal to 1 while b and c are equal to 0. This is graphed in blue in the image. The shape difference between these two parent functions is pretty big. Again, the linear function is straight, while the quadratic is curved.

Parent Graphs


No matter what numbers are plugged in for the coefficients, the graphs will always have the same basic shapes. In the picture below, you can see graphs for y = -4x - 1 in red and y = -4x2 + 6x + 4 in blue. The linear graph is flipped because of the negative m value, and the quadratic graph is upside down because of the negative a value. But, overall, the two graphs are just variations of the parent graph, but just transformed in some way.

Examples of Linear and Quadratic Graphs

How to Graph a Linear Function

Now that we have seen the differences between the shapes of the graph, let's talk about how we can actually graph them. When graphing a linear function, it's a good idea to start with the y-intercept, which is the point where the graph crosses the vertical y-axis. When you have a linear function in the form of y = mx + b, use the b value to find out where the graph crosses the vertical axis. In the example of y = -4x - 1, the b value is equal to -1. This means the graph crosses the y-axis at -1. That point would equate to (0,-1). Thus, we can put a point there.

Next we can use the slope, or the m value, to find another point. The m value is right in front of the x. In our example, the slope is -4. This means our next point will be down 4 boxes and right 1 box from the y-intercept. This will graph as (1,-5). Once we have two points, we can simply draw a line through them and beyond.

Graphing a Linear Function

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