Graphing Linear and Quadratic Functions

Graphing Functions Using a Table of Value

Using a table of values is the easiest way to generate accurate points when graphing a function. You create a table of ordered pairs [ i.e. (x, y) ] by first selecting values of x over the required domain and then computing corresponding values of y. Since we are usually free to select any x values we like, we pick “easy” integer values. We then plot the set of ordered pairs on an x-y plane (Cartesian plane). Our first graph will be of a first degree function, represented by a straight line. All first degree functions produce straight lines, this is why they’re called linear functions as well.

Question: Graph the linear function y = f(x) = 3x + 1 for values of x from –1 to +3.

  • Question 2 in the video explores how second degree (quadratic) functions are graphed. Notice how this time a parabola is produced rather than a straight line.
  • While we know that first degree equations produce straight lines and second degree produce parabolas, sometimes you’ll be given a table of values without being shown the function. In that case, the first and second difference test are performed (as shown).
    • If the first differences are the same, it’s linear.
    • If the second differences are the same, it’s quadratic.
    • You cannot perform this test if the x coordinates are separated by inconsistent intervals.
  • If you’d like to watch a second example of a quadratic function being graphed using a table of values, follow this link.

Now given what you already know about functions and how to generate tables of values, here’s how you can determine whether a table represents a function:

Graphing Linear Functions without a Table of Value

You just learned at how to plot ordered pairs to form a graph, but what if you wanted to take a shortcut by not creating a table of values. This can be done by carefully analyzing the linear function for certain features. Take for example the following linear equations / functions:

  1. y = 4x + 2
  2. y – 4x = 2
  3. –2 – 4x + y = 0

You may have not realized it but all three equations are identical, except that their terms have been rearranged. However, when a linear equation is written in the same format as equation (1), it’s known as slope y-intercept form. All slope y-intercept form equations have the pattern y = mx + b, where m represents a special property of a line called slope, while b is where the line crosses the y axis, called the y-intercept. The slope is a measure of steepness, and it’s evident from the equation that 4 represents the slope and 2 represents the y-intercept. Technically, these are the only two things you need to graph any linear equation – more on this below.

As mentioned, slope in an important analytic measure of straight lines. The slopes of various lines are shown below. Notice how lines moving up to the right always have a positive slope, whereas those that move down to the right have a negative slope. In addition, the larger the slope is in magnitude, the steeper the line becomes.

Let’s focus on y = 4x + 2 again. You discovered that the slope is 4 and y-intercept is 2. To graph without a table of values, start by plotting the y-intercept, 2, whose ordered pair is (0, 2). Remember, at any y-intercept, the x-coordinate is always 0. Once this point is graphed, your slope being 4 can be written as a fraction (if your slope is already a fraction, skip this step):


The numerator, 4, represents the rise, and the denominator represents the run. Starting from the y-intercept, your next point will be 4 units upwards, and 1 unit to the right.

The video below shows four more examples that are in equation form:

A more analytical approach to calculate slope is by using the formula:


As mentioned in the video, finding the slope this way involves picking two points along a line, decoding their (x, y) coordinates, and substituting each coordinate into the formula. To create a complete equation in slope y-intercept form (y = mx + b), you also need to locate where the line crosses the y-axis then substitute it into b. The last video demonstrates how this is done from start to finish.