Equations and Their Graphs

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The picture obtained by plotting all the points of an equation is called the graph of that equation. If there is only one variable in an equation, the graph is on a number line. If there are two variables, the graph is on the coordinate plane (also known as Cartesian Plane). If there are three variables, the graph is in three-dimensional coordinates. In general, for n variables, the graph is in n dimensions.

One can create graphs of equations using Desmos Graphical Calculator. Desmos is an advanced graphing calculator implemented as a web application and a mobile application written in JavaScript. It was founded by Eli Luberoff, math, and physics double major from Yale University, and was launched as a startup at TechCrunch’s Disrupt New York conference in 2011. In addition to graphing both equations and inequalities, it also features lists, plots, regressions, interactive variables, graph restriction, simultaneous graphing, piecewise function graphing, polar function graphing, two types of graphing grids – among other computational features commonly found in a programmable calculator. It can also be used in several different languages.

Users can create accounts and save the graphs and plots that they have created to them. A permalink can then be generated which allows users to share their graphs and elect to be considered for staff picks. The tool comes pre-programmed with 36 different example graphs to teach new users about the tool and the mathematics involved.

In this article, we will look into some of the basic equations and their graphs:

1. Linear

y = mx + b is the slope-intercept form of writing the equation of a straight line. In the equation ‘y = mx + b‘, ‘b‘ is the point, where the line intersects the ‘y axis’ and ‘m‘ denotes the slope of the line. The slope or gradient of a line describes how steep a line is. It can have either a positive or a negative value. When m is positive, we get an increasing line, whereas when m is negative, we get a decreasing line.

Important points to remember:

The equation of the slope-intercept form of a line whose slope is ‘m‘ and whose y-intercept is ‘b‘ or (0,b) is y = mx + b.
The equation of a horizontal line passing through (a,b) is of the form y = b.
The equation of a vertical line passing through (a,b) is of the form x = a.
m is calculated using the formula rise over run or (change in y)/ (change in x)

Effect of change in the value of m and b on the graph of an equation

2. Quadratic

The quadratic function y – k = a(x – h)², a not equal to zero, is said to be in standard form. If a is positive, the graph opens upward, and if a is negative, then it opens downward. The line of symmetry is the vertical line x = h, and the vertex is the point (h, k). When a quadratic function is in standard form, then it is easy to sketch its graph by reflecting, shifting and stretching/shrinking the parabola y = x².

Important points to remember:

In the vertex form (h, k) represents the vertex of the parabola where parabola has either maximum or minimum value.
If a > 0, the parabola has minimum at (h, k)
If a < 0, the parabola has maximum at (h, k)

Effect of change in the value of a, h, and k on the graph of an equation

3. Exponential

An exponential function is a Mathematical function in the form y = a^x, where “x” is a variable and “a” is a constant which is called the base of the function and it should be greater than 0. The most commonly used exponential function base is the transcendental number e, which is approximately equal to 2.71828. In such a case, the equation becomes y = e^x.

Effect of change in the value of a on the graph of an equation

4. Logarithmic

As you well know that, a logarithm is a mathematical operation that is the inverse of exponentiation. The logarithm of a number is abbreviated as “log.”

An exponential equation is an equation in which the variable appears in an exponent. A logarithmic equation is an equation that involves the logarithm of an expression containing a variable. The general form of a logarithmic equation is y – k = log((x + h).

Effect of change in the value of h and k on the graph of an equation

5. Absolute Value

General form of the absolute value equation is y – k = a|x – h|. The variable a tells us how far the graph stretches vertically, and whether the graph opens up or down. The variables h and k tell us how far the graph shifts horizontally or vertically.

Effect of change in the value of a, h, and k on the graph of an equation

6. Sine

A general equation for the sine function is y = A sin Bx. The A and B are numbers that affect the amplitude and period of the basic sine function, respectively.

The graph of the function y = A sin Bx has an amplitude of A and a period of (2π/B). The amplitude, A, is the distance measured from the y-value of a horizontal line drawn through the middle of the graph (or the average value) to the y-value of the highest point of the sine curve, and B is the number of times the sine curve repeats itself within 2π or 360 degrees.

Effect of change in the value of A and B on the graph of an equation

7. Cosine

A general equation for the sine function is y = A cos Bx. The A and B are numbers that affect the amplitude and period of the basic sine function, respectively.

The graph of the function y = A cos Bx has an amplitude of A and a period of (2π/B). The amplitude, A, is the distance measured from the y-value of a horizontal line drawn through the middle of the graph (or the average value) to the y-value of the highest point of the sine curve, and B is the number of times the sine curve repeats itself within 2π or 360 degrees.

Effect of change in the value of A and B on the graph of an equation

8. Tangent

A general equation for the tangent function is y = A tan Bx. The A and B are numbers that affect the amplitude and period of the basic sine function, respectively.

The graph of the function y = A tan Bx has an amplitude of A and a period of (2π/B). The amplitude, A, is the distance measured from the y-value of a horizontal line drawn through the middle of the graph (or the average value) to the y-value of the highest point of the sine curve, and B is the number of times the sine curve repeats itself within 2π or 360 degrees.

Effect of change in the value of A and B on the graph of an equation

Do you know, you can create wonderful images using graphs of equations! Check out for some of the fun graphs in Desmos

Equations and Their Graphs Decoded in a Fun Way