How to Create Equations From a Graph

By Kathryn White

Pre-algebra and algebra I classes focus on linear equations—equations that can be visually represented with a line when graphed on the coordinate plane. While it is important to learn how to graph a linear equation when it is given in algebraic form, working backwards to write an equation when given a graph will help enhance your understanding of the concept. In practicing how to relate the graph and equation to each other, you also develop the ability to recognize the ways in which word problems and graphs go together. Furthermore, these skills can be applied in science and statistics where equations can be formed from gathered data and used to predict future situations.

Identify two distinct points on the graph and label them as coordinate pairs using the markings on the y-axis and x-axis as guides. For example, if you were to draw an imaginary line from the point you picked down to the x-axis, and it were to hit at a value of negative three, the x part of the point would be -3. If you were to draw an imaginary horizontal line from the point over to the y-axis, and it would hit at positive four, the point would be labeled (-3, 4).

Label one of your points "point one" and the other one "point two" so that you don't get them mixed up.

Use the slope formula to figure out the slope or "steepness" of the line. Subtract the y coordinate of point two from the y coordinate of point one. Subtract the x coordinate of point two from the x coordinate of point one. Divide the first number by the second number. If the numbers do not divide evenly, leave them as a reduced fraction. Label this number as your slope.

Pick either of your two points and circle it. From now on, you will ignore the other point.

Write out the equation in "point-slope" form. On the left, write the letter "y" minus the y coordinate of your circled point. If the coordinate is negative, and you have two minus signs, change them to one plus sign. On the left, write the slope multiplied by a set of parentheses. Inside the parentheses, write the letter "x" minus the x coordinate of the circled point. Again, change two negatives to a positive. For instance, you might end up with y - 4 = 5 (x + 3).

If the directions ask for the equation in slope intercept form, you must then get the y alone. Do this by distributing the slope (multiply it by both the x and the number in the parenthesis). Then, add or subtract the number from the left side to isolate the "y." In the example of y - 4 = 5(x + 3), you would end up with y = 5x + 23.


To make the mathematical work easier on yourself, try to identify points that use round integers and avoid fractions or decimals. It does not matter which point you start from when you calculate the slope, as long as you use the same order for both the x coordinates and the y coordinates.

About the Author

Kathryn White has over 11 years of experience tutoring a range of subjects at the kindergarten through college level. Her writing reflects her instructional ability as well as her belief in making all concepts understandable and approachable. White earned a Bachelor of Arts in history from Illinois Wesleyan University.