Shapes of Common Equations

As you progress in math, it is important that you know what the final graph should look like. This will help you know if the answers you have gotten are correct. It will also often give you an idea of how many answers you should expect for questions that ask about zeros or x-intercepts.

I’ve listed out here what the most common equations are and what they look like. Memorize these combinations and your math career will be much easier.

Linear

This is the equation type we graph most often. You will see linear equations everywhere including on the SAT/ ACT. 

A linear equation generally takes one of these forms 

y = mx + b

ax + by = c

y – y₀ = m (x – x₀)

Notice how the x term is only a constant times a single x. It isn’t x² or x³, it is just x. That is what makes these equations linear.

The base linear equation is y = x, which looks like this:

Adding numbers to the equation will change the slope (how steep it is) and may shift the line up or down. Don’t believe me? Check it out! In Desmos, graph y = x, y = 2x, y = 0.5x, and y = x+2. See what changes.

Quadratic

Quadratic equations are the second most likely equations you will see. They have an x² term and look like a U. Because they are shaped like a U, they can cross the x-axis once, twice, or no times. The crossings are called zeros, roots, solutions, and x-intercepts.

If you want to see how they change when you add more terms or multiply numbers with the x²  term, head over to Desmos to try some equations and check it out.

Cubic

Cubic equations are rarer than linear and quadratic equations, but you still need to know what they look like and how many times they cross the x-axis. A common question on the SAT is to find the roots of a cubic equation. Just knowing that cubic equations cross the x-axis 3 times is often enough to get the correct answer without calculating. 

The cubic base equation is y = x³ and looks like this:

Feel free to explore it in Desmos in more depth.

Exponential

Exponential equations have the variable in the exponent. So they look like this: y = 2^x . When the variable is in the exponent, the graph tends to climb very quickly. Exponential graphs look like this:

Absolute value

The absolute value function is the last one you are likely to see in Algebra 1. It looks like a V and has the equation y = | x |

These 5 equations are the ones you are likely to encounter in Algebra 1.

You should know what they look like and how they shift around. Being able to identify these equations will help you on standardized tests like the SAT and ACT and will make courses like Geometry and Algebra 2 easier.

Take some time to memorize these forms and be able to identify the equations that cause them. A good way to memorize these equations is to play around with them in Desmos. See what you can make each base equation do, and have some fun with it.