Leading coefficients can help you predict what a graph will look like before you actually see it. In this lesson, you’ll learn about leading coefficients and how to use them.

## What are Leading Coefficients?

**Leading coefficients** are the numbers written in front of the variable with the largest exponent.

Just like regular coefficients, they can be positive, negative, real, or imaginary as well as whole numbers, fractions or decimals. For example, in the equation *-7x^4 + 2x^3 – 11*, the highest exponent is 4. The coefficient for that term is -7, which means that -7 is the leading coefficient.

## Leading Coefficients and Graphs

The leading coefficient can tell you two things about a graph. To begin, it can tell you what direction the graph is facing, which is determined by whether the leading coefficient is positive or negative. To see how this works, let’s compare the graphs of *y = x^2 + 2* and *y = -x^2 + 2*.

We see that the graph of *y = x^2 + 2*, which has a positive leading coefficient, looks like the letter ‘U,’ and the graph of *y = -x^2 + 2*, with a negative leading coefficient, looks like an upside down ‘U.’ Changing the sign on the leading coefficient changed the direction of the graph.

Now, let’s take a look at a linear example. Compare the graphs of *y = x – 4* and *y = -x – 4*.

The graph of *y = x – 4* has a positive leading coefficient and grows as the graph moves from left to right while the graph of *y = -x – 4* has a negative leading coefficient and decreases as the graph moves from left to right. A different sign on the leading coefficient changed the direction of this graph as well.The second thing that the leading coefficient can tell you is how wide or skinny a quadratic graph will be or how steep a linear equation will be.

Let’s first take a look at a quadratic equation. Compare the graph of *y = x^2 – 1* to the graph of *y = 4x^2 – 1*.

The graph of *y = 4x^2 – 1*, the skinnier graph, has a larger coefficient than the graph of *y = x^2 – 1*. Therefore, for quadratic equations, a larger leading coefficient will cause the graph to be skinnier, and a smaller leading coefficient will cause the graph to be wider.For a linear example, let’s compare the graphs of *y = 2x – 3* to the graph of *y = 6x – 3*.

The graph of *y = 6x – 3*, the steeper graph, has a larger coefficient than the graph of *y = 2x – 3*.

Therefore, in linear equations, a larger leading coefficient will create a steeper graph than an equation with a smaller leading coefficient.

## Lesson Summary

A **leading coefficient** is the coefficient preceding the term with the largest exponent. The direction of a graph is determined by whether the leading coefficient is positive or negative, and the width or steepness of a graph is determined by how large or small the leading coefficient is.