Leading Coefficient: Definition & Explanation

Leading Coefficient: Definition & Explanation
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Lesson Transcript
Instructor: DaQuita Hester

DaQuita has taught high school mathematics for six years and has a master's degree in secondary mathematics education.

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.

Graph: y = x^2 + 2

Graph: 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.

Graph: y = x - 4

Graph: 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.

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