Zeros and Intercepts

Finding Zeros and Intercepts of Linear Functions

You can apply both algebraic and graphic strategies to find the zeros and intercepts of linear functions. As you progress through the examples in this interactivity, you will explore techniques to determine the zeros and intercepts both algebraically and by using the graphing calculator. Click the player button to begin.

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Finding Zeros and Intercepts and Quadratic Functions

When given a quadratic function, you can determine the zeros and intercepts both algebraically and graphically. In this interactivity, you will discover the relationship between zeros and x-intercepts, and learn how to use factoring to determine these values algebraically. You will also learn how to use the graphing calculator to identify the zeros and x- and y-intercepts graphically. Click the player button to begin.

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Zeros and Factors

A relationship exists between the zeros and the factors of a polynomial function.

If a is a zero, then (x − a) is a factor.

Consider the polynomial function f(x) = x² + 2x − 15, represented by the graph above.

• 3 is a zero, therefore (x − 3) is a factor
• −5 is a zero, therefore (x + 5) is a factor

Zeros and Roots

A zero of a polynomial function f(x) is also a solution, or root, of the polynomial equation f(x) = 0.

For example, as shown in the graph above, 3 and −5 are zeros of f(x) = x² + 2x − 15.

Therefore, they are solutions, or roots, of x² + 2x − 15 = 0. You can apply the Substitution Property to verify.

	x2 + 2x − 15 = 0		x2 + 2x − 15= 0
	(3)2 + 2(3) − 15 = 0		(−5)2 + 2(−5) − 15 = 0
	9 + 6 − 15 = 0		25 − 10 − 15 = 0
	0 = 0		0 = 0

Note: 3 and −5 may also be referred to as real roots, as they are real number values.

Practical Problem

During a local high school football game, the height h feet of a football after t seconds can be represented by the function h(t) = −16t² + 58t + 2. Find the y-intercept and explain its meaning in the scenario.

To find the y-intercept, find the value of the h when t = 0.

h(0)		= −16(0)2 + 58(0) + 2
		= 0 + 0 + 2
h(0)		= 2

The y-intercept is 2. You can conclude that the football was kicked from a height of 2 feet.

 

Zeros and Intercepts Review

Now that you have learned about zeros and intercepts, it is time to test your knowledge. This interactivity will help you review the information covered throughout this topic. Click the player button to get started.