The graph of the even function [tex]f(x)[/tex] has five [tex]x[/tex]-intercepts. If [tex](6,0)[/tex] is one of the intercepts, which set of points can be the other [tex]x[/tex]-intercepts of the graph of [tex]f(x)[/tex] ?
A. [tex](-6,0),(-2,0)[/tex], and [tex](0,0)[/tex]
B. [tex](-6,0),(-2,0)[/tex], and [tex](4,0)[/tex]
C. [tex](-4,0),(0,0)[/tex], and [tex](2,0)[/tex]
D. [tex](-4,0),(-2,0)[/tex], and [tex](0,0)[/tex]
Answer (2)
An even function f ( x ) satisfies f ( x ) = f ( − x ) , meaning if ( a , 0 ) is an x-intercept, ( − a , 0 ) is also an x-intercept.
Given ( 6 , 0 ) is an x-intercept, ( − 6 , 0 ) must also be an x-intercept.
Check each option to see which set of points, along with their symmetric counterparts, results in a total of five x-intercepts.
Option 1, ( − 6 , 0 ) , ( − 2 , 0 ) , and ( 0 , 0 ) , yields the x-intercepts ( 6 , 0 ) , ( − 6 , 0 ) , ( 2 , 0 ) , ( − 2 , 0 ) , and ( 0 , 0 ) , totaling five intercepts. Thus, the answer is ( − 6 , 0 ) , ( − 2 , 0 ) , and ( 0 , 0 ) .
Explanation
Understanding Even Functions and Intercepts Since f ( x ) is an even function, we know that f ( x ) = f ( − x ) . This means that if ( a , 0 ) is an x -intercept, then ( − a , 0 ) must also be an x -intercept. We are given that ( 6 , 0 ) is an x -intercept, so ( − 6 , 0 ) must also be an x -intercept. We are also told that the graph has five x -intercepts.
Analyzing Option 1 Let's examine the first option: ( − 6 , 0 ) , ( − 2 , 0 ) , and ( 0 , 0 ) . Since we already have ( 6 , 0 ) and ( − 6 , 0 ) , we need to check if adding these points results in a total of five x -intercepts. If ( − 2 , 0 ) is an intercept, then ( 2 , 0 ) must also be an intercept. If ( 0 , 0 ) is an intercept, then ( − 0 , 0 ) = ( 0 , 0 ) must also be an intercept. So, the x -intercepts would be ( 6 , 0 ) , ( − 6 , 0 ) , ( 2 , 0 ) , ( − 2 , 0 ) , and ( 0 , 0 ) . This gives us a total of five intercepts, which matches the given information.
Analyzing Option 2 Now let's examine the second option: ( − 6 , 0 ) , ( − 2 , 0 ) , and ( 4 , 0 ) . We already have ( 6 , 0 ) and ( − 6 , 0 ) . If ( − 2 , 0 ) is an intercept, then ( 2 , 0 ) must also be an intercept. If ( 4 , 0 ) is an intercept, then ( − 4 , 0 ) must also be an intercept. So, the x -intercepts would be ( 6 , 0 ) , ( − 6 , 0 ) , ( 2 , 0 ) , ( − 2 , 0 ) , ( 4 , 0 ) , and ( − 4 , 0 ) . This gives us a total of six intercepts, which does not match the given information.
Analyzing Option 3 Now let's examine the third option: ( − 4 , 0 ) , ( 0 , 0 ) , and ( 2 , 0 ) . We already have ( 6 , 0 ) and ( − 6 , 0 ) . If ( − 4 , 0 ) is an intercept, then ( 4 , 0 ) must also be an intercept. If ( 2 , 0 ) is an intercept, then ( − 2 , 0 ) must also be an intercept. If ( 0 , 0 ) is an intercept, then ( − 0 , 0 ) = ( 0 , 0 ) must also be an intercept. So, the x -intercepts would be ( 6 , 0 ) , ( − 6 , 0 ) , ( 4 , 0 ) , ( − 4 , 0 ) , ( 2 , 0 ) , ( − 2 , 0 ) , and ( 0 , 0 ) . This gives us a total of seven intercepts, which does not match the given information.
Analyzing Option 4 Now let's examine the fourth option: ( − 4 , 0 ) , ( − 2 , 0 ) , and ( 0 , 0 ) . We already have ( 6 , 0 ) and ( − 6 , 0 ) . If ( − 4 , 0 ) is an intercept, then ( 4 , 0 ) must also be an intercept. If ( − 2 , 0 ) is an intercept, then ( 2 , 0 ) must also be an intercept. If ( 0 , 0 ) is an intercept, then ( − 0 , 0 ) = ( 0 , 0 ) must also be an intercept. So, the x -intercepts would be ( 6 , 0 ) , ( − 6 , 0 ) , ( 4 , 0 ) , ( − 4 , 0 ) , ( 2 , 0 ) , ( − 2 , 0 ) , and ( 0 , 0 ) . This gives us a total of seven intercepts, which does not match the given information.
Conclusion Therefore, the only set of points that can be the other x -intercepts of the graph of f ( x ) is ( − 6 , 0 ) , ( − 2 , 0 ) , and ( 0 , 0 ) .
Examples
Even functions are symmetric about the y-axis, which means they have a mirror-image property. This concept is used in physics to describe symmetrical phenomena, such as the distribution of electric charge around a symmetrical molecule. Understanding even functions helps in simplifying complex calculations and making predictions about the behavior of symmetrical systems. For example, if you know the behavior of an even function on one side of the y-axis, you automatically know its behavior on the other side. This symmetry simplifies analysis and problem-solving in various scientific and engineering applications.
The correct set of other x-intercepts for the even function with one known x-intercept of (6,0) is (-6,0), (-2,0), and (0,0). This option maintains the required total of five x-intercepts. Therefore, the answer is Option A.
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