Which point is an [tex]$x$[/tex]-intercept of the quadratic function [tex]$f(x)=(x-8)(x+9)$[/tex]?
A. [tex]$(0,8)$[/tex]
B. [tex]$(0,-8)$[/tex]
C. [tex]$(9,0)$[/tex]
D. [tex]$(-9,0)$[/tex]
Answer (2)
Set the quadratic function f ( x ) = ( x − 8 ) ( x + 9 ) to zero to find the x -intercept.
Solve the equation ( x − 8 ) ( x + 9 ) = 0 , which gives x = 8 and x = − 9 .
The x -intercepts are the points ( 8 , 0 ) and ( − 9 , 0 ) .
The correct option is ( − 9 , 0 ) , so the final answer is ( − 9 , 0 ) .
Explanation
Understanding the Problem We are given the quadratic function f ( x ) = ( x − 8 ) ( x + 9 ) and asked to find its x -intercept. The x -intercept is the point where the graph of the function intersects the x -axis, which means f ( x ) = 0 .
Setting up the Equation To find the x -intercept, we set f ( x ) = 0 and solve for x :
( x − 8 ) ( x + 9 ) = 0
Solving for x This equation is satisfied if either x − 8 = 0 or x + 9 = 0 . Solving these equations gives us: x − 8 = 0 ⇒ x = 8 x + 9 = 0 ⇒ x = − 9
Identifying the Intercepts Therefore, the x -intercepts are the points ( 8 , 0 ) and ( − 9 , 0 ) . We check the given options to see which one matches.
Finding the Correct Option The point ( − 9 , 0 ) is among the given options.
Examples
Understanding x-intercepts is crucial in many real-world applications. For example, if f ( x ) represents the profit of a company as a function of the number of units sold ( x ), the x-intercepts would represent the break-even points where the company makes zero profit. Similarly, in physics, if f ( x ) represents the height of a projectile as a function of time ( x ), the x-intercepts would represent the times when the projectile hits the ground. Finding x-intercepts helps in analyzing and understanding the behavior of functions in various practical scenarios.
The x-intercepts of the quadratic function f ( x ) = ( x − 8 ) ( x + 9 ) are found by setting the function to zero, leading to the x-intercepts ( 8 , 0 ) and ( − 9 , 0 ) . Among the options given, the point ( − 9 , 0 ) is the correct choice. Thus, the answer is ( − 9 , 0 ) .
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