What are the features of the quadratic function [tex]$f(x)=(x-3)(x+7)$[/tex]?
A) Vertex [tex]$=(-2,25)$[/tex], y-intercept [tex]$=(0,21)$[/tex], x-intercepts [tex]$=(7,0)$[/tex] and [tex]$(3,0)$[/tex], axis of symmetry is [tex]$x=-2$[/tex]
B) Vertex [tex]$=(-2,-25)$[/tex], y-intercept [tex]$=(0,-21)$[/tex], x-intercepts [tex]$=(-7,0)$[/tex] and [tex]$(3,0)$[/tex], axis of symmetry is [tex]$x=-2$[/tex]
C) Vertex [tex]$=(-2,-25)$[/tex], y-intercept [tex]$=(0,-21)$[/tex], x-intercepts [tex]$=(-7,0)$[/tex] and [tex]$(3,0)$[/tex], axis of symmetry is [tex]$x=2$[/tex]
D) Vertex [tex]$=(2,25)$[/tex], y-intercept [tex]$=(0,21)$[/tex], x-intercepts [tex]$=(7,0)$[/tex] and [tex]$(3,0)$[/tex], axis of symmetry is [tex]$x=2$[/tex]
Answer (1)
Find the x-intercepts by setting f ( x ) = 0 and solving for x , resulting in x = 3 and x = − 7 .
Find the y-intercept by setting x = 0 and evaluating f ( 0 ) , which gives f ( 0 ) = − 21 .
Calculate the vertex by finding the x-coordinate as the average of the x-intercepts, x v = − 2 , and then finding the y-coordinate by evaluating f ( − 2 ) = − 25 .
Determine the axis of symmetry as the vertical line passing through the vertex, x = − 2 . The correct answer is B: Vertex = ( − 2 , − 25 ) , y -intercept = ( 0 , − 21 ) , x -intercepts = ( − 7 , 0 ) and ( 3 , 0 ) , axis of symmetry is x = − 2 . B
Explanation
Understanding the Problem The problem asks us to identify the key features of the quadratic function f ( x ) = ( x − 3 ) ( x + 7 ) . These features include the vertex, y -intercept, x -intercepts, and the axis of symmetry. We will find each of these features and then compare them to the given options to select the correct one.
Finding the x-intercepts First, let's find the x -intercepts. The x -intercepts occur when f ( x ) = 0 . So we have ( x − 3 ) ( x + 7 ) = 0 . This gives us x = 3 or x = − 7 . Thus, the x -intercepts are ( 3 , 0 ) and ( − 7 , 0 ) .
Finding the y-intercept Next, let's find the y -intercept. The y -intercept occurs when x = 0 . So we have f ( 0 ) = ( 0 − 3 ) ( 0 + 7 ) = ( − 3 ) ( 7 ) = − 21 . Thus, the y -intercept is ( 0 , − 21 ) .
Finding the Vertex Now, let's find the vertex. The x -coordinate of the vertex is the average of the x -intercepts: x v = 2 3 + ( − 7 ) = 2 − 4 = − 2 . To find the y -coordinate of the vertex, we evaluate f ( x v ) = f ( − 2 ) = ( − 2 − 3 ) ( − 2 + 7 ) = ( − 5 ) ( 5 ) = − 25 . Thus, the vertex is ( − 2 , − 25 ) .
Finding the Axis of Symmetry The axis of symmetry is a vertical line that passes through the vertex. Since the x -coordinate of the vertex is − 2 , the equation of the axis of symmetry is x = − 2 .
Conclusion Comparing our findings with the given options:
Vertex: ( − 2 , − 25 )
y -intercept: ( 0 , − 21 )
x -intercepts: ( − 7 , 0 ) and ( 3 , 0 )
Axis of symmetry: x = − 2
Option B matches all these features. Therefore, the correct answer is B.
Examples
Understanding the features of a quadratic function, such as its vertex, intercepts, and axis of symmetry, is crucial in various real-world applications. For instance, consider designing a parabolic arch for a bridge. Knowing the vertex helps determine the maximum height of the arch, while the x-intercepts can define the base's width. Engineers use these parameters to ensure structural stability and aesthetic appeal. Similarly, in projectile motion, understanding the quadratic trajectory allows us to calculate the maximum height and range of a projectile, which is vital in sports and military applications. By analyzing the quadratic function, we can optimize designs and predict outcomes in numerous practical scenarios.
