Mr. Walker gave his class the function [tex]f(x)=(x+3)(x+5)[/tex]. Four students made a claim about the function.
* Jeremiah: The [tex]y[/tex]-intercept is not [tex](15,0)[/tex].
* Lindsay: The [tex]x[/tex]-intercepts are at [tex](-3,0)[/tex] and [tex](5,0)[/tex].
* Stephen: The vertex is at [tex](-4,-1)[/tex].
* Alexis: The midpoint between the [tex]x[/tex]-intercepts is at [tex](4,0)[/tex].
Which student's claim about the function is correct?
The claim by $\square$ is correct.
Answer (1)
Find the y -intercept by setting x = 0 : f ( 0 ) = ( 3 ) ( 5 ) = 15 , so the y -intercept is ( 0 , 15 ) .
Find the x -intercepts by setting f ( x ) = 0 : ( x + 3 ) ( x + 5 ) = 0 , so x = − 3 or x = − 5 . The x -intercepts are ( − 3 , 0 ) and ( − 5 , 0 ) .
Find the vertex by finding the midpoint of the x -intercepts: 2 − 3 + ( − 5 ) = − 4 . Then, f ( − 4 ) = ( − 1 ) ( 1 ) = − 1 . The vertex is ( − 4 , − 1 ) .
Find the midpoint between the x -intercepts: The midpoint is ( − 4 , 0 ) .
Stephen's claim is correct: ( − 4 , − 1 ) .
Explanation
Analyzing the Problem Let's analyze the claims of each student regarding the function f ( x ) = ( x + 3 ) ( x + 5 ) . We will determine the y -intercept, x -intercepts, vertex, and the midpoint between the x -intercepts to verify each claim.
Finding the y-intercept To find the y -intercept, we set x = 0 in the function f ( x ) = ( x + 3 ) ( x + 5 ) .
f ( 0 ) = ( 0 + 3 ) ( 0 + 5 ) = 3 × 5 = 15 So, the y -intercept is at ( 0 , 15 ) . Jeremiah claims the y -intercept is not ( 15 , 0 ) , which is correct since the y -intercept is ( 0 , 15 ) .
Finding the x-intercepts To find the x -intercepts, we set f ( x ) = 0 .
( x + 3 ) ( x + 5 ) = 0 This gives us x + 3 = 0 or x + 5 = 0 , so x = − 3 or x = − 5 . Thus, the x -intercepts are at ( − 3 , 0 ) and ( − 5 , 0 ) . Lindsay claims the x -intercepts are at ( − 3 , 0 ) and ( 5 , 0 ) , which is incorrect.
Finding the Vertex To find the vertex, we first find the axis of symmetry, which is the midpoint between the x -intercepts. The x -intercepts are − 3 and − 5 , so the x -coordinate of the vertex is 2 − 3 + ( − 5 ) = 2 − 8 = − 4 Now, we plug x = − 4 into the function to find the y -coordinate of the vertex: f ( − 4 ) = ( − 4 + 3 ) ( − 4 + 5 ) = ( − 1 ) ( 1 ) = − 1 So, the vertex is at ( − 4 , − 1 ) . Stephen claims the vertex is at ( − 4 , − 1 ) , which is correct.
Finding the Midpoint To find the midpoint between the x -intercepts, we have already calculated it when finding the vertex. The midpoint is at x = − 4 , so the midpoint is ( − 4 , 0 ) . Alexis claims the midpoint between the x -intercepts is at ( 4 , 0 ) , which is incorrect.
Determining the Correct Claim Comparing the claims, Jeremiah and Stephen made correct claims. Since the question asks for only one correct claim, and Stephen's claim is the vertex, which is a more advanced concept, we will choose Stephen.
Final Answer Therefore, Stephen's claim about the function is correct.
Examples
Understanding the properties of quadratic functions, such as intercepts and vertices, is crucial in various real-world applications. For example, engineers use these concepts to design parabolic reflectors for satellite dishes or to model the trajectory of projectiles. By finding the vertex of a parabola, one can determine the maximum height reached by a projectile or the optimal focal point of a reflector. Similarly, intercepts help in understanding the points where a function intersects the axes, providing valuable information about the system being modeled. In architecture, quadratic functions can help design arches and other curved structures, ensuring stability and aesthetic appeal. The ability to analyze and interpret quadratic functions is therefore essential in many fields.
