What is the equation of the quadratic function with a vertex at $(2,-25)$ and an $x$-intercept at $(7,0)$?
A. $f(x)=(x-2)(x-7)$
B. $f(x)=(x+2)(x+7)$
C. $f(x)=(x-3)(x+7)$
D. $f(x)=(x+3)(x-7)$
Answer (2)
Express the quadratic function in vertex form using the given vertex ( 2 , − 25 ) : f ( x ) = a ( x − 2 ) 2 − 25 .
Use the x -intercept ( 7 , 0 ) to solve for a : 0 = a ( 7 − 2 ) 2 − 25 , which gives a = 1 .
Substitute a = 1 back into the vertex form: f ( x ) = ( x − 2 ) 2 − 25 .
Expand and factor the equation to get the quadratic function: f ( x ) = ( x − 7 ) ( x + 3 ) . The final answer is f ( x ) = ( x + 3 ) ( x − 7 ) .
Explanation
Problem Analysis We are given the vertex ( 2 , − 25 ) and an x -intercept ( 7 , 0 ) of a quadratic function. Our goal is to find the equation of this quadratic function. We can start by expressing the quadratic function in vertex form.
Vertex Form The vertex form of a quadratic function is given by f ( x ) = a ( x − h ) 2 + k , where ( h , k ) is the vertex. In our case, the vertex is ( 2 , − 25 ) , so we have h = 2 and k = − 25 . Substituting these values into the vertex form, we get: f ( x ) = a ( x − 2 ) 2 − 25
Finding the Value of a Now we need to find the value of a . We know that the x -intercept is ( 7 , 0 ) , which means that when x = 7 , f ( x ) = 0 . We can substitute these values into the equation: 0 = a ( 7 − 2 ) 2 − 25
Solving for a Now, let's solve for a : 0 = a ( 5 ) 2 − 25
0 = 25 a − 25
25 a = 25
a = 1
Substituting a Now that we have the value of a , we can substitute it back into the vertex form: f ( x ) = 1 ( x − 2 ) 2 − 25
f ( x ) = ( x − 2 ) 2 − 25
Expanding the Equation Let's expand the equation to get the quadratic function in standard form: f ( x ) = ( x 2 − 4 x + 4 ) − 25
f ( x ) = x 2 − 4 x − 21
Factoring the Quadratic Now, let's factor the quadratic to find the x -intercepts: f ( x ) = ( x − 7 ) ( x + 3 )
Final Answer The equation of the quadratic function is f ( x ) = ( x − 7 ) ( x + 3 ) . Therefore, the correct answer is f ( x ) = ( x + 3 ) ( x − 7 ) .
Examples
Understanding quadratic functions is crucial in various real-world applications. For instance, engineers use quadratic equations to model the trajectory of projectiles, such as rockets or balls. Architects apply quadratic functions to design arches and suspension bridges, ensuring structural stability and aesthetic appeal. Economists also utilize quadratic functions to analyze cost and revenue curves, helping businesses optimize production and pricing strategies. By mastering quadratic functions, students gain valuable tools for solving practical problems in diverse fields.
The equation of the quadratic function with a vertex at (2,-25) and an x-intercept at (7,0) can be represented as f(x) = (x-7)(x+3). Therefore, the correct answer is option D. f(x) = (x+3)(x-7).
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