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)
Use the vertex form of a quadratic equation: f ( x ) = a ( x − h ) 2 + k , where ( h , k ) is the vertex.
Substitute the given vertex ( 2 , − 25 ) into the vertex form: 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 and expand: f ( x ) = ( x − 2 ) 2 − 25 = x 2 − 4 x − 21 = ( x + 3 ) ( x − 7 ) .
The equation of the quadratic function is f ( x ) = ( x + 3 ) ( x − 7 ) .
Explanation
Understanding the Problem We are given the vertex ( 2 , − 25 ) and an x -intercept ( 7 , 0 ) of a quadratic function, and we need to find the equation of this quadratic function.
Using the Vertex Form The general 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 can write the equation as f ( x ) = a ( x − 2 ) 2 − 25 .
Substituting the x-intercept We know that the x -intercept is ( 7 , 0 ) , which means f ( 7 ) = 0 . We can use this information to find the value of a . Substituting x = 7 into the equation, we get 0 = a ( 7 − 2 ) 2 − 25 .
Solving for a Now we solve for a :
0 = a ( 5 ) 2 − 25
0 = 25 a − 25
25 a = 25
a = 1
Finding the Quadratic Function Now that we have a = 1 , we can write the quadratic function as f ( x ) = 1 ( x − 2 ) 2 − 25 . Expanding this, we get:
f ( x ) = ( x 2 − 4 x + 4 ) − 25
f ( x ) = x 2 − 4 x − 21
Factoring the Quadratic Equation We can factor the quadratic equation to find the x -intercepts:
f ( x ) = x 2 − 4 x − 21 = ( x − 7 ) ( x + 3 )
Final Answer The equation of the quadratic function is f ( x ) = ( x − 7 ) ( x + 3 ) . Comparing this to the given options, we see that 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 use quadratic functions to analyze cost and revenue curves, helping businesses optimize their production and pricing strategies. These examples demonstrate the practical significance of quadratic functions in diverse fields.
The equation of the quadratic function with a vertex at (2,-25) and an x-intercept at (7,0) is derived using the vertex form. After determining the coefficient and expanding, we find that the function factors to (x+3)(x-7). Therefore, the correct choice is option D: f(x) = (x+3)(x-7).
;
