To determine the equation of the quadratic function with a vertex at [tex]\( (2, -25) \)[/tex] and an x-intercept at [tex]\( (7, 0) \)[/tex], we can follow these steps:
### Step 1: Understand the Vertex Form of a Quadratic Function
The vertex form of a quadratic function is given by:
[tex]\[ f(x) = a(x - h)^2 + k \][/tex]
where [tex]\((h, k)\)[/tex] is the vertex of the parabola.
Given the vertex [tex]\((2, -25)\)[/tex], we have:
[tex]\[ h = 2 \][/tex]
[tex]\[ k = -25 \][/tex]
So, the quadratic function in vertex form will start as:
[tex]\[ f(x) = a(x - 2)^2 - 25 \][/tex]
### Step 2: Use the x-Intercept to Find the Coefficient [tex]\(a\)[/tex]
The x-intercept provided is [tex]\( (7, 0) \)[/tex], which means when [tex]\( x = 7 \)[/tex], [tex]\( f(x) = 0 \)[/tex].
Substitute [tex]\( x = 7 \)[/tex] and [tex]\( f(x) = 0 \)[/tex] into the vertex form to find [tex]\( a \)[/tex]:
[tex]\[ 0 = a(7 - 2)^2 - 25 \][/tex]
[tex]\[ 0 = a(5)^2 - 25 \][/tex]
[tex]\[ 0 = 25a - 25 \][/tex]
[tex]\[ 25 = 25a \][/tex]
[tex]\[ a = 1 \][/tex]
Thus, the equation in vertex form becomes:
[tex]\[ f(x) = (x - 2)^2 - 25 \][/tex]
### Step 3: Convert to Standard Form
To convert the function to the standard form, we expand the vertex form and simplify:
[tex]\[ f(x) = (x - 2)^2 - 25 \][/tex]
[tex]\[ f(x) = (x^2 - 4x + 4) - 25 \][/tex]
[tex]\[ f(x) = x^2 - 4x + 4 - 25 \][/tex]
[tex]\[ f(x) = x^2 - 4x - 21 \][/tex]
### Step 4: Identify Intercepts from Factored Form
Since we are asked for the factored form and we notice that the factored form of a quadratic with intercepts [tex]\( x = 2 \)[/tex] and [tex]\( x = 7 \)[/tex] can be written as:
[tex]\[ f(x) = (x - 2)(x - 7) \][/tex]
Given the potential answer choices:
[tex]\[ f(x) = (x-2)(x-7) \][/tex]
[tex]\[ f(x) = (x+2)(x+7) \][/tex]
[tex]\[ f(x) = (x-3)(x+7) \][/tex]
[tex]\[ f(x) = (x+3)(x-7) \][/tex]
The correct equation is clearly:
[tex]\[ f(x) = (x-2)(x-7) \][/tex]
Thus, the equation of the quadratic function is:
[tex]\[ f(x) = (x-2)(x-7) \][/tex]
