To find the equation of a quadratic function given a vertex and an x-intercept, we can use the vertex form of a quadratic equation and the properties of intercepts. Here's a detailed, step-by-step solution for this problem:
### Step 1: Identify the vertex
The vertex of the quadratic function is given as [tex]\((2, -25)\)[/tex]. The vertex form of a quadratic function is:
[tex]\[ f(x) = a(x - h)^2 + k \][/tex]
where [tex]\((h, k)\)[/tex] is the vertex of the parabola. For this problem, [tex]\(h = 2\)[/tex] and [tex]\(k = -25\)[/tex].
### Step 2: Identify the x-intercept
The x-intercept is given as [tex]\((7, 0)\)[/tex]. This means that [tex]\(f(7) = 0\)[/tex].
### Step 3: Set up the vertex form
Plugging the vertex [tex]\((2, -25)\)[/tex] into the vertex form, we get:
[tex]\[ f(x) = a(x - 2)^2 - 25 \][/tex]
### Step 4: Use the x-intercept to solve for [tex]\(a\)[/tex]
Using the x-intercept [tex]\((7, 0)\)[/tex] into the function [tex]\(f(x)\)[/tex], we obtain:
[tex]\[ 0 = a(7 - 2)^2 - 25 \][/tex]
[tex]\[ 0 = a(5)^2 - 25 \][/tex]
[tex]\[ 0 = 25a - 25 \][/tex]
[tex]\[ 25a = 25 \][/tex]
[tex]\[ a = 1 \][/tex]
### Step 5: Write the quadratic equation in vertex form
Now that we know [tex]\(a = 1\)[/tex], we can write the equation as:
[tex]\[ f(x) = (x - 2)^2 - 25 \][/tex]
### Step 6: Expand the equation to standard form
Alternatively, we can also express the quadratic equation in its standard or factored form. Since we have the x-intercept [tex]\((7, 0)\)[/tex] and recognize that the vertex form will make it easy to factor, we express [tex]\(f(x)\)[/tex] as:
[tex]\[ f(x) = (x - 2)(x - 7) \][/tex]
### Step 7: Verify the solution
The factored form now aligns with the given vertex and x-intercept. The expressions match the properties given in the problem. Thus, the equation of the quadratic function is:
[tex]\[ f(x) = (x - 2)(x - 7) \][/tex]
### Conclusion
The correct option from the ones provided is:
[tex]\[ f(x) = (x - 2)(x - 7) \][/tex]
So, the correct answer is:
[tex]\[ \boxed{f(x) = (x - 2)(x - 7)} \][/tex]
