To find the equation of a quadratic function given a vertex and an x-intercept, we should use the vertex form of a quadratic equation and the given information to determine the parameters of the equation.
1. Identify the given information:
- The vertex of the quadratic function is [tex]\( (2, -25) \)[/tex].
- An x-intercept (where the function intersects the x-axis) is [tex]\( (7, 0) \)[/tex].
2. Start with the vertex form of the quadratic equation:
[tex]\[ f(x) = a(x - h)^2 + k \][/tex]
where [tex]\( (h, k) \)[/tex] is the vertex.
In this case, [tex]\( h = 2 \)[/tex] and [tex]\( k = -25 \)[/tex]. So the equation becomes:
[tex]\[ f(x) = a(x - 2)^2 - 25 \][/tex]
3. Use the x-intercept to find the value of [tex]\( a \)[/tex]:
We know that the function passes through the point [tex]\( (7, 0) \)[/tex]. Thus, we substitute [tex]\( x = 7 \)[/tex] and [tex]\( f(x) = 0 \)[/tex] into the equation:
[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]
Now, substitute [tex]\( a = 1 \)[/tex] back into the vertex form of the equation:
[tex]\[ f(x) = (x - 2)^2 - 25 \][/tex]
4. Expand the vertex form to get the standard form:
Expand the equation [tex]\( f(x) = (x - 2)^2 - 25 \)[/tex]:
[tex]\[ f(x) = (x - 2)(x - 2) - 25 \][/tex]
[tex]\[ f(x) = x^2 - 4x + 4 - 25 \][/tex]
[tex]\[ f(x) = x^2 - 4x - 21 \][/tex]
So, the quadratic function in standard form is:
[tex]\[ f(x) = x^2 - 4x - 21 \][/tex]
5. Compare with the given options:
The given options are:
1. [tex]\( f(x) = (x-2)(x-7) \)[/tex]
2. [tex]\( f(x) = (x+2)(x+7) \)[/tex]
3. [tex]\( f(x) = (x-3)(x+7) \)[/tex]
4. [tex]\( f(x) = (x+3)(x-7) \)[/tex]
None of the given options seem to match the equation [tex]\( f(x) = x^2 - 4x - 21 \)[/tex] directly, as they are written in factored form. However, we can factor our equation to verify if it matches one of the options.
Factor [tex]\( x^2 - 4x - 21 \)[/tex]:
[tex]\[ x^2 - 4x - 21 = (x - 7)(x + 3) \][/tex]
So, in factored form:
[tex]\[ f(x) = (x - 7)(x + 3) \][/tex]
Therefore, the correct answer is:
[tex]\[ f(x) = (x - 7)(x + 3) \][/tex]
So, among the provided options, the correct equation is:
[tex]\[ f(x) = (x + 3)(x - 7) \][/tex]
Hence, the correct option is:
[tex]\[ f(x) = (x + 3)(x - 7) \][/tex]
