To rewrite the quadratic function [tex]\( f(x) = x^2 - 12x + 46 \)[/tex] from standard form to vertex form, follow these steps:
### Step 1: Identify the coefficients
The standard form of a quadratic function is given by [tex]\( f(x) = ax^2 + bx + c \)[/tex], where:
- [tex]\( a = 1 \)[/tex]
- [tex]\( b = -12 \)[/tex]
- [tex]\( c = 46 \)[/tex]
### Step 2: Find the vertex
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. The vertex can be determined using the formulas for [tex]\( h \)[/tex] and [tex]\( k \)[/tex].
#### Finding [tex]\( h \)[/tex]:
[tex]\[ h = -\frac{b}{2a} \][/tex]
Substitute [tex]\( a = 1 \)[/tex] and [tex]\( b = -12 \)[/tex]:
[tex]\[ h = -\frac{-12}{2 \cdot 1} = \frac{12}{2} = 6 \][/tex]
#### Finding [tex]\( k \)[/tex]:
Calculate [tex]\( f(h) \)[/tex] by substituting [tex]\( h = 6 \)[/tex] back into the original function:
[tex]\[ f(6) = (1 \cdot 6^2) + (-12 \cdot 6) + 46 \][/tex]
[tex]\[ f(6) = 36 - 72 + 46 \][/tex]
[tex]\[ f(6) = 10 \][/tex]
So, [tex]\( k = 10 \)[/tex].
### Step 3: Write the vertex form
Now that we have [tex]\( h = 6 \)[/tex] and [tex]\( k = 10 \)[/tex], substitute these values along with [tex]\( a = 1 \)[/tex] into the vertex form:
[tex]\[ f(x) = 1(x - 6)^2 + 10 \][/tex]
Thus, the function [tex]\( f(x) = x^2 - 12x + 46 \)[/tex] rewritten in vertex form is:
[tex]\[ f(x) = (x - 6)^2 + 10 \][/tex]
And that is the desired vertex form of the quadratic function.
