To rewrite the given quadratic function [tex]\( f(x) = x^2 - 4x + 49 \)[/tex] from standard form to vertex form, we should follow these steps:
1. Identify the coefficients: In the standard form [tex]\( f(x) = ax^2 + bx + c \)[/tex], we have:
- [tex]\( a = 1 \)[/tex]
- [tex]\( b = -4 \)[/tex]
- [tex]\( c = 49 \)[/tex]
2. Find the vertex coordinates (h, k): The vertex form of a quadratic function is [tex]\( f(x) = a(x - h)^2 + k \)[/tex]. To find the vertex [tex]\((h, k)\)[/tex], we use the formula:
- [tex]\( h = -\frac{b}{2a} \)[/tex]
- [tex]\( k \)[/tex] is found by substituting [tex]\( h \)[/tex] back into the original function.
3. Calculate [tex]\( h \)[/tex]:
[tex]\[
h = -\frac{b}{2a} = -\frac{-4}{2 \cdot 1} = \frac{4}{2} = 2
\][/tex]
4. Calculate [tex]\( k \)[/tex] by substituting [tex]\( h \)[/tex] into the original function [tex]\( f(x) \)[/tex]:
[tex]\[
k = f(h) = f(2) = (2)^2 - 4 \cdot 2 + 49 = 4 - 8 + 49 = 45
\][/tex]
5. Write the vertex form of the function: Now that we have [tex]\( h = 2 \)[/tex] and [tex]\( k = 45 \)[/tex], we can rewrite the quadratic function in vertex form:
[tex]\[
f(x) = a(x - h)^2 + k = 1(x - 2)^2 + 45
\][/tex]
So, the quadratic function [tex]\( f(x) = x^2 - 4x + 49 \)[/tex] in vertex form is:
[tex]\[
f(x) = (x - 2)^2 + 45
\][/tex]
