Sure! Let’s rewrite the quadratic function [tex]\( f(x) = 2x^2 - 20x + 26 \)[/tex] from its standard form to its vertex form.
### Step 1: Identify the coefficients
In the quadratic function [tex]\( f(x) = 2x^2 - 20x + 26 \)[/tex], the coefficients are:
- [tex]\( a = 2 \)[/tex]
- [tex]\( b = -20 \)[/tex]
- [tex]\( c = 26 \)[/tex]
### Step 2: Find the x-coordinate of the vertex [tex]\( h \)[/tex]
The x-coordinate of the vertex, [tex]\( h \)[/tex], is found using the formula:
[tex]\[
h = -\frac{b}{2a}
\][/tex]
Plugging in the values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex]:
[tex]\[
h = -\frac{-20}{2 \cdot 2} = \frac{20}{4} = 5.0
\][/tex]
### Step 3: Find the y-coordinate of the vertex [tex]\( k \)[/tex]
The y-coordinate of the vertex, [tex]\( k \)[/tex], is found by substituting [tex]\( h \)[/tex] back into the original quadratic function:
[tex]\[
k = a(h^2) + b(h) + c
\][/tex]
Substituting [tex]\( h = 5.0 \)[/tex]:
[tex]\[
k = 2(5.0^2) - 20(5.0) + 26
\][/tex]
[tex]\[
k = 2(25.0) - 100.0 + 26
\][/tex]
[tex]\[
k = 50.0 - 100.0 + 26 = -24.0
\][/tex]
### Step 4: Write the vertex form
The vertex form of a quadratic function is given by:
[tex]\[
f(x) = a(x - h)^2 + k
\][/tex]
Substitute [tex]\( a = 2 \)[/tex], [tex]\( h = 5.0 \)[/tex], and [tex]\( k = -24.0 \)[/tex]:
[tex]\[
f(x) = 2(x - 5.0)^2 - 24.0
\][/tex]
So, the quadratic function [tex]\( f(x) = 2x^2 - 20x + 26 \)[/tex] written in vertex form is:
[tex]\[
f(x) = 2(x - 5.0)^2 - 24.0
\][/tex]
