To convert the quadratic function [tex]\( f(x) = -0.5 x^2 + 6 x - 10 \)[/tex] into vertex form, follow these steps:
### Step 1: Identify the coefficients
The given quadratic function is:
[tex]\[ f(x) = -0.5 x^2 + 6 x - 10 \][/tex]
Here, the coefficients are:
- [tex]\( a = -0.5 \)[/tex]
- [tex]\( b = 6 \)[/tex]
- [tex]\( c = -10 \)[/tex]
### Step 2: Find the vertex (h, k)
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 formula to find [tex]\( h \)[/tex] (the x-coordinate of the vertex) is:
[tex]\[ h = -\frac{b}{2a} \][/tex]
Substitute [tex]\( a = -0.5 \)[/tex] and [tex]\( b = 6 \)[/tex] into the formula:
[tex]\[ h = -\frac{6}{2(-0.5)} = -\frac{6}{-1} = 6 \][/tex]
Now, we need to find [tex]\( k \)[/tex] (the y-coordinate of the vertex) by substituting [tex]\( h \)[/tex] back into the original function:
[tex]\[ k = f(h) = f(6) \][/tex]
Calculate [tex]\( f(6) \)[/tex]:
[tex]\[ f(6) = -0.5(6)^2 + 6(6) - 10 \][/tex]
[tex]\[ f(6) = -0.5(36) + 36 - 10 \][/tex]
[tex]\[ f(6) = -18 + 36 - 10 \][/tex]
[tex]\[ f(6) = 8 \][/tex]
So, [tex]\( k = 8 \)[/tex].
The vertex is [tex]\((h, k) = (6, 8)\)[/tex].
### Step 3: Write the function in vertex form
Using the vertex [tex]\((6, 8)\)[/tex] and the coefficient [tex]\( a = -0.5 \)[/tex], the vertex form of the function is:
[tex]\[ f(x) = -0.5(x - 6)^2 + 8 \][/tex]
Hence, the quadratic function [tex]\( f(x) = -0.5 x^2 + 6 x - 10 \)[/tex] written in vertex form is:
[tex]\[ f(x) = -0.5(x - 6)^2 + 8 \][/tex]
