To rewrite the quadratic function from standard form to vertex form, we need to use the method of completing the square. The given quadratic function is:
[tex]\[ f(x) = 2x^2 + 8x + 70 \][/tex]
### Step-by-Step Solution:
1. Identify the coefficients: In the standard quadratic form [tex]\(ax^2 + bx + c\)[/tex], identify [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex].
[tex]\[ a = 2 \][/tex]
[tex]\[ b = 8 \][/tex]
[tex]\[ c = 70 \][/tex]
2. Calculate the vertex coordinates (h, k):
- The x-coordinate of the vertex [tex]\( h \)[/tex] can be found using the formula:
[tex]\[ h = \frac{-b}{2a} \][/tex]
Substituting the values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex]:
[tex]\[ h = \frac{-8}{2 \cdot 2} \][/tex]
[tex]\[ h = \frac{-8}{4} \][/tex]
[tex]\[ h = -2 \][/tex]
- To find the y-coordinate [tex]\( k \)[/tex], substitute [tex]\( h \)[/tex] back into the original equation:
[tex]\[ k = f(h) = 2(h^2) + 8h + 70 \][/tex]
Substituting [tex]\( h = -2 \)[/tex]:
[tex]\[ k = 2(-2)^2 + 8(-2) + 70 \][/tex]
[tex]\[ k = 2(4) - 16 + 70 \][/tex]
[tex]\[ k = 8 - 16 + 70 \][/tex]
[tex]\[ k = 62 \][/tex]
The coordinates of the vertex are [tex]\( (-2, 62) \)[/tex].
3. Rewrite the function in vertex form:
- The vertex form of a quadratic function is given by:
[tex]\[ f(x) = a(x - h)^2 + k \][/tex]
- Substitute the values of [tex]\( a \)[/tex], [tex]\( h \)[/tex], and [tex]\( k \)[/tex]:
[tex]\[ f(x) = 2(x - (-2))^2 + 62 \][/tex]
[tex]\[ f(x) = 2(x + 2)^2 + 62 \][/tex]
### Conclusion:
The quadratic function [tex]\( f(x) = 2x^2 + 8x + 70 \)[/tex] in vertex form is:
[tex]\[ f(x) = 2(x + 2)^2 + 62 \][/tex]
