To determine the vertex form of the quadratic function [tex]\( f(x) = 2x^2 + 8x + 3 \)[/tex] and find the maximum or minimum value of the function, we can follow these steps:
1. Identify the standard form of the quadratic function:
[tex]\[
f(x) = ax^2 + bx + c
\][/tex]
Here, [tex]\( a = 2 \)[/tex], [tex]\( b = 8 \)[/tex], and [tex]\( c = 3 \)[/tex].
2. Find the vertex (h, k):
The vertex form of a quadratic function [tex]\( f(x) = a(x-h)^2 + k \)[/tex] represents the quadratic function in a way that makes the vertex [tex]\((h, k)\)[/tex] easily identifiable. The vertex can be found using the formulas:
[tex]\[
h = -\frac{b}{2a}
\][/tex]
[tex]\[
k = f(h) = a(h)^2 + b(h) + c
\][/tex]
Plugging in the given values:
[tex]\[
h = -\frac{8}{2 \cdot 2} = -\frac{8}{4} = -2
\][/tex]
Substitute [tex]\( h = -2 \)[/tex] back into the original function to find [tex]\( k \)[/tex]:
[tex]\[
k = 2(-2)^2 + 8(-2) + 3 = 2(4) - 16 + 3 = 8 - 16 + 3 = -5
\][/tex]
So, the vertex is [tex]\((-2, -5)\)[/tex].
3. Write the function in vertex form:
Using the values of [tex]\( a \)[/tex], [tex]\( h \)[/tex], and [tex]\( k \)[/tex], we can write the vertex form of the function:
[tex]\[
f(x) = a(x-h)^2 + k
\][/tex]
[tex]\[
f(x) = 2(x - (-2))^2 + (-5)
\][/tex]
[tex]\[
f(x) = 2(x + 2)^2 - 5
\][/tex]
Hence, the vertex form of the function is:
[tex]\[
f(x) = 2(x + 2)^2 - 5
\][/tex]
4. Determine if the vertex is a maximum or minimum:
The sign of the coefficient [tex]\( a \)[/tex] in the quadratic function determines whether the parabola opens upwards or downwards:
- If [tex]\( a > 0 \)[/tex], the parabola opens upwards and the vertex is a minimum point.
- If [tex]\( a < 0 \)[/tex], the parabola opens downwards and the vertex is a maximum point.
Since [tex]\( a = 2 \)[/tex] (which is greater than 0), the parabola opens upwards. Therefore, the vertex [tex]\((-2, -5)\)[/tex] is a minimum point.
5. State the minimum value of the function:
The minimum value of the function [tex]\( f(x) \)[/tex] is the [tex]\( y \)[/tex]-value of the vertex, which is [tex]\( k = -5 \)[/tex].
Hence, the detailed solution is:
- The vertex form of the function is:
[tex]\[
f(x) = 2(x + 2)^2 - 5
\][/tex]
- The type of vertex is a minimum point.
- The minimum value of the function is:
[tex]\[
-5
\][/tex]
