Given the function [tex]\( F(x) = 4x^2 + 2 \)[/tex], let's break down and understand the components and steps involved in evaluating or analyzing this quadratic function.
1. Identify the Function:
The function is a quadratic expression in the form:
[tex]\[
F(x) = 4x^2 + 2
\][/tex]
2. Standard Form of a Quadratic Function:
A quadratic function is generally expressed in the standard form:
[tex]\[
ax^2 + bx + c
\][/tex]
Here, [tex]\( a = 4 \)[/tex], [tex]\( b = 0 \)[/tex], and [tex]\( c = 2 \)[/tex]. This identifies the coefficients and constants associated with the quadratic function.
3. Vertex of the Quadratic Function:
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. For our function [tex]\( F(x) = 4x^2 + 2 \)[/tex]:
- Since there is no [tex]\( x \)[/tex] term (i.e., [tex]\( b = 0 \)[/tex]), the vertex is at [tex]\( x = 0 \)[/tex].
- Plugging [tex]\( x = 0 \)[/tex] into the function, we get the vertex:
[tex]\[
F(0) = 4(0)^2 + 2 = 2
\][/tex]
Thus, the vertex is [tex]\((0, 2)\)[/tex].
4. Y-Intercept:
The y-intercept occurs when [tex]\( x = 0 \)[/tex]. Evaluating the function at [tex]\( x = 0 \)[/tex]:
[tex]\[
F(0) = 2
\][/tex]
So, the y-intercept is [tex]\( (0, 2) \)[/tex].
5. Axis of Symmetry:
The axis of symmetry for a quadratic function [tex]\( ax^2 + bx + c \)[/tex] is given by:
[tex]\[
x = -\frac{b}{2a}
\][/tex]
Since [tex]\( b = 0 \)[/tex], the axis of symmetry is:
[tex]\[
x = -\frac{0}{2(4)} = 0
\][/tex]
6. Direction of the Parabola:
Since the coefficient [tex]\( a = 4 \)[/tex] is positive, the parabola opens upwards.
7. General Shape and Behavior:
- The parabola opens upward.
- The vertex at [tex]\( (0, 2) \)[/tex] is the minimum point of the function.
- The y-intercept is at [tex]\( (0, 2) \)[/tex].
8. Graphical Representation:
- Plot the vertex at [tex]\( (0, 2) \)[/tex].
- Draw the axis of symmetry along [tex]\( x = 0 \)[/tex].
- Sketch the parabola opening upward from the vertex.
By following these steps, you can fully understand and visualize the quadratic function [tex]\( F(x) = 4x^2 + 2 \)[/tex].
