Let's analyze the function [tex]\( y = -2x^2 + 4 \)[/tex] to determine which statements correctly describe its graph.
1. Maximum value of the function:
The given function is a parabola in the form [tex]\( y = ax^2 + bx + c \)[/tex]. Here, [tex]\( a = -2 \)[/tex], [tex]\( b = 0 \)[/tex], and [tex]\( c = 4 \)[/tex]. Since [tex]\( a < 0 \)[/tex], the parabola opens downward, meaning it has a maximum value at its vertex.
The vertex of a parabola [tex]\( y = ax^2 + bx + c \)[/tex] occurs at [tex]\( x = -\frac{b}{2a} \)[/tex]. For this function, since [tex]\( b = 0 \)[/tex], the vertex is at [tex]\( x = 0 \)[/tex]. Plugging [tex]\( x = 0 \)[/tex] into the function, we get:
[tex]\[
y = -2(0)^2 + 4 = 4
\][/tex]
Thus, the maximum value of the function is [tex]\( 4 \)[/tex].
2. Increasing when [tex]\( x > 0 \)[/tex]:
For a parabola in the form [tex]\( y = ax^2 + bx + c \)[/tex] with [tex]\( a < 0 \)[/tex], it is always decreasing on both sides of its vertex. So, for [tex]\( x > 0 \)[/tex], this function is decreasing, not increasing.
Thus, the statement about the function increasing when [tex]\( x > 0 \)[/tex] is incorrect.
3. Intersection with the [tex]\( y \)[/tex]-axis:
The graph of the function intersects the [tex]\( y \)[/tex]-axis where [tex]\( x = 0 \)[/tex]. Plugging [tex]\( x = 0 \)[/tex] into the function, we get:
[tex]\[
y = -2(0)^2 + 4 = 4
\][/tex]
So, it intersects the [tex]\( y \)[/tex]-axis at [tex]\( y = 4 \)[/tex], not at [tex]\( x = \sqrt{2} \)[/tex].
4. Intersections with the [tex]\( x \)[/tex]-axis:
The graph intersects the [tex]\( x \)[/tex]-axis where [tex]\( y = 0 \)[/tex]. Setting [tex]\( y = 0 \)[/tex] and solving for [tex]\( x \)[/tex]:
[tex]\[
0 = -2x^2 + 4 \implies 2x^2 = 4 \implies x^2 = 2 \implies x = \pm \sqrt{2}
\][/tex]
Hence, the graph intersects the [tex]\( x \)[/tex]-axis at [tex]\( x = \sqrt{2} \)[/tex] and [tex]\( x = -\sqrt{2} \)[/tex].
5. Axis of symmetry:
The axis of symmetry for a parabola [tex]\( y = ax^2 + bx + c \)[/tex] is given by [tex]\( x = -\frac{b}{2a} \)[/tex]. Since [tex]\( b = 0 \)[/tex], the axis of symmetry is [tex]\( x = 0 \)[/tex].
After this detailed analysis, the correct statements are:
- The function has a maximum value of [tex]\( 4 \)[/tex].
- The graph of the function intersects the [tex]\( x \)[/tex]-axis at [tex]\( x = \pm \sqrt{2} \)[/tex].
- The graph of the function has an axis of symmetry at [tex]\( x = 0 \)[/tex].
