Alright, let's analyze the quadratic function [tex]\( y = -2(x-2)^2 + 5 \)[/tex] step-by-step.
### Step 1: Identify the Form of the Quadratic Function
This function is given in the vertex form:
[tex]\[ y = a(x-h)^2 + k \][/tex]
where [tex]\((h, k)\)[/tex] is the vertex of the parabola, and [tex]\(a\)[/tex] is the coefficient that determines the direction and the width of the parabola.
### Step 2: Determine the Vertex
From the given function [tex]\( y = -2(x-2)^2 + 5 \)[/tex], we can identify that:
- [tex]\( a = -2 \)[/tex]
- [tex]\( h = 2 \)[/tex]
- [tex]\( k = 5 \)[/tex]
In general, the vertex form [tex]\( y = -2(x-2)^2 + 5 \)[/tex] directly tells us that the vertex of the parabola is [tex]\((h, k) = (2, 5)\)[/tex].
### Step 3: Assess the Direction of the Parabola
The coefficient [tex]\( a = -2 \)[/tex] is negative, which means the parabola opens downwards. The negative sign indicates that it is an inverted parabola.
### Step 4: Locate the Axis of Symmetry
The axis of symmetry of a parabola in vertex form [tex]\( y = a(x-h)^2 + k \)[/tex] is a vertical line that passes through the vertex. It is given by the equation:
[tex]\[ x = h \][/tex]
For our function, since [tex]\( h = 2 \)[/tex], the axis of symmetry is:
[tex]\[ x = 2 \][/tex]
### Step 5: Write the Final Analysis
Putting all the information together:
1. Vertex: The vertex of the parabola is [tex]\((2, 5)\)[/tex].
2. Axis of Symmetry: The axis of symmetry is the vertical line [tex]\( x = 2 \)[/tex].
3. Direction of Opening: The parabola opens downwards because the coefficient [tex]\( a \)[/tex] is negative.
### Conclusion
For the quadratic function [tex]\( y = -2(x-2)^2 + 5 \)[/tex]:
- The vertex is [tex]\((2, 5)\)[/tex].
- The axis of symmetry is [tex]\( x = 2 \)[/tex].
- The function describes a parabola that opens downwards.
So the final results are:
[tex]\[ y = -2(x-2)^2 + 5 \][/tex]
[tex]\[
\text{Vertex: } (2, 5)
\][/tex]
[tex]\[
\text{Axis of Symmetry: } x = 2
\][/tex]
