To graph the quadratic function [tex]\( g(x) = -(x + 2)^2 + 5 \)[/tex] by transforming the graph of [tex]\( f(x) = x^2 \)[/tex], we can follow these steps:
1. Start with the basic quadratic function [tex]\( f(x) = x^2 \)[/tex]:
The parent function [tex]\( f(x) = x^2 \)[/tex] is an upward-opening parabola with its vertex at the origin (0,0).
2. Identify the transformations:
The given function [tex]\( g(x) = -(x + 2)^2 + 5 \)[/tex] can be seen as a transformation of [tex]\( f(x) = x^2 \)[/tex]. We can break down the steps of transformation:
- The term [tex]\( x + 2 \)[/tex] indicates a horizontal shift. Specifically, the graph will be shifted left by 2 units. Therefore, the vertex moves from (0, 0) to (-2, 0).
- The negative sign in front of the squared term indicates a vertical reflection over the x-axis, causing the parabola to open downwards instead of upwards.
- The +5 at the end indicates a vertical shift upwards by 5 units. So, the vertex of the parabola will move from (-2, 0) to (-2, 5).
3. Determine the vertex:
From the transformations identified, the vertex of the function [tex]\( g(x) = -(x + 2)^2 + 5 \)[/tex] is located at [tex]\((-2, 5)\)[/tex].
4. Determine the axis of symmetry:
The axis of symmetry is a vertical line that passes through the vertex of the parabola. This will be the x-coordinate of the vertex. For the equation [tex]\( g(x) = -(x + 2)^2 + 5 \)[/tex], the axis of symmetry is the line [tex]\( x = -2 \)[/tex].
5. Determine whether it's a maximum or minimum value:
Since the parabola is opening downwards (as indicated by the negative sign in front of the squared term), the vertex represents the maximum value of the function. Therefore, the maximum value of [tex]\( g(x) \)[/tex] is [tex]\( 5 \)[/tex] at [tex]\( x = -2 \)[/tex].
### Summary:
- Vertex: The vertex of the function [tex]\( g(x) = -(x + 2)^2 + 5 \)[/tex] is at [tex]\((-2, 5)\)[/tex].
- Axis of Symmetry: The axis of symmetry is the vertical line [tex]\( x = -2 \)[/tex].
- Maximum or Minimum Value: The function has a maximum value of 5 at the vertex [tex]\((-2, 5)\)[/tex].
