Let's address each part of the problem step-by-step.
### (a) Identify the parent function [tex]\( f \)[/tex].
The parent function [tex]\( f \)[/tex] is given as:
[tex]\[ f(x) = x^2 \][/tex]
### (b) Describe the sequence of transformations from [tex]\( f \)[/tex] to [tex]\( g \)[/tex].
The function [tex]\( g(x) \)[/tex] is given as:
[tex]\[ g(x) = -4 - (x + 1)^2 \][/tex]
Let's break this down step-by-step:
1. Horizontal shift of 1 unit to the left:
- The term [tex]\((x + 1)\)[/tex] inside the squared function indicates a horizontal shift of the function [tex]\( f(x) = x^2 \)[/tex] one unit to the left. The function becomes [tex]\((x + 1)^2\)[/tex].
2. Reflection in the [tex]\( y \)[/tex]-axis:
- There is no reflection in the [tex]\( y \)[/tex]-axis for this transformation as there are no terms of the form [tex]\((-x)\)[/tex] inside the square.
3. Reflection in the [tex]\( x \)[/tex]-axis:
- The negative sign in front of the squared term [tex]\(-(x + 1)^2\)[/tex] indicates a reflection in the [tex]\( x \)[/tex]-axis. So, the function [tex]\( (x + 1)^2 \)[/tex] becomes [tex]\( -(x + 1)^2 \)[/tex].
4. Vertical shift of 4 units downward:
- The term [tex]\(-4\)[/tex] outside of the squared term indicates a vertical shift downward by 4 units. So, the function [tex]\( -(x + 1)^2 \)[/tex] becomes [tex]\( -4 - (x + 1)^2 \)[/tex].
### (c) Sketch the graph of [tex]\( g \)[/tex].
To sketch [tex]\( g(x) = -4 - (x + 1)^2 \)[/tex]:
1. Start with the graph of [tex]\( f(x) = x^2 \)[/tex], which is a parabola opening upwards with its vertex at the origin [tex]\((0, 0)\)[/tex].
2. Shift this parabola 1 unit to the left: The vertex moves to [tex]\((-1, 0)\)[/tex].
3. Reflect this modified parabola in the [tex]\( x \)[/tex]-axis: The parabola will now open downwards with the vertex still at [tex]\((-1, 0)\)[/tex].
4. Finally, shift the entire parabola 4 units downward: The vertex will move to [tex]\((-1, -4)\)[/tex].
Thus, the vertex of the parabola described by [tex]\( g(x) \)[/tex] is at [tex]\((-1, -4)\)[/tex], and the parabola opens downwards.
### (d) Use function notation to write [tex]\( g \)[/tex] in terms of [tex]\( f \)[/tex].
Given [tex]\( f(x) = x^2 \)[/tex]:
[tex]\[ g(x) = -4 - (x + 1)^2 \][/tex]
We need to express [tex]\( g \)[/tex] using [tex]\( f \)[/tex]:
Since [tex]\( f(x) = x^2 \)[/tex], we have [tex]\( f(x + 1) = (x + 1)^2 \)[/tex].
Therefore, we can rewrite [tex]\( g(x) \)[/tex] as:
[tex]\[ g(x) = -4 - f(x + 1) \][/tex]
So the function notation to write [tex]\( g \)[/tex] in terms of [tex]\( f \)[/tex] is:
[tex]\[ g(x) = -4 - f(x + 1) \][/tex]
