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In order to understand how the parent function [tex]\( f(x) = x^3 \)[/tex] is transformed into [tex]\( g(x) = x^3 + 2 \)[/tex] and [tex]\( h(x) = (x + 2)^3 \)[/tex], let's break down each transformation and how it affects the graph.
### Parent Function: [tex]\( f(x) = x^3 \)[/tex]
First, let's recall the graph of the parent function [tex]\( f(x) = x^3 \)[/tex]. It is a cubic curve that passes through the origin (0,0). It has a point of inflection at the origin, where the curve changes concavity.
### Transformation 1: [tex]\( g(x) = x^3 + 2 \)[/tex]
This transformation involves adding a constant [tex]\( 2 \)[/tex] to the entire function. This is a vertical shift.
1. Vertical Shift Upwards: By adding 2 to the whole function, every point on the graph of [tex]\( f(x) = x^3 \)[/tex] is shifted upwards by 2 units.
- So, the new function [tex]\( g(x) = x^3 + 2 \)[/tex] will have the same shape as [tex]\( f(x) = x^3 \)[/tex], but will be shifted vertically upwards.
- For example, the original point (0,0) on [tex]\( f(x) \)[/tex] will be at (0,2) on [tex]\( g(x) \)[/tex].
### Transformation 2: [tex]\( h(x) = (x + 2)^3 \)[/tex]
This transformation involves adding [tex]\( 2 \)[/tex] inside the function argument. This is a horizontal shift.
2. Horizontal Shift Left: By adding 2 inside the function argument, the graph of [tex]\( f(x) = x^3 \)[/tex] is shifted to the left by 2 units.
- The new function [tex]\( h(x) = (x + 2)^3 \)[/tex] will have the same shape as [tex]\( f(x) = x^3 \)[/tex], but shifted horizontally to the left.
- For example, the original point (0,0) on [tex]\( f(x) \)[/tex] will be at (-2,0) on [tex]\( h(x) \)[/tex].
### Graph Comparison
Let's sketch the graphs to visualize these transformations:
1. Graph of [tex]\( f(x) = x^3 \)[/tex]:
- The base graph is plotted with points like (-2,-8), (-1,-1), (0,0), (1,1), (2,8).
2. Graph of [tex]\( g(x) = x^3 + 2 \)[/tex]:
- Each point on the parent function is shifted upwards by 2 units. For instance:
- (-2, -8) becomes (-2, -6).
- (0, 0) becomes (0, 2).
- (2, 8) becomes (2, 10).
3. Graph of [tex]\( h(x) = (x + 2)^3 \)[/tex]:
- Each point on the parent function is shifted left by 2 units. For instance:
- (-2, -8) becomes (-4, -8).
- (0, 0) becomes (-2, 0).
- (2, 8) becomes (0, 8).
### Sketch of the graphs
```
Graph of f(x) = x³:
5 | .
4 | .
3 | .
2 | .
1 | .
| .
-1 | .
-2 | .
-3 | .
-4 | .
-5 | .
--------------------------------
-5 -4 -3 -2 -1 0 1 2 3 4 5
Graph of g(x) = x³ + 2:
7 |
6 | .
5 | .
4 | .
3 | .
2 | .
1 |
|
-1 |
-2 |
-3 |
-4 |
--------------------------------
-5 -4 -3 -2 -1 0 1 2 3 4 5
Graph of h(x) = (x + 2)³:
3 |
2 |
1 |
| .
-1 | .
-2 | .
-3 | .
-4 | .
-5 | .
-6 | .
-7 | .
-8 |
--------------------------------
-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7
Note: The values on the y-axis are approximations to indicate the shifts visually.
```
### Conclusion
- [tex]\( g(x) = x^3 + 2 \)[/tex] is obtained by shifting the parent function [tex]\( f(x) = x^3 \)[/tex] upwards by 2 units.
- [tex]\( h(x) = (x + 2)^3 \)[/tex] is obtained by shifting the parent function [tex]\( f(x) = x^3 \)[/tex] left by 2 units.
These visual and step-by-step transformations demonstrate the effects on the graph of the cubic function when a constant is added either vertically (to the function) or horizontally (to the variable inside the function).
### Parent Function: [tex]\( f(x) = x^3 \)[/tex]
First, let's recall the graph of the parent function [tex]\( f(x) = x^3 \)[/tex]. It is a cubic curve that passes through the origin (0,0). It has a point of inflection at the origin, where the curve changes concavity.
### Transformation 1: [tex]\( g(x) = x^3 + 2 \)[/tex]
This transformation involves adding a constant [tex]\( 2 \)[/tex] to the entire function. This is a vertical shift.
1. Vertical Shift Upwards: By adding 2 to the whole function, every point on the graph of [tex]\( f(x) = x^3 \)[/tex] is shifted upwards by 2 units.
- So, the new function [tex]\( g(x) = x^3 + 2 \)[/tex] will have the same shape as [tex]\( f(x) = x^3 \)[/tex], but will be shifted vertically upwards.
- For example, the original point (0,0) on [tex]\( f(x) \)[/tex] will be at (0,2) on [tex]\( g(x) \)[/tex].
### Transformation 2: [tex]\( h(x) = (x + 2)^3 \)[/tex]
This transformation involves adding [tex]\( 2 \)[/tex] inside the function argument. This is a horizontal shift.
2. Horizontal Shift Left: By adding 2 inside the function argument, the graph of [tex]\( f(x) = x^3 \)[/tex] is shifted to the left by 2 units.
- The new function [tex]\( h(x) = (x + 2)^3 \)[/tex] will have the same shape as [tex]\( f(x) = x^3 \)[/tex], but shifted horizontally to the left.
- For example, the original point (0,0) on [tex]\( f(x) \)[/tex] will be at (-2,0) on [tex]\( h(x) \)[/tex].
### Graph Comparison
Let's sketch the graphs to visualize these transformations:
1. Graph of [tex]\( f(x) = x^3 \)[/tex]:
- The base graph is plotted with points like (-2,-8), (-1,-1), (0,0), (1,1), (2,8).
2. Graph of [tex]\( g(x) = x^3 + 2 \)[/tex]:
- Each point on the parent function is shifted upwards by 2 units. For instance:
- (-2, -8) becomes (-2, -6).
- (0, 0) becomes (0, 2).
- (2, 8) becomes (2, 10).
3. Graph of [tex]\( h(x) = (x + 2)^3 \)[/tex]:
- Each point on the parent function is shifted left by 2 units. For instance:
- (-2, -8) becomes (-4, -8).
- (0, 0) becomes (-2, 0).
- (2, 8) becomes (0, 8).
### Sketch of the graphs
```
Graph of f(x) = x³:
5 | .
4 | .
3 | .
2 | .
1 | .
| .
-1 | .
-2 | .
-3 | .
-4 | .
-5 | .
--------------------------------
-5 -4 -3 -2 -1 0 1 2 3 4 5
Graph of g(x) = x³ + 2:
7 |
6 | .
5 | .
4 | .
3 | .
2 | .
1 |
|
-1 |
-2 |
-3 |
-4 |
--------------------------------
-5 -4 -3 -2 -1 0 1 2 3 4 5
Graph of h(x) = (x + 2)³:
3 |
2 |
1 |
| .
-1 | .
-2 | .
-3 | .
-4 | .
-5 | .
-6 | .
-7 | .
-8 |
--------------------------------
-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7
Note: The values on the y-axis are approximations to indicate the shifts visually.
```
### Conclusion
- [tex]\( g(x) = x^3 + 2 \)[/tex] is obtained by shifting the parent function [tex]\( f(x) = x^3 \)[/tex] upwards by 2 units.
- [tex]\( h(x) = (x + 2)^3 \)[/tex] is obtained by shifting the parent function [tex]\( f(x) = x^3 \)[/tex] left by 2 units.
These visual and step-by-step transformations demonstrate the effects on the graph of the cubic function when a constant is added either vertically (to the function) or horizontally (to the variable inside the function).
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