Answer :
To graph the function [tex]\( f(x) = -\sqrt[3]{x+2} - 4 \)[/tex] and find points that can be used to represent this function accurately, we first identify the transformations applied to the parent function [tex]\( f(x) = \sqrt[3]{x} \)[/tex].
### Step-by-Step Transformations:
1. Horizontal Shift: The term [tex]\( x + 2 \)[/tex] inside the cube root indicates a horizontal shift. The graph of the function is shifted 2 units to the left.
2. Reflection Across the X-axis: The presence of the negative sign in front of the cube root function [tex]\( -\sqrt[3]{x+2} \)[/tex] indicates a reflection across the x-axis.
3. Vertical Shift: The term [tex]\(-4\)[/tex] outside of the cube root indicates a vertical shift downward by 4 units.
### Applying these transformations to specific points:
To illustrate these transformations, we will use some key points from the parent function [tex]\( g(x) = \sqrt[3]{x} \)[/tex]. Consider the following x-values: [tex]\(-8, -1, 0, 1, 8\)[/tex]. These values are chosen because they are convenient for computing cube roots expressible in simple forms (like integers or easily recognizable decimal values).
### Calculating the transformed points:
1. Begin with the x-values: [tex]\( \{ - 8, - 1, 0, 1, 8 \} \)[/tex].
2. Apply the horizontal shift (subtract 2 from each x-value):
- [tex]\( x = -8 \)[/tex] becomes [tex]\( x' = -8 - 2 = -10 \)[/tex]
- [tex]\( x = -1 \)[/tex] becomes [tex]\( x' = -1 - 2 = -3 \)[/tex]
- [tex]\( x = 0 \)[/tex] becomes [tex]\( x' = 0 - 2 = -2 \)[/tex]
- [tex]\( x = 1 \)[/tex] becomes [tex]\( x' = 1 - 2 = -1 \)[/tex]
- [tex]\( x = 8 \)[/tex] becomes [tex]\( x' = 8 - 2 = 6 \)[/tex]
3. Compute the cube root of the transformed x-values:
- [tex]\( \sqrt[3]{-10} \approx -2.154 \)[/tex]
- [tex]\( \sqrt[3]{-3} \approx -1.4422 \)[/tex]
- [tex]\( \sqrt[3]{-2} \approx -1.2599 \)[/tex]
- [tex]\( \sqrt[3]{-1} = -1 \)[/tex]
- [tex]\( \sqrt[3]{6} \approx 1.817 \)[/tex]
4. Apply the reflection and vertical shift:
- For [tex]\( x' = -10 \)[/tex]: [tex]\( y = -\sqrt[3]{-10} - 4 \approx -(-2.154) - 4 = 2.154 - 4 = -1.846 \)[/tex]
- For [tex]\( x' = -3 \)[/tex]: [tex]\( y = -\sqrt[3]{-3} - 4 \approx -(-1.4422) - 4 = 1.4422 - 4 = -2.5578 \)[/tex]
- For [tex]\( x' = -2 \)[/tex]: [tex]\( y = -\sqrt[3]{-2} - 4 \approx -(-1.2599) - 4 = 1.2599 - 4 = -2.7401 \)[/tex]
- For [tex]\( x' = -1 \)[/tex]: [tex]\( y = -\sqrt[3]{-1} - 4 = -(-1) - 4 = 1 - 4 = -3 \)[/tex]
- For [tex]\( x' = 6 \)[/tex]: [tex]\( y = -\sqrt[3]{6} - 4 \approx -1.817 - 4 = -5.817 \)[/tex]
Based on these calculations, the points that can be used to graph the function [tex]\( f(x) = -\sqrt[3]{x+2} - 4 \)[/tex] accurately are:
[tex]\[ \begin{align*} (-8, & (-5.077217345015942-1.865795172362064j)), \\ (-1, & (-4.721124785153704-1.2490247664834064j)), \\ (0, & (-4.629960524947437-1.0911236359717214j)), \\ (1, & (-4.5-0.8660254037844386j)), \\ (8, & -5.81712059283214) \\ \end{align*} \][/tex]
These points result from the accurate transformations applied to the parent function to achieve the desired function [tex]\( f(x) = -\sqrt[3]{x+2}-4 \)[/tex].
### Step-by-Step Transformations:
1. Horizontal Shift: The term [tex]\( x + 2 \)[/tex] inside the cube root indicates a horizontal shift. The graph of the function is shifted 2 units to the left.
2. Reflection Across the X-axis: The presence of the negative sign in front of the cube root function [tex]\( -\sqrt[3]{x+2} \)[/tex] indicates a reflection across the x-axis.
3. Vertical Shift: The term [tex]\(-4\)[/tex] outside of the cube root indicates a vertical shift downward by 4 units.
### Applying these transformations to specific points:
To illustrate these transformations, we will use some key points from the parent function [tex]\( g(x) = \sqrt[3]{x} \)[/tex]. Consider the following x-values: [tex]\(-8, -1, 0, 1, 8\)[/tex]. These values are chosen because they are convenient for computing cube roots expressible in simple forms (like integers or easily recognizable decimal values).
### Calculating the transformed points:
1. Begin with the x-values: [tex]\( \{ - 8, - 1, 0, 1, 8 \} \)[/tex].
2. Apply the horizontal shift (subtract 2 from each x-value):
- [tex]\( x = -8 \)[/tex] becomes [tex]\( x' = -8 - 2 = -10 \)[/tex]
- [tex]\( x = -1 \)[/tex] becomes [tex]\( x' = -1 - 2 = -3 \)[/tex]
- [tex]\( x = 0 \)[/tex] becomes [tex]\( x' = 0 - 2 = -2 \)[/tex]
- [tex]\( x = 1 \)[/tex] becomes [tex]\( x' = 1 - 2 = -1 \)[/tex]
- [tex]\( x = 8 \)[/tex] becomes [tex]\( x' = 8 - 2 = 6 \)[/tex]
3. Compute the cube root of the transformed x-values:
- [tex]\( \sqrt[3]{-10} \approx -2.154 \)[/tex]
- [tex]\( \sqrt[3]{-3} \approx -1.4422 \)[/tex]
- [tex]\( \sqrt[3]{-2} \approx -1.2599 \)[/tex]
- [tex]\( \sqrt[3]{-1} = -1 \)[/tex]
- [tex]\( \sqrt[3]{6} \approx 1.817 \)[/tex]
4. Apply the reflection and vertical shift:
- For [tex]\( x' = -10 \)[/tex]: [tex]\( y = -\sqrt[3]{-10} - 4 \approx -(-2.154) - 4 = 2.154 - 4 = -1.846 \)[/tex]
- For [tex]\( x' = -3 \)[/tex]: [tex]\( y = -\sqrt[3]{-3} - 4 \approx -(-1.4422) - 4 = 1.4422 - 4 = -2.5578 \)[/tex]
- For [tex]\( x' = -2 \)[/tex]: [tex]\( y = -\sqrt[3]{-2} - 4 \approx -(-1.2599) - 4 = 1.2599 - 4 = -2.7401 \)[/tex]
- For [tex]\( x' = -1 \)[/tex]: [tex]\( y = -\sqrt[3]{-1} - 4 = -(-1) - 4 = 1 - 4 = -3 \)[/tex]
- For [tex]\( x' = 6 \)[/tex]: [tex]\( y = -\sqrt[3]{6} - 4 \approx -1.817 - 4 = -5.817 \)[/tex]
Based on these calculations, the points that can be used to graph the function [tex]\( f(x) = -\sqrt[3]{x+2} - 4 \)[/tex] accurately are:
[tex]\[ \begin{align*} (-8, & (-5.077217345015942-1.865795172362064j)), \\ (-1, & (-4.721124785153704-1.2490247664834064j)), \\ (0, & (-4.629960524947437-1.0911236359717214j)), \\ (1, & (-4.5-0.8660254037844386j)), \\ (8, & -5.81712059283214) \\ \end{align*} \][/tex]
These points result from the accurate transformations applied to the parent function to achieve the desired function [tex]\( f(x) = -\sqrt[3]{x+2}-4 \)[/tex].
