Question Statement

Draw the graph of the function $y=x^3$ and then sketch the graphs of the following functions using translations. Verify the results using a graphical calculator:

(a) $y=x^3+1$ (b) $y=x^3-1$ (c) $y=(x-1{)}^3$ (d) $y=(x+1{)}^3$

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Background and Explanation

To sketch these graphs, we use the concept of function transformations. A vertical translation $y=f(x)+k$ shifts the graph up or down, while a horizontal translation $y=f(x-h)$ shifts the graph left or right.

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Solution

To begin, we calculate the coordinates for the parent function $y=x^3$ and the transformed functions to see how the points shift.

Part (a) $y=x^3+1$

This function is in the form $y=f(x)+k$, where $k=1$. This represents a vertical translation 1 unit upward. Every $y$-coordinate of the parent function $y=x^3$ is increased by 1. For example, the origin $(0,0)$ moves to $(0,1)$.

Part (b) $y=x^3-1$

This function is in the form $y=f(x)+k$, where $k=-1$. This represents a vertical translation 1 unit downward. Every $y$-coordinate of the parent function $y=x^3$ is decreased by 1. For example, the origin $(0,0)$ moves to $(0,-1)$.

Part (c) $y=(x-1{)}^3$

This function is in the form $y=f(x-h)$, where $h=1$. This represents a horizontal translation 1 unit to the right. Note that the graph moves in the opposite direction of the sign inside the parentheses. Every $x$-coordinate is shifted right by 1. For example, the point $(1,1)$ on the original graph corresponds to $(2,1)$ on this graph.

Part (d) $y=(x+1{)}^3$

This function is in the form $y=f(x-h)$, where $h=-1$. This represents a horizontal translation 1 unit to the left. Every $x$-coordinate of the parent function is shifted left by 1. For example, the origin $(0,0)$ moves to $(-1,0)$.

Graphical Representation

Below are the sketches of the vertical and horizontal translations:

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Key Formulas or Methods Used

• Parent Function: $f(x)=x^3$ (Cubic function).
• Vertical Translation: $y=f(x)+k$
  • If $k>0$, shift up.
  • If $k<0$, shift down.
• Horizontal Translation: $y=f(x-h)$
  • If $h>0$ (e.g., $(x-1)$), shift right.
  • If $h<0$ (e.g., $(x+1)$), shift left.

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Summary of Steps

1. Plot the base graph: Create a table of values for $y=x^3$ and draw the characteristic "S" shape passing through $(0,0)$, $(1,1)$, and $(-1,-1)$.
2. Identify Vertical Shifts: For $y=x^3\pm k$, move the entire curve up or down by $k$ units.
3. Identify Horizontal Shifts: For $y=(x\pm h{)}^3$, move the entire curve left or right. Remember that $(x-1)$ moves right and $(x+1)$ moves left.
4. Verify Points: Check specific points (like the inflection point at the origin) to ensure the translation is applied correctly.