Question Statement

Draw the graph of the function $y=x^2$ and then sketch the graphs of the following functions using translation. Verify the results using a graphical calculator:

(a) $y=x^2+4$ (b) $y=x^2-4$ (c) $y=(x-4{)}^2$ (d) $y=(x+4{)}^2$

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Background and Explanation

This problem explores function transformations, specifically translations. A vertical translation occurs when a constant is added to or subtracted from the entire function, while a horizontal translation occurs when a constant is added to or subtracted from the independent variable $x$ before the function is applied.

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Solution

To sketch these graphs, we first establish a table of values for the base function $y=x^2$ and the transformed functions.

Part (a) $y=x^2+4$

This function is in the form $y=f(x)+k$, where $k=4$. Adding 4 to the output of the function results in a vertical translation. Every point on the graph of $y=x^2$ is shifted 4 units upward. For example, the vertex moves from $(0,0)$ to $(0,4)$.

Part (b) $y=x^2-4$

This function is in the form $y=f(x)-k$, where $k=4$. Subtracting 4 from the output results in a vertical translation. Every point on the graph of $y=x^2$ is shifted 4 units downward. The vertex moves from $(0,0)$ to $(0,-4)$.

Part (c) $y=(x-4{)}^2$

This function is in the form $y=f(x-h)$, where $h=4$. When we subtract a value from $x$ inside the function, it results in a horizontal translation. The graph of $y=x^2$ is shifted 4 units to the right. The vertex moves from $(0,0)$ to $(4,0)$.

Part (d) $y=(x+4{)}^2$

This function is in the form $y=f(x+h)$, which can be thought of as $f(x-(-4))$. This results in a horizontal translation. The graph of $y=x^2$ is shifted 4 units to the left. The vertex moves from $(0,0)$ to $(-4,0)$.

Visual Representation

Below are the sketches of the vertical translations (left) and horizontal translations (right):

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Key Formulas or Methods Used

• Vertical Translation: $y=f(x)+k$
  • If $k>0$, shift up $k$ units.
  • If $k<0$, shift down $|k|$ units.
• Horizontal Translation: $y=f(x-h)$
  • If $h>0$ (e.g., $(x-4)$), shift right $h$ units.
  • If $h<0$ (e.g., $(x+4)$), shift left $|h|$ units.
• Base Function: $y=x^2$ (The standard parabola).

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Summary of Steps

1. Plot the Base Graph: Start by plotting the parent function $y=x^2$ using standard points like $(-2,4),(-1,1),(0,0),(1,1),(2,4)$.
2. Apply Vertical Shifts: For $y=x^2\pm k$, move every point on the parent graph up or down by $k$ units.
3. Apply Horizontal Shifts: For $y=(x\pm h{)}^2$, move every point on the parent graph left or right by $h$ units.
4. Identify New Vertices: Note the new position of the vertex $(0,0)$ to ensure the translation is accurate.
5. Verify: Use a table of values or a graphing calculator to confirm the coordinates of the new curves.