Question Statement

Draw the graphs of the function $y=\sqrt{x}$ and then sketch the graphs of the following functions using translation. Verify the results using a graphical calculator:

(a) $y=\sqrt{x+3}$ (b) $y=\sqrt{x-3}$ (c) $y=\sqrt{x}+3$ (d) $y=\sqrt{x}-3$

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Background and Explanation

This problem explores function transformations, specifically translations. Adding or subtracting a constant inside the function argument, $f(x\pm c)$, results in a horizontal shift, while adding or subtracting a constant outside the function, $f(x)\pm c$, results in a vertical shift.

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Solution

To graph these functions, we first establish a table of values for the parent function $y=\sqrt{x}$ and the transformed functions. Note that the square root is only defined for non-negative values ($x\ge 0$ for the parent function).

Table of Values

Part (a): $y=\sqrt{x+3}$

This function represents a horizontal translation. Since the constant $+3$ is inside the square root (added to $x$), the graph of $y=\sqrt{x}$ is shifted 3 units to the left. The domain starts at $x=-3$.

Part (b): $y=\sqrt{x-3}$

This function also represents a horizontal translation. Since the constant $3$ is subtracted from $x$ inside the square root, the graph of $y=\sqrt{x}$ is shifted 3 units to the right. The domain starts at $x=3$.

Part (c): $y=\sqrt{x}+3$

This function represents a vertical translation. Since the constant $+3$ is outside the square root, the $y$-values of the parent function are increased by 3. This shifts the graph 3 units up.

Part (d): $y=\sqrt{x}-3$

This function represents a vertical translation. Since the constant $3$ is subtracted outside the square root, the $y$-values of the parent function are decreased by 3. This shifts the graph 3 units down.

Graphical Representation

The following images illustrate the translations described above:

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Key Formulas or Methods Used

• Horizontal Translation: For $y=f(x+h)$, the graph shifts left by $h$ units if $h>0$, and right if $h<0$.
• Vertical Translation: For $y=f(x)+k$, the graph shifts up by $k$ units if $k>0$, and down if $k<0$.
• Parent Function: $y=\sqrt{x}$, which has a starting point at $(0,0)$ and exists for $x\ge 0$.

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Summary of Steps

1. Plot the parent function: Calculate and plot points for $y=\sqrt{x}$ starting from $x=0$.
2. Identify Horizontal Shifts: For (a) and (b), adjust the $x$-coordinates of the parent points (left for $+3$, right for $-3$).
3. Identify Vertical Shifts: For (c) and (d), adjust the $y$-coordinates of the parent points (up for $+3$, down for $-3$).
4. Sketch and Verify: Draw the smooth curves through the new points and use a graphical calculator to check the accuracy of the translations.