Question Statement

Draw the graphs of the given functions and then sketch the graphs of other functions using translation. Verify the results using a graphical calculator:

Base function: $y=|x|$

(a) $y=|x+2|$ (b) $y=|x-2|$ (c) $y=|x|+2$ (d) $y=|x|-2$

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Background and Explanation

The absolute value function $y=|x|$ forms a characteristic "V" shape with its vertex at the origin $(0,0)$. Transformations of this function involve shifting the graph horizontally or vertically: $y=|x\pm h|$ results in a horizontal shift, while $y=|x|\pm k$ results in a vertical shift.

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Solution

To graph these functions, we first calculate a table of values for $x$ ranging from $-4$ to $4$.

Part (a) $y=|x+2|$

The function $y=|x+2|$ represents a horizontal translation of the parent function $y=|x|$. Because the constant $+2$ is inside the absolute value, the graph shifts 2 units to the left. For example, the vertex moves from $(0,0)$ to $(-2,0)$.

Part (b) $y=|x-2|$

The function $y=|x-2|$ represents a horizontal translation. Because the constant $-2$ is inside the absolute value, the graph shifts 2 units to the right. The vertex moves from $(0,0)$ to $(2,0)$.

Part (c) $y=|x|+2$

The function $y=|x|+2$ represents a vertical translation. Because the constant $+2$ is outside the absolute value, the entire graph of $y=|x|$ shifts 2 units upward. The vertex moves from $(0,0)$ to $(0,2)$.

Part (d) $y=|x|-2$

The function $y=|x|-2$ represents a vertical translation. Because the constant $-2$ is outside the absolute value, the entire graph of $y=|x|$ shifts 2 units downward. The vertex moves from $(0,0)$ to $(0,-2)$.

Graphical Representation

Below are the sketches of the translated functions:

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Key Formulas or Methods Used

• Parent Function: $y=|x|$ (V-shape with vertex at origin).
• Horizontal Translation: $y=f(x+h)$ shifts the graph left by $h$ units; $y=f(x-h)$ shifts the graph right by $h$ units.
• Vertical Translation: $y=f(x)+k$ shifts the graph up by $k$ units; $y=f(x)-k$ shifts the graph down by $k$ units.

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Summary of Steps

1. Create a Table of Values: Calculate the $y$-coordinates for the parent function $y=|x|$ and each transformed function across a range of $x$ values.
2. Identify the Transformation: Determine if the change is inside the absolute value (horizontal) or outside (vertical).
3. Plot the Parent Function: Draw the basic V-shape starting at $(0,0)$.
4. Apply Translations: Shift the points of the parent graph according to the identified horizontal or vertical rules.
5. Sketch and Verify: Connect the points to form the new V-shapes and use a graphical calculator to confirm the accuracy of the shifts.