Graphing and Analyzing Absolute Value Functions

Q1. Graph the function $g(x) = 2|6 - x|$ using shifting, compressing, stretching, and/or reflecting. Start with the graph of the basic function. Show at least three key points. Find the domain and range of the function.

Background

Topic: Transformations of Absolute Value Functions

This question tests your understanding of how to graph absolute value functions and apply transformations such as shifting, stretching, compressing, and reflecting. You are also asked to identify the domain and range.

Key Terms and Formulas

• Absolute Value Function: $f(x) = |x|$

• Vertical Stretch: Multiplying by a constant $a$ ($a > 1$) stretches the graph vertically.

• Horizontal Shift: $|x - h|$ shifts the graph right by $h$ units; $|x + h|$ shifts left by $h$ units.

• Reflection: $| - x |$ reflects the graph across the y-axis.

• Domain: The set of all possible input values ($x$).

• Range: The set of all possible output values ($g(x)$).

Step-by-Step Guidance

1. Start with the basic absolute value function $f(x) = |x|$. This graph is a "V" shape with its vertex at the origin (0,0).

2. Apply the horizontal shift and reflection: $|6 - x|$ can be rewritten as $|-(x - 6)|$, which means the graph is reflected across the y-axis and shifted right by 6 units. The vertex moves to $(6, 0)$.

3. Apply the vertical stretch: Multiply the function by 2, so $g(x) = 2|6 - x|$. This makes the "V" shape steeper, stretching it vertically by a factor of 2.

4. Identify three key points: Try plugging in $x = 6$, $x = 5$, and $x = 7$ to find their corresponding $g(x)$ values. This will help you plot the graph accurately.

5. Consider the domain and range: Think about what values $x$ can take (domain) and what values $g(x)$ can output (range) for this function.

Try solving on your own before revealing the answer!