If f(x) = a|x - 3|, what is the effect on the graph of f(x) as a increases?

The function \( f(x) = a|x - 3| \) represents a vertically scaled absolute value function with a horizontal shift.

The effect on the graph of \( f(x) \) as \( a \) increases is as follows:

1. **Vertical Stretch/Compression**: The coefficient \( a \) affects the vertical stretch or compression of the graph.
   - If \( a > 1 \), the graph is **stretched vertically**, making it narrower because the values of \( f(x) \) increase faster as \( x \) moves away from \( x = 3 \).
   - If \( 0 < a < 1 \), the graph is **compressed vertically**, making it wider because the values of \( f(x) \) increase more slowly as \( x \) moves away from \( x = 3 \).
   
2. **Reflection**: If \( a \) is negative, the graph is reflected across the x-axis.

Since the absolute value function is V-shaped, increasing \( a \) makes the graph steeper.

Would you like more details or have any questions?

Here are 5 related questions for further understanding:
1. How does the graph change if \( a \) is negative?
2. What happens to the vertex of the graph as \( a \) changes?
3. How would the graph of \( f(x) = a|x + 3| \) differ from \( f(x) = a|x - 3| \)?
4. How does the value of \( a \) affect the domain and range of \( f(x) \)?
5. Can the graph ever intersect the x-axis for different values of \( a \)?

**Tip**: When analyzing the effect of a coefficient on a function's graph, think of how it affects the "steepness" or "flatness" of the curve.

Discover how increasing the coefficient 'a' in the function f(x) = a|x - 3| affects the graph. Learn about vertical stretching, compression, and reflections, essential concepts in algebra and graph transformations for students in Grades 9-12.

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