Math Problem Statement

Describe how the graph of g(x) is related to the graph of the parent function.

g(x)=−3|x|

Solution

The function g(x)=3xg(x) = -3|x| is a transformation of the parent function f(x)=xf(x) = |x|, the absolute value function. Here's how the graph of g(x)g(x) relates to the parent function:

  1. Reflection: The negative sign in front of 3x3|x| reflects the graph across the x-axis. Instead of opening upwards like f(x)=xf(x) = |x|, g(x)g(x) opens downwards.

  2. Vertical Stretch: The coefficient 33 in 3x-3|x| causes a vertical stretch by a factor of 3. This means that for each xx-value, g(x)g(x) is 3 times as far from the x-axis compared to f(x)=xf(x) = |x|, making it steeper.

In summary, the graph of g(x)=3xg(x) = -3|x| is a reflected and vertically stretched version of the parent function f(x)=xf(x) = |x|, opening downward with a steeper slope.

Would you like more details or have any questions?

  1. How would the graph change if the coefficient was -2 instead of -3?
  2. What if the coefficient was positive but greater than 1, like 3x3|x|?
  3. How does the graph change if the absolute value term includes a horizontal shift, such as xh|x - h|?
  4. How would a vertical shift affect g(x)=3xg(x) = -3|x|?
  5. What is the effect of adding a constant outside the absolute value, like g(x)=3x+kg(x) = -3|x| + k?

Tip: Transformations can be applied in sequence, and understanding each transformation's effect can make analyzing complex functions easier.

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Math Problem Analysis

Mathematical Concepts

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Formulas

g(x) = a|x| where 'a' affects vertical stretches and reflections

Theorems

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Suitable Grade Level

Grades 9-10