Math Problem Statement

The graph of y = g(x) is shown. Draw the graph of y = 2g(x) + 4.

Solution

The problem asks you to draw the graph of y=2g(x)+4y = 2g(x) + 4 based on the given graph of y=g(x)y = g(x).

To help you adjust the graph correctly, here’s how to modify g(x)g(x):

  1. Vertical Stretch by a Factor of 2:

    • Since the equation includes 2g(x)2g(x), the graph of g(x)g(x) will be stretched vertically by a factor of 2. This means that all points on the graph will be twice as far from the x-axis. For example, if g(x)=1g(x) = 1, then 2g(x)=22g(x) = 2; if g(x)=2g(x) = -2, then 2g(x)=42g(x) = -4.
  2. Vertical Shift by 4 Units Up:

    • The "+4" in the equation shifts the graph 4 units upward. So, after applying the vertical stretch, you'll move all points up by 4 units.

Steps to draw y=2g(x)+4y = 2g(x) + 4:

  • First, take each point on the graph of g(x)g(x) and multiply the y-coordinate by 2.
  • Then, shift each resulting point up by 4 units on the y-axis.

Would you like more details or have any questions? Here are five related questions that might help you:

  1. How does a vertical stretch change the graph's appearance compared to a vertical compression?
  2. What would happen if the function had a negative sign in front of g(x)g(x), like g(x)-g(x)?
  3. How does shifting a graph horizontally differ from shifting it vertically?
  4. Can you explain how to combine multiple transformations like scaling and shifting on a graph?
  5. What happens if both the x- and y-coordinates are transformed simultaneously?

Tip: Always apply stretching or shrinking transformations before shifting when working with function graphs.

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Math Problem Analysis

Mathematical Concepts

Function Transformations
Graphing
Vertical Stretch
Vertical Shift

Formulas

y = 2g(x) + 4

Theorems

Vertical Stretch
Vertical Shift

Suitable Grade Level

Grades 9-11