Math Problem Statement

Match the equation with its graph: (a) y = f(x - 4), (b) y = f(x) + 3, (c) y = 2f(x + 6), (d) y = -f(2x)

Solution

To solve this problem, we need to match each equation with the corresponding graph, using the transformations of the base function y=f(x)y = f(x). Here are the steps to do that:

  1. Equation (a): y=f(x4)y = f(x - 4)

    • This represents a horizontal shift to the right by 4 units.
    • Look for the graph where the function f(x)f(x) is shifted to the right.
  2. Equation (b): y=f(x)+3y = f(x) + 3

    • This represents a vertical shift upwards by 3 units.
    • Look for the graph where the function is moved upward by 3 units.
  3. Equation (c): y=2f(x+6)y = 2f(x + 6)

    • This represents a horizontal shift to the left by 6 units and a vertical stretch by a factor of 2.
    • Look for the graph that is stretched and shifted to the left.
  4. Equation (d): y=f(2x)y = -f(2x)

    • This represents a reflection over the x-axis and a horizontal compression by a factor of 2.
    • Look for the graph that is flipped upside down and horizontally compressed.

Now, let's analyze the graphs:

  1. Graph (1):

    • This graph appears to be horizontally shifted to the right by 4 units.
    • This corresponds to equation (a): y=f(x4)y = f(x - 4).
  2. Graph (2):

    • This graph is stretched vertically and shifted to the left by 6 units.
    • This corresponds to equation (c): y=2f(x+6)y = 2f(x + 6).
  3. Graph (3):

    • This graph is shifted upwards by 3 units.
    • This corresponds to equation (b): y=f(x)+3y = f(x) + 3.
  4. Graph (4):

    • This graph is reflected over the x-axis and horizontally compressed.
    • This corresponds to equation (d): y=f(2x)y = -f(2x).

Final Matching:

  • (a) y=f(x4)y = f(x - 4): Graph 1
  • (b) y=f(x)+3y = f(x) + 3: Graph 3
  • (c) y=2f(x+6)y = 2f(x + 6): Graph 2
  • (d) y=f(2x)y = -f(2x): Graph 4

Would you like more details or have any questions?

Here are 5 related questions you might explore next:

  1. How do horizontal shifts affect the graph of a function?
  2. What is the effect of vertical stretching on a graph?
  3. How does a reflection over the x-axis change the graph of a function?
  4. Can you determine the transformation if a function is compressed horizontally?
  5. What is the combined effect of multiple transformations on a graph?

Tip: Always check the order of transformations, as some transformations (like shifts) are affected by others (like reflections or stretches).

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Math Problem Analysis

Mathematical Concepts

Graph transformations
Shifting and stretching functions

Formulas

Horizontal shift: y = f(x - h)
Vertical shift: y = f(x) + k
Vertical stretch: y = a*f(x)
Reflection: y = -f(x)
Horizontal compression: y = f(bx)

Theorems

Transformation rules for functions

Suitable Grade Level

High School (Grades 9-12)