Math Problem Statement

Function f(x) = log₂x is shown on the graph. What effect would changing it to f(x) = log₂(x - 5) + 2 have on the graph? Select two that apply.

Solution

The original function shown on the graph is f(x)=log2xf(x) = \log_2 x. The modified function given in the problem is f(x)=log2(x5)+2f(x) = \log_2(x - 5) + 2.

To analyze how this modification affects the graph, let's break down each part:

  1. Horizontal Shift: The expression x5x - 5 inside the logarithm shifts the graph horizontally. Specifically, it shifts the graph 5 units to the right, since subtracting a constant from xx inside a function moves the graph in the positive xx-direction.

  2. Vertical Shift: The +2+2 added outside the logarithmic function shifts the graph vertically. This shifts the graph 2 units up, as adding a constant to the function moves it in the positive yy-direction.

Therefore, the correct answers are:

  • A: Move 5 spaces right.
  • C: Move 2 spaces up.

Would you like more details or have any questions?


Related Questions:

  1. How would changing f(x)=log2(x)f(x) = \log_2(x) to f(x)=log2(x+3)f(x) = \log_2(x + 3) affect the graph?
  2. What happens to the graph of f(x)=log2(x)f(x) = \log_2(x) if we change it to f(x)=log2(x)f(x) = -\log_2(x)?
  3. How does multiplying the logarithmic function by a constant (e.g., 3log2(x)3 \log_2(x)) affect its graph?
  4. What transformation occurs if we add a negative constant outside the function, such as f(x)=log2(x)4f(x) = \log_2(x) - 4?
  5. How would the graph of f(x)=log2(x)f(x) = \log_2(x) change if it was modified to f(x)=log2(x)f(x) = \log_2(-x)?

Tip:

When analyzing function transformations, remember that changes inside the function argument (like x5x - 5) affect horizontal shifts, while changes outside (like +2+2) affect vertical shifts.

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Math Problem Analysis

Mathematical Concepts

Logarithmic Functions
Transformations of Functions

Formulas

f(x) = log₂(x - h) + k

Theorems

Properties of logarithmic transformations

Suitable Grade Level

Grades 10-12