Math Problem Statement

Perhatikan grafik fungsi eksponen dengan bilangan pokok 2 berikut. Grafik f(x) ditranslasi oleh (a, b) menghasilkan peta grafik g(x). Tentukan nilai dari 2a - b.

Solution

The problem involves the transformation of two exponential functions, f(x)f(x) and g(x)g(x), with a base of 2. Based on the graph and the question, we are asked to determine the value of 2ab2a - b, where the function f(x)f(x) is translated by a vector (ab)\begin{pmatrix} a \\ b \end{pmatrix} to form g(x)g(x).

Key Observations:

  • The graph of f(x)f(x) appears to be f(x)=2xf(x) = 2^x.
  • The graph of g(x)g(x) appears to be the same function but shifted to the right and upward.
  • A translation (ab)\begin{pmatrix} a \\ b \end{pmatrix} means the graph of f(x)f(x) has shifted horizontally by aa units and vertically by bb units to produce g(x)g(x).

Determining the values of aa and bb:

  1. From the graph, we see that:
    • The graph of f(x)f(x) passes through the point (0,1)(0, 1), which corresponds to 20=12^0 = 1.
    • The graph of g(x)g(x) passes through (1,3)(1, 3). This suggests that g(x)g(x) is a translation of f(x)f(x) by 1 unit to the right (horizontally) and 2 units up (vertically), because f(1)=2f(1) = 2, but g(1)=4g(1) = 4, which means it is shifted upwards by 2.

Thus, the translation vector is (12)\begin{pmatrix} 1 \\ 2 \end{pmatrix}, where:

  • a=1a = 1
  • b=2b = 2

Final Calculation:

Now, we need to find the value of 2ab2a - b: 2ab=2(1)2=22=02a - b = 2(1) - 2 = 2 - 2 = 0

Answer:

The value of 2ab2a - b is 0\boxed{0}.

Let me know if you need any further details or have any questions!

Here are five related questions you might find helpful:

  1. How do translations affect the equation of a function algebraically?
  2. What is the general form of an exponential function, and how do shifts modify it?
  3. Can you describe the effect of a horizontal and vertical shift on any graph?
  4. How can we determine translations from the coordinates of key points on a graph?
  5. What other transformations (e.g., scaling or reflection) can be applied to exponential functions?

Tip: When solving graph translation problems, always check how specific points, like the intercepts or key points, shift between the original and transformed graphs.

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Math Problem Analysis

Mathematical Concepts

Exponential Functions
Graph Translation
Coordinate Geometry

Formulas

Translation of function: g(x) = f(x - a) + b
Exponential function: f(x) = 2^x

Theorems

Graph translation theorem

Suitable Grade Level

Grades 10-12