Math Problem Statement
Solution
The problem involves the transformation of two exponential functions, and , with a base of 2. Based on the graph and the question, we are asked to determine the value of , where the function is translated by a vector to form .
Key Observations:
- The graph of appears to be .
- The graph of appears to be the same function but shifted to the right and upward.
- A translation means the graph of has shifted horizontally by units and vertically by units to produce .
Determining the values of and :
- From the graph, we see that:
- The graph of passes through the point , which corresponds to .
- The graph of passes through . This suggests that is a translation of by 1 unit to the right (horizontally) and 2 units up (vertically), because , but , which means it is shifted upwards by 2.
Thus, the translation vector is , where:
Final Calculation:
Now, we need to find the value of :
Answer:
The value of is .
Let me know if you need any further details or have any questions!
Here are five related questions you might find helpful:
- How do translations affect the equation of a function algebraically?
- What is the general form of an exponential function, and how do shifts modify it?
- Can you describe the effect of a horizontal and vertical shift on any graph?
- How can we determine translations from the coordinates of key points on a graph?
- 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
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