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Chapter 3: Problem 85

Suppose \(f(x)=2^{x}\). Explain why shifting the graph of \(f\) left 3 units produces the same graph as vertically stretching the graph of \(f\) by a factor of 8 .

Short Answer

Expert verified
In conclusion, shifting the graph of \(f(x) = 2^x\) 3 units to the left produces the same graph as vertically stretching it by a factor of 8 because both transformations result in the same equation: \(f_1(x) = f_2(x) = 8 \cdot 2^x\), which have identical graphs.

Step by step solution

01

Compute the shifted function.

To shift the graph of \(f(x) = 2^x\) left by 3 units, we need to replace 'x' with 'x + 3'. This will give us: \(f_1(x) = 2^{x + 3}\).
02

Compute the vertically stretched function.

To vertically stretch the graph of \(f(x) = 2^x\) by a factor of 8, we need to multiply the function by 8, which results in: \(f_2(x) = 8 \cdot 2^x\).
03

Compare the shifted and vertically stretched functions.

Now we'll compare the shifted function \(f_1(x) = 2^{x + 3}\) and the vertically stretched function \(f_2(x) = 8 \cdot 2^x\). By using the properties of exponents, we can rewrite the shifted function as follows: \(f_1(x) = 2^{x + 3} = 2^x \cdot 2^3 = 8 \cdot 2^x\). Notice that the shifted function equation is the same as the vertically stretched function equation: \(f_1(x) = 8 \cdot 2^x = f_2(x)\). Since both functions have the same equation, their graphs will be identical.
04

Conclusion

In conclusion, shifting the graph of the function \(f(x) = 2^x\) 3 units to the left yields the same graph as vertically stretching it by a factor of 8. This is because both transformations, when applied to the original function, result in the same equation: \(f_1(x) = f_2(x) = 8 \cdot 2^x\).

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