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Given 𝑔(𝑥)=𝑎𝑏^(𝑘(𝑥−𝑑))+𝑐, we have to consider the different transformations:

1. horizontal translation

2. horizontal stretch or compression

3. reflection in the y-axis

4. vertical stretch or compression

5. reflection in the x-axis

6. vertical translation

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The basis function will vary depending on the equation being dealt with.

It has the same rules as quatratic function to apply. We first graph the base function and then apply the transformations.

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Domain is not affected by transformations, it stays as D: {x∈R}

Range is affected by transformations.

-if the transformed graph is above the asymptote y=c, R: {y>c|y∈R}

-if the transformed graph is below the asymptote y=c, R:{y<c|y∈R}

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For this transformation, state the base function and then describe the transformations in the order it could be applied.

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A cup of hot liquid was left to cool in a room whose temperature was 20 °C. The temperature changes with time(seconds) according to the function in the picture above.

a) How would the equation change if the time is 30s?

b) What are the domain and range of this function in the context of this problem?

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Integer Exponents

Integer exponents represent repeated multiplication. For example, 2³=2x2x2=8

Exponential Growth: A function like f(x) = 2^x shows exponential growth. As x increases, the value of f(x) increases rapidly.

Exponential Decay: For example, f(x) = (1/2)^x represents exponential decay. The value of f(x) decreases as x increases.

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Horizontal Stretch and Compression: If f(x) = 2^x and g(x) = 2^(2x), g(x) is a horizontal compression of f(x) by a factor of 1/2.

Vertical Stretch and Compression: h(x) = 3 × 2^x is a vertical stretch of f(x) by a factor of 3.

Reflection in the y-axis: k(x) = 2^(-x) is the reflection of f(x) in the y-axis.

Reflection in the x-axis: m(x) = -2^x is the reflection of f(x) in the x-axis.

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Question 1: The function g(x) = 2 × 4^x undergoes a vertical compression by a factor of 1/2. What is the new function?
Solution: The new function is h(x) = 1/2 × (2 × 4^x) = 4^x

Question 2: Find the domain and range of f(x) = 2^(x - 3)
Solution: The domain is all real numbers. For the range, as x approaches negative infinity, f(x) approaches 0. So, the range is (0, ∞).