Question: 2. The function y = 4 × 2^(-x-2) can be though of as comprising of a vertical stretch, a,…

The function y = 4 × 2^(-x-2) can be though of as comprising of a vertical stretch, a, reflection in the y axis and a horizontal translation:

a. Determine the vertical stretch factor. [1 mark]

b. Determine whether the function has been translated left or right and by what amount. [1 mark]

Here is the same function from question 2. y = 4 × 2^(-x-2) This function can be considered as a function that has only been reflected in the y axis. Explain your answer by providing a mathematical argument. [2 marks]

Solution

The function given is \( y = 4 \times 2^{-x-2} \). a. To determine the vertical stretch factor: The vertical stretch factor is the coefficient of the function. Here, it is 4. b. To determine whether the function has been translated left or right and by what amount: The expression inside the exponent is \(-x-2\), which can be rewritten as \(-1(x+2)\). This indicates a horizontal translation 2 units to the left. 3. Here is the same function \( y = 4 \times 2^{-x-2} \). To consider this function as having only reflected in the y-axis: The reflection in the y-axis occurs when we replace \(x\) with \(-x\) in the function. The function \( y = 4 \times 2^{-x}\) would be a reflection in the y-axis alone. The exponent \(-x-2\) does show a reflection in the y-axis because of the \(-x\), but it also includes a horizontal shift of 2 units to the left due to \(-2\). Thus, mathematically, the function incorporates both a reflection and a translation, not only a reflection.