Question: Sketch the graph of the function and check the graph with a graphing calculator. Before doing so,…

Sketch the graph of the function and check the graph with a graphing calculator. Before doing so, describe how the graph of the function can be obtained from the graph of a basic exponential function.

f(x) = 2^{x + 3} - 1

Describe how the graph of the function can be obtained from the graph of a basic exponential function.

Start with the graph of y = 2^x. Shift it 3 units and then shift it 1 unit.

Solution

The function given is \[ f(x) = 2^{x+3} - 1 \] To obtain the graph of this function from a basic exponential function, follow these steps: Start with the graph of \[ y = 2^x \] Shift the graph horizontally to the left by 3 units. This is because of the \(x+3\) in the exponent. Then shift the graph vertically downward by 1 unit. This is due to the \(-1\) in the equation. So, describe it as: Start with the graph of \(y = 2^x\), shift it left 3 units, and then shift it down 1 unit.