Question

1. given the exponential function: $f(x) = 5 \cdot 2^{x-2} + 1$, explain what the (5), (-2) and the (+1) will do to the function:

Explanation:

1. The coefficient 5 is a vertical stretch factor: it multiplies all output values of the parent exponential function $2^x$ by 5, making the function grow 5 times as steeply.
2. The $-2$ inside the exponent is a horizontal shift: it shifts the parent function $2^x$ 2 units to the right (since it is of the form $x - h$ where $h=2$).
3. The $+1$ at the end is a vertical shift: it moves every point of the transformed function 1 unit upward, shifting the horizontal asymptote from $y=0$ to $y=1$.

Answer:

• The factor 5 vertically stretches the parent exponential function $2^x$ by a factor of 5.
• The $-2$ in the exponent shifts the parent function 2 units to the right.
• The $+1$ shifts the entire function 1 unit upward.