Functions - Exponents – exponential function

Given the function $f(x) = 5 \cdot 2^{x} - 10$, find the horizontal asymptote, the $y$-intercept, and the $x$-intercept.

1. Horizontal Asymptote: The constant term added to the exponential part is $-10$. Thus, the horizontal asymptote is $y = -10$.
2. $y$-intercept: Set $x = 0$. $f(0) = 5 \cdot 2^{0} - 10 = 5(1) - 10 = -5$ The $y$-intercept is $(0, -5)$.
3. $x$-intercept: Set $f(x) = 0$. $5 \cdot 2^{x} - 10 = 0$ $5 \cdot 2^{x} = 10$ $2^{x} = 2$ $x = 1$ The $x$-intercept is $(1, 0)$.

To find intercepts, we alternate setting $x$ and $y$ to zero. The horizontal asymptote is determined by the vertical shift of the parent exponential function.

A population of bacteria grows according to the model $P(t) = 1000e^{0.05t}$, where $t$ is time in hours. Find the initial population and the population after 10 hours.

1. Initial population: Set $t = 0$. $P(0) = 1000e^{0.05(0)} = 1000e^{0} = 1000 \cdot 1 = 1000$
2. Population after 10 hours: $P(10) = 1000e^{0.05(10)} = 1000e^{0.5}$ Using a calculator: $P(10) \approx 1000 \cdot 1.6487 = 1648.7$ Population $\approx 1649$ bacteria.

The coefficient $1000$ represents the initial value $P_0$. The exponent $0.05$ represents the continuous growth rate of $5\%$.

Solve for $x$: $9^{x-1} = \left(\frac{1}{3}\right)^{x+2}$

Write both sides with the same base $3$: $(3^{2})^{x-1} = (3^{-1})^{x+2}$ $3^{2x-2} = 3^{-x-2}$ Equate the exponents: $2x - 2 = -x - 2$ $3x = 0$ $x = 0$

When solving exponential equations without logarithms, try to express both sides as powers of the same base and then set the exponents equal to each other.

1. Asymptote: The constant term is $+3$, so the horizontal asymptote is $y = 3$.
2. y-intercept: Set $x = 0$: $f(0) = 2^{0-1} + 3 = 2^{-1} + 3 = 0.5 + 3 = 3.5$ The $y$-intercept is $(0, 3.5)$.
3. Horizontal Shift: The term $(x-1)$ shifts the graph of $2^x$ one unit to the right.

To graph an exponential function, first identify the horizontal asymptote, then calculate the $y$-intercept and one or two additional points to determine the shape.

1. Substitute the point $(2, 12)$ into the equation: $12 = 3 \cdot e^{2k}$
2. Divide by $3$: $4 = e^{2k}$
3. Take the natural logarithm of both sides: $\ln(4) = \ln(e^{2k})$ $\ln(4) = 2k$
4. Solve for $k$: $k = \frac{\ln(4)}{2} = \frac{\ln(2^2)}{2} = \frac{2 \ln(2)}{2} = \ln(2) \approx 0.693$

When given a point on an exponential curve, substitute the coordinates to solve for the unknown parameter using logarithms.