Exponential Functions

The usual slips: reading $0 < b < 1$ as "negative"; calling $a b^x$ increasing whenever $b > 1$, even with $a < 0$; keeping the asymptote at $y = 0$ after a vertical translation; putting $k$ itself in the range of $a b^x + k$; and claiming a point of inflection, which an exponential function never has.

On the AP exam, write end behavior with limits, $\lim_{x \to -\infty} (a b^x + k) = k$, and say what it means: "so $y = k$ is a horizontal asymptote". Justify concavity through rates: "the average rates of change over equal-length intervals increase, so the graph is concave up".

1. Reading the behavior from $a$ and $b$. Let $f(x) = -4(0.6)^x$. Is $f$ increasing or decreasing? Is its graph concave up or concave down? Give its range and its end behavior.

Here $a = -4 < 0$ and $0 < b = 0.6 < 1$. The factor $0.6^{x}$ shrinks as $x$ increases, so $-4(0.6)^x$ gets closer to 0 from below: $f$ is increasing. Because $a < 0$ the graph is concave down. The average rates of change over $[0, 1]$, $[1, 2]$, $[2, 3]$ confirm both:

The rates are positive (increasing) and get smaller (concave down). Every output has the sign of $a$, so the range is $y < 0$. End behavior:

The usual slip is to call $f$ decreasing because $b < 1$: with $a < 0$ the reflection reverses that.

2. A shifted exponential from a table. The table gives values of a function $f$ of the form $f(x) = ab^{x} + k$, and $\lim_{x \to -\infty} f(x) = 3$. Find $a$, $b$ and $k$.

The limit says the horizontal asymptote is $y = 3$, so $k = 3$. The outputs themselves are not proportional ($\frac{ {11} }{ {7} } \ne \frac{ {19} }{ {11} }$), but the outputs minus 3 are: each is $2$ times the one before. So $b = 2$, and at $x = 0$, $a b^0 = a = {4}$:

Check at $x = 3$: ${4} \cdot {2}^3 + {3} = {35}$, which is the table's value.

3. Asymptote and range after a translation. Let $g(x) = -2\left(\frac{1}{3}\right)^x + 6$. Find the end behavior of $g$, its horizontal asymptote and its range.

As $x$ increases, $\left(\frac{1}{3}\right)^x$ shrinks toward 0, so $g(x)$ approaches $6$: the line $y = 6$ is a horizontal asymptote, on the right this time. As $x$ decreases, $\left(\frac{1}{3}\right)^x$ grows without bound and the factor $-2$ sends $g(x)$ down without bound. Since $-2\left(\frac{1}{3}\right)^x$ is always negative, $g(x) < 6$ for every $x$: the range is $y < 6$.

4. Greatest and least values on a closed interval. A pie is taken out of an oven. Its temperature $t$ hours later is $T(t) = 200(0.75)^t + 70$ degrees Fahrenheit, for $0 \le t \le 6$. Find the greatest and least temperatures on that interval.

Here $a = 200 > 0$ and $b = 0.75 < 1$, so $T$ is decreasing on the whole interval. A function that only decreases has its greatest value at the left end and its least at the right end:

The greatest temperature is $270$ degrees Fahrenheit, when the pie comes out, and the least is about $105.596$ degrees Fahrenheit, after 6 hours. There is no relative minimum inside the interval: the temperature keeps falling toward the room's 70 degrees, the horizontal asymptote, without reaching it.

5. Repeated multiplication. A culture starts with 500 bacteria, and the number triples every hour. How many are there after 6 hours? Write the number after $t$ hours as an exponential function.

Each hour multiplies the number by 3, so after 6 hours it has been multiplied by 3 six times:

After $t$ hours, $N(t) = 500 \cdot 3^t$: the initial value is $a = 500$ and the base is $b = 3$, the factor for each hour. The formula also gives values between whole hours, such as $N(2.5) \approx {7794.229}$, assuming the growth goes on smoothly. Writing $500 \cdot 3t$ or $500 + 3t$ treats repeated multiplication as a single product or as repeated addition.