Exponential Graphs Flashcards

All 5 cards in this deck

What does the graph of $y = ab^x$ (with $a > 0$ and $b > 1$) look like?

An increasing curve through $(0, a)$, rising steeply to the right and approaching the $x$-axis on the left.

e.g. $y = 3 \times 2^x$ passes through $(0, 3)$

How is the graph of $y = ab^{-x}$ related to the graph of $y = ab^{x}$?

It is its reflection in the $y$-axis, since $ab^{-x} = a\left(\frac{1}{b}\right)^x$.

e.g. $y = 3 \times 2^{-x}$ is $y = 3 \times 2^{x}$ reflected in the $y$-axis

True or false? The graph of $y = 3 \times 2^{-x}$ falls as $x$ increases and eventually crosses the $x$-axis.

False. It gets closer and closer to the $x$-axis but never touches or crosses it.

What are the steps to find $a$ and $b$ for a curve $y = ab^x$ passing through two given points?

e.g. the curve passes through $(0, 5)$ and $(2, 45)$

1. Use the point with $x = 0$ to get $a$: $a = 5$.
2. Substitute the other point: $5b^2 = 45$, so $b^2 = 9$.
3. Take the positive root: $b = 3$, so $y = 5 \times 3^x$.

What are the steps to find the $x$-coordinate of a point on $y = ab^x$ with a known $y$-value?

e.g. find $x$ when $y = 96$ on $y = 3 \times 2^x$

1. Divide by $a$ to isolate the power: $2^x = 32$.
2. Write the number as a power of $b$: $32 = 2^5$.
3. Equate the exponents: $x = 5$.