7.1 Notes: Growth and Decay Functions

What is an exponential function? The variable is in the exponent rather than the

base. Exponential growth increases slowly at first,

then drastically increases as time continues. The basic graph looks like:

Basic Graph info: With a “go-to” point (always passes through) at

(0, 1) Horizontal asymptote at y = 0. Wait…what’s an “asymptote?”…….it’s a line

that the graph will never cross, only approach forever.

How do we evaluate an exponential growth function?

We pick some values for “x” and plug them in

EX: EX A: y = 2x x f(x)

Graphing: Pick at least 3 values for x, plug them in to find y. Graph the points

y = 3x x y

Transformations of growth functions: Vertical (y – direction): y = bx + k If it is + k, the graph moves up “k” times If it is – k, the graph moves down “k” times Soooo……. Y = 3x + 7 moves the graph and the asymptote

up 7 on the y – axis Y = 2x – 4 moves the graph and the asymptote

down 4 on the y - axis

Transformations of growth functions: Horizontal (x – direction): y = bx-h **Note, the letter/number is up with the variable! **Also, “x’s” are liars, so it moves opposite the

direction you think. If it is – h, the graph moves RIGHT h units on the x-axis. If it is + h, the graph moves LEFT h units on the x –

axis. Sooo….. Y = 4x-3, moves the graph 3 units to the right. The

asymptote does not technically move. Y = 6x+9, moves the graph 9 units to the left. The

asymptote does not technically move.

Transformations of growth functions: Reflections reflect the graph over the y – axis: y

= -bx So, this makes all the y – values opposite their

original sign. The go – to point of (0, 1) will now be at (0, -1).

The new graph would look like:

Multiple transformations in one equation: Always work transformations from left to right! EX A: f(x) = -4x+3 – 2 This flips the graph upside

down, moves it to the left 3 and down two. Let’s draw the new graph.

EX B: y = 5x-2 + 4 What happens to this?

Natural Base “e” The natural base’s symbol is “e,” and is an

irrational number (similar to pi). It is approximately 2.7183. You can find this on your calculator.

The graph for the natural base is the same shape as the exponential growth graphs, with the same go-to point and asymptote.

The transformations follow the same rules that we just came over.

EX: f(x) = ex-1 - 3

Exponential Decay What is an exponential decay function? There is a

rapid decrease initially and then the decrease becomes more gradual.

The basic graph has the same go – to point at (0, 1) and asymptote at y = 0.

The base is between 0 and 1. The graph looks like:

Evaluating exponential decay functions: EX A: f(x) = (½)x x f(x)

Graphing: Pick at least 3 values for x, plug them in to find y. Graph the points

y = x yx

3

1

Transformations of decay functions: They follow the same rules as growth functions. EX A: y = -(¼)x-3 – 5 What happens?

Graph it

EX B: f(x) = (½)x+4 + 2 What happens? Graph it