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 = 2 x  x f(x)  Graphing: Pick at least 3 values for x, plug them in to find y. Graph the points  y = 3 x x y

Transformations of growth functions:  Vertical (y – direction): y = b x + k  If it is + k, the graph moves up “k” times  If it is – k, the graph moves down “k” times  Soooo…….  Y = 3 x + 7 moves the graph and the asymptote up 7 on the y – axis  Y = 2 x – 4 moves the graph and the asymptote down 4 on the y - axis

Transformations of growth functions:  Horizontal (x – direction): y = b x-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 = 4 x-3, moves the graph 3 units to the right. The asymptote does not technically move.  Y = 6 x+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 = -b x  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) = -4 x+3 – 2This flips the graph upside down, moves it to the left 3 and down two. Let’s draw the new graph.  EX B: y = 5 x 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 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) = e x-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 y

Transformations of decay functions:  They follow the same rules as growth functions.  EX A: y = -(¼) x-3 – 5What happens? Graph it  EX B: f(x) = (½) x+4 + 2What happens? Graph it