Exponential function

Exponential functions are in the form of f(x)=a^x, where a is a positive constant and a \neq 1 (because if a=1 then the function would be a constant).

Asymptotes

An asymptote is a line that a graph approaches but never touches.

• They have a horizontal asymptote at y=0 (the x-axis) because the result of a^x will never be 0 or negative.

Asymptote at y=0

Growth

As you increase the value of x of an exponential graph, the y value increases very quickly. That’s because an increase of just 1 in x means that the y value is multiplied by a.

For example, if a=2 and x increases from 3 to 4, the y value increases from 2^3=8 to 2^4=16 - it multiplies by 2.

Domain

The domain of an exponential graph is (-\infty, \infty) (anything) because you can go as far left or right as you want on the x-axis (all values of x will give a valid output to the function).

Domain: (-\infty, \infty)

Range

The range is (0, \infty) (positive numbers) because the output of a^x will always be positive.

range: (0, \infty)

Solving where y=0

At no point on the graph is y=0 because the output of a^x is always positive (never 0).

That means there are no solutions to the equation a^x=0.

There can be solutions to the equation a^x+c=0 (where c is a negative constant) because the graph of a^x+c is just the graph of a^x shifted down by c units, so it can cross the x-axis.

But the normal graph of a^x has no solutions to a^x=0.

Solving for a given y value

• We have our equation:
  • y=a^x
• If we want to find the value of x for a given y value, we can rearrange the equation into a logarithm:
  • a^x=y
  • x=\log_a(y)

the solution to y=a^x is x=\log_a(y).

Intercept

The y intercept is the y value when x=0.

f(0) = a^0 = 1 (because any number to the power of 0 is 1)

So the y intercept is at (0, 1).

Growth or decay?

• The value of y will increase with the value of x if a>1 (in the function f(x)=a^x). That’s called growth.
• The value of y will decrease with the value of x if 0<a<1 (in the function f(x)=a^x). That’s called decay.

a>1: growth
0<a<1: decay

Proportionality of exponential graphs

• For y=a^x, the gradient \frac{dy}{dx} of the graph at any point is directly proportional to the value of y at that point.

Graph of y=e^x

• The graph of y=e^x passes through the point (0,1).
• The graph of y=e^x increases faster than any other exponential graph y=a^x where a>1.
• The graph of y=e^x decreases slower than any other exponential graph y=a^x where 0<a<1.
• The derivative of y=e^x is equal to e^x itself:
  • \frac{dy}{dx} e^x = e^x

Any exponential function can be expressed in terms of e:

• a^x = e^{(\ln a)x}

Approaching infinity

• For y=a^x:
  • As x \to +\infty, y \to +\infty.
  • As x \to -\infty, y \to 0.
• The rate of increase of y becomes faster as x increases.
• The rate of decrease of y becomes slower as x decreases.
• The function never actually reaches y=0; it only approaches it as an asymptote.

Graph of an inverse function

• The inverse of a graph is the reflection of the graph in the line y=x.