Exponential graph

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

flashcards

Question	Answer
What is the general form of an exponential function?	f(x)=a^x, where a is a positive constant and a \neq 1
Why can a not be 1 in the exponential function f(x)=a^x?	If a=1, the function would be a constant.
What is an asymptote?	A line that a graph approaches but never touches.
What is the horizontal asymptote of the graph of f(x)=a^x?	y=0 (the x-axis)
Why do exponential graphs have a horizontal asymptote at y=0?	Because the result of a^x will never be 0 or negative.
How does the y-value of an exponential graph change as x increases by 1?	The y value is multiplied by a.
What is the domain of an exponential graph?	(-\infty, \infty) (all real numbers)
Why is the domain of an exponential graph (-\infty, \infty)?	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).
What is the range of an exponential graph?	(0, \infty) (positive numbers)
Why is the range of an exponential graph (0, \infty)?	Because the output of a^x will always be positive.
How many solutions does the equation a^x=0 have?	No solutions, because the output of a^x is always positive.
When can the equation a^x+c=0 (where c is a negative constant) have a solution?	When the graph of a^x is shifted down by c units, it can cross the x-axis.
How do you solve for x in the equation y=a^x?	x=\log_a(y)
What is the y-intercept of the graph of f(x)=a^x?	(0, 1)
Why is the y-intercept of f(x)=a^x at y=1?	Because any number to the power of 0 is 1 (a^0=1).
In the function f(x)=a^x, what type of behaviour is described as ‘growth’?	The value of y increases with the value of x, which occurs when a>1.
In the function f(x)=a^x, what type of behaviour is described as ‘decay’?	The value of y decreases with the value of x, which occurs when 0<a<1.