Exponential function
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 ofy 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 graphy=a^x wherea>1 . - The graph of
y=e^x decreases slower than any other exponential graphy=a^x where0<a<1 . - The derivative of
y=e^x is equal toe^x itself:\frac{dy}{dx} e^x = e^x
Any exponential function can be expressed in terms of
a^x = e^{(\ln a)x}
Asymptotes
For the function
- There is a horizontal asymptote at
y=0 (the x-axis).
Approaching infinity
- For
y=a^x :- As
x \to +\infty ,y \to +\infty . - As
x \to -\infty ,y \to 0 .
- As
- The rate of increase of
y becomes faster asx increases. - The rate of decrease of
y becomes slower asx 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 .
flashcards
| Question | Answer |
|---|---|
| What is the relationship for | The gradient is directly proportional to the value of |
| Through which point does the graph of | The point |
| How does the graph of | It increases faster than any other exponential graph |
| How does the graph of | It decreases slower than any other exponential graph |
| What is the derivative of | |
| How can any exponential function | |
| What is the horizontal asymptote for the function | There is a horizontal asymptote at |
| For | |
| For | |
| How does the rate of increase of | The rate of increase becomes faster as |
| How does the rate of decrease of | The rate of decrease becomes slower as |
| Does the function | No, the function never actually reaches |
| What is the graph of an inverse function in relation to the original graph? | The inverse of a graph is the reflection of the graph in the line |