Indices and exponential functions · Years 11–12
In Indices and exponential functions (Years 11–12) you practise 4 skills: The laws of indices · Properties of the exponential function · Exponential equations · Shifts and reflections of an exponential function. When you multiply powers of the same base, add the indices: 2 to the power of 3 times 2 to the power of 4 is 2 to the power of 7. When you divide, subtract the indices, and for a power of a power, multiply them. Any positive number to the power of 0 is 1, and a negative index turns the number upside down: 2 to the power of minus 3 is one eighth. An index that is a fraction is a root: the denominator is the order of the root, and the numerator is the power. For example, 8 to the power of two thirds is the cube root of 8, squared, which is 4.
In an exponential function the variable is in the index, and the base is a positive number other than 1. The function is defined for every number, all its values are positive, and its graph crosses the vertical axis at height 1. The horizontal axis is an asymptote: the graph gets close to it and never touches it. When the base is greater than 1 the function is increasing, and the larger the base, the faster it rises to the right of the vertical axis. When the base is between 0 and 1 the function is decreasing. To the left of the vertical axis it is the other way round: there the smaller base gives the larger value. To solve an exponential equation, write both sides as powers of the same base and set the indices equal. An inequality works the same way, but when the base is between 0 and 1 the inequality sign turns round.
Tip: Before solving, look for a common base: 4, 8 and 16 are all powers of 2.
What you learn
The laws of indices
For example: Write the expression as a single power: 5⁶ × 5^(−2) What is the index of that power?
Answer and solution
The answer: 4
Add the indices: 6 + (−2) = 4 So: 5⁶ × 5^(−2) = 5⁴
Properties of the exponential function
For example: Here is the function: f(x) = 2ˣ What is the value of f(−3) ?
Answer and solution
The answer: 1/8
Substitute for x: f(−3) = 2^(−3) = 1/8
Exponential equations
For example: Solve the equation: 2^(x − 3) = 1/32 What is the solution?
Answer and solution
The answer: −2
Write the right side as a power of 2 : 2^(x − 3) = 2^(−5) Set the indices equal: x − 3 = −5 x = −2
Shifts and reflections of an exponential function
For example: Here is the function: y = 2ˣ − 1 Which of these graphs is its graph? Graph A:
Graph B: Graph C: Graph D:- Graph B
- Graph C
- Graph D
- Graph A
Answer and solution
The answer: Graph A
Adding a number to the whole function moves the graph up or down, and the asymptote with it: y = −1 . That is graph A.
The exercises are made by rules and the answers are worked out in code, so there are new exercises every time. The topic follows the national curriculum in England: mathematics (key stages 3 and 4) and selected AS and A level mathematics practice in England (the GCE subject content).