# Indices and exponential functions · Grade 12 · 4 units

> Explanation, worked examples and free practice following the Israeli curriculum.

In Indices and exponential functions (Grade 12 · 4 units) you practise 5 skills: The laws of indices · Properties of the exponential function · Exponential equations · Exponential inequalities · 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

- Example: Write the expression as a single power: 5⁶ × 5^(−2) What is the index of that power?
- Answer: 4
- Solution: Add the indices: 6 + (−2) = 4 So: 5⁶ × 5^(−2) = 5⁴

### Properties of the exponential function

- Example: Here is the function: f(x) = 2ˣ What is the value of f(−3) ?
- Answer: 1/8
- Solution: Substitute for x: f(−3) = 2^(−3) = 1/8

### Exponential equations

- Example: Solve the equation: 2^(x − 3) = 1/32 What is the solution?
- Answer: −2
- Solution: Write the right side as a power of 2 : 2^(x − 3) = 2^(−5) Set the indices equal: x − 3 = −5 x = −2

### Exponential inequalities

- Example: Solve the inequality: 2ˣ < 16 What are all its solutions?
- Answer: x < 4
- Solution: Write the right side as a power of 2 : 2ˣ < 2⁴ The base is greater than 1, so the sign stays the same: x < 4

### Shifts and reflections of an exponential function

- Example: Here is the function: y = 2ˣ − 1 Which of these graphs is its graph? Graph A: (Graph A: a rising curve; the horizontal asymptote is at height −1.) Graph B: (Graph B: a rising curve; the horizontal asymptote is at height 1.) Graph C: (Graph C: a rising curve; the horizontal axis is the asymptote, and at x = 1 the height is 1.) Graph D: (Graph D: a rising curve; the horizontal axis is the asymptote, and at x = −1 the height is 1.)
- Options: Graph B · Graph C · Graph D · Graph A
- Answer: Graph A
- Solution: Adding a number to the whole function moves the graph up or down, and the asymptote with it: y = −1 . That is graph A.

## Not on the site yet

- Sketching the graph of an exponential function by hand
- Stretches and the absolute value of an exponential function, which are in the review of families of functions

The exercises are made by rules and the answers are worked out in code. The topic follows the Israeli Ministry of Education’s high school maths curriculum at 4 units (grades 10 to 12).

[Practise on the site](https://kixplo.com/il/en/) · [All Grade 12 · 4 units topics](https://kixplo.com/il/en/math/grade-12-4u/)