# Precalculus: families of functions · Year 11

> Precalculus: families of functions, Year 11: Translating a graph and more. Explanation, worked examples and free practice for the national curriculum in England.

In Precalculus: families of functions (Year 11) you practise 2 skills: Translating a graph · Reflecting a graph.

The domain of a function is every value you are allowed to put in. Under a square root the expression is not negative, and in a fraction the denominator is not zero. An even function stays the same when x is replaced by its opposite, and its graph is symmetric in the y axis. An odd function changes sign, and its graph is symmetric about the origin.

Adding a number to a function moves its graph up or down, and adding a number to x inside the function moves it left or right, in the direction opposite to the sign. Multiplying the function by minus one reflects the graph in the x axis, and replacing x by its opposite reflects it in the y axis. Multiplying the function by a number greater than one stretches the graph away from the x axis.

The absolute value of a function keeps the parts above the x axis and reflects upwards the parts below it. Where a question has a picture, it shows both graphs: the original dashed and the new one solid.

**Tip:** Check one point: take a point on the original graph and see where it goes. That is enough to know which way the graph moved.

## What you learn

### Translating a graph

- Example: The point (5, −4) is on the graph of y = f(x) . Which point matches it on the graph of y = f(x) + 3 ?
- Answer: (5, −1)
- Solution: The y coordinate goes up by 3, and the x coordinate stays the same: (5, −1)

### Reflecting a graph

- Example: The point (5, −4) is on the graph of y = f(x) . Which point matches it on the graph of y = −f(x) ?
- Answer: (5, 4)
- Solution: The y coordinate changes sign, and the x coordinate stays the same: (5, 4)

The exercises are made by rules and the answers are worked out in code. The topic follows the national curriculum in England: mathematics (key stages 3 and 4).

[Practise on the site](https://kixplo.com/uk/en/) · [All Year 11 topics](https://kixplo.com/uk/en/maths/year-11/)