# Quadratic models · Year 11

> Quadratic models, Year 11: Reading a quadratic graph in context and more. Explanation, worked examples and free practice for the national curriculum in England.

In Quadratic models (Year 11) you practise 3 skills: Reading a quadratic graph in context · Where the graph meets the x axis · A quadratic model of a situation.

The graph of a quadratic function is a parabola. If the coefficient of x squared is positive, the parabola opens upwards and its vertex is a minimum point. If the coefficient is negative, it opens downwards and its vertex is a maximum point. The constant term is the height where the graph meets the y axis, and the x coordinate of the vertex is minus the coefficient of x, divided by twice the coefficient of x squared.

The points where the graph meets the x axis are the solutions of the equation where the function equals zero, and the quadratic formula solves it. The x coordinate of the vertex is exactly halfway between the x coordinates of those two points. Every point on the parabola has a symmetric point on the other side of the axis of symmetry, at the same height.

In a quadratic model of a situation, such as the height of a ball against time or a total against the number of people, not every solution makes sense. A time, a number of people and a price are never negative, so a negative solution is rejected, and the graph describes the situation only where it makes sense. The greatest or the least value of the model is at the vertex.

**Tip:** Before you choose an answer, check that it makes sense in the situation: a negative time or a negative number of people is rejected.

## What you learn

### Reading a quadratic graph in context

- Example: A model rocket is launched upwards. The graph shows the height of the rocket, y , in metres, against the time x , in seconds. (A graph of the height of the rocket against time: an arc of a parabola that starts at a height of 0 metres, rises to the marked vertex at 1 second and 5 metres, and comes down to the time axis) What is the greatest height that the rocket reaches?
- Answer: 5 m
- Solution: The highest point of the graph is the marked vertex, at the time x = 1 and the height y = 5 . So the rocket reaches a height of 5 metres.

### Where the graph meets the x axis

- Example: What are the solutions of the equation 2x² + 2x − 4 = 0 ?
- Answer: x = −2 or x = 1
- Solution: By the quadratic formula, the discriminant is 36 and its square root is 6 . This gives x = −2 or x = 1 . These are the x coordinates of the points where the graph meets the x axis.

### A quadratic model of a situation

- Example: A group of 10 friends goes on a trip, and each pays £40. For each extra friend who joins, everyone’s price falls by £4. Which of these rules gives the total y when x extra friends join?
- Options: y = (10 − x)(40 − 4x) · y = (10 + x)(40 + 4x) · y = 400 − 4x · y = (10 + x)(40 − 4x)
- Answer: y = (10 + x)(40 − 4x)
- Solution: The number of friends is 10 + x , and the price for each is 40 − 4x . So y = (10 + x)(40 − 4x) .

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).

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