# Quadratic models · Grade 12 · 3 units

> Quadratic models, Grade 12 · 3 units: Coefficients and the parabola and more. Explanation, worked examples and free practice following the Israeli curriculum.

In Quadratic models (Grade 12 · 3 units) you practise 5 skills: Coefficients and the parabola · Matching a formula to its graph · 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

### Coefficients and the parabola

- Example: Is the turning point of the parabola y = −2x² + 7 a minimum point or a maximum point?
- Options: A minimum point · A maximum point
- Answer: A maximum point
- Solution: The coefficient of the squared term is −2 , which is negative: the parabola opens downwards, and the turning point is a maximum point.

### Matching a formula to its graph

- Example: The function y = −x² − 4x − 7 is given. Which of the three graphs is its graph? Graph A: (Graph A: A parabola that opens downwards, with its vertex where x is −2, meeting the y axis at −7) Graph B: (Graph B: A parabola that opens upwards, with its vertex where x is 2, meeting the y axis at −7) Graph C: (Graph C: A parabola that opens downwards, with its vertex where x is −2, meeting the y axis at −9)
- Options: Graph B · Graph C · Graph A
- Answer: Graph A
- Solution: The parabola opens downwards, so the coefficient of x² is negative. It meets the y axis at (0, −7) , and its vertex is where x = −2 . That is graph A.

### 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 shekels. For each extra friend who joins, everyone’s price falls by 4 shekels. 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) .

## Not on the site yet

- Sketching a parabola by hand
- Explaining the model in words
- Quadratic equations with an algebraic denominator

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 3 units (grades 10 and 11).

[Practise on the site](https://kixplo.com/il/en/) · [All Grade 12 · 3 units topics](https://kixplo.com/il/en/math/grade-12-3u/)