Chapter 2: Algebra and functions

The chapter problem: will the lorry fit under the arch?

A new road passes through an arched tunnel. The road across the tunnel is level, and the middle of the road is at $x = 0$ . The height of the arch above the road is modelled by

y = 4 - \frac{x^2}{4}, \qquad -4 \le x \le 4,

where $x$ is the distance in metres from the middle of the road and $y$ is the height in metres.

A lorry is $2.5$ m wide and $3.2$ m tall. It drives along the middle of the road. The drivers must have at least $0.2$ m of clear space above the roof of the lorry. Does the lorry fit?

By the end of the chapter you will answer that question exactly. You will find where the arch meets the road, how high the arch rises, and how much room the lorry really has. You will then change one feature of the design at a time and see how the answer moves.

First contact — no working needed yet. Put $x = 0$ and $x = 2$ into the model and work out the two heights. Sketch the arch roughly on paper. At which value of $x$ is the arch highest? Write one sentence saying what you would need to work out before you could be sure that the lorry fits.

If you are not sure where to start

The lorry is $2.5$ m wide and sits in the middle, so its sides are at $x = 1.25$ and $x = -1.25$ . Substituting $x = 0$ gives the height at the centre of the road. Substituting $x = 4$ gives the height at the edge of the arch.

For $x = 2$ the model gives $y = 4 - \frac{2^2}{4} = 4 - 1 = 3$ . The road is level because the height does not change as you cross it.

By the end of this chapter you will be able to

use the laws of indices for fractional and negative exponents;
simplify surds and rationalise a denominator;
complete the square, read the vertex of a parabola, and use the discriminant;
solve quadratic equations, including equations that are quadratic in a function of the unknown;
solve simultaneous equations by elimination and by substitution, including one linear and one quadratic equation;
solve linear and quadratic inequalities and write the solution in set notation;
divide polynomials, use the factor theorem, and factorise a cubic;
simplify rational expressions and split one into partial fractions;
sketch graphs including the modulus function and the asymptotes of $y = \frac{a}{x}$ and $y = \frac{a}{x^2}$ ;
form composite and inverse functions and describe their domains;
apply the graph transformations $y = af(x)$ , $y = f(x) + a$ , $y = f(x + a)$ and $y = f(ax)$ ;
build and refine a function model, and say what its limitations are.

Where it is used

Algebra is the language the rest of the course is written in. Differentiation needs the index laws for every negative and fractional power. Integration needs partial fractions. Coordinates need simultaneous equations. Statistics and mechanics both start by turning a situation into a function and then asking what that function does. The tasks in this chapter — solve, factorise, complete the square, choose a method — are the tasks that appear on every later page.

Your turn to connect it

Think of a quantity in your own life that changes when another one changes. It could be the cost of a journey against distance, the time a phone battery lasts against use, or the height of a thrown ball against time. Write down how the two quantities are linked, and say what one of them does at the start and at the end.

Chapter 2: Algebra and functions

The chapter problem: will the lorry fit under the arch?

By the end of this chapter you will be able to

Where it is used

Your turn to connect it

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