Lecture 2 · Barnett ch. 1 to 2 · about 45 min

Linear and Quadratic Functions

Straight-line cost and revenue models, break-even analysis, and the parabola that appears whenever a linear price meets a linear cost: how to find its vertex and its roots without a graph.

By the end you can

Write a linear function from a slope and a point, or from two points, and interpret the slope economically
Find the break-even quantity where a linear cost and revenue function cross
Put a quadratic into vertex form and read off its maximum or minimum
Use the quadratic formula to find break-even points for a quadratic profit function

Linear functions

A linear function has a constant rate of change: every extra unit of $x$ changes $y$ by the same amount, $m$ .

Slope-intercept formFormula sheet →

y = mx + b

$m$: slope, the change in y per unit change in x
$b$: y-intercept, the value of y when x = 0

Use it when you know the rate of change and the starting value. In cost functions, m is the variable cost per unit and b is the fixed cost.

Economically, $m$ is a marginal quantity (cost per extra unit, or the change in price per extra unit sold) and $b$ is a baseline quantity (fixed cost, or a starting price).

Break-even analysis with two lines

Break-even is the output level where total revenue equals total cost: $R(x) = C(x)$ . Below it the firm loses money; above it, profit is positive.

Break-even pointConditionFormula sheet →

R(x) = C(x)

$R(x)$: px = price times quantity
$C(x)$: variable cost + fixed cost

Use it when revenue and cost are both linear in x. Solve the resulting linear equation for x.

Quadratic functions

A quadratic function has a squared term and graphs as a parabola. In business, a quadratic often appears when a linear demand curve meets a linear cost: revenue $= p(x)\cdot x$ becomes quadratic in $x$ once price itself depends on $x$ .

Standard formFormula sheet →

f(x) = ax^2 + bx + c

$a$: shapes the curve (a > 0 opens upward, a < 0 opens downward)
$b, c$: shift and position the curve

Use it when you are given a quadratic in expanded form and need to identify its shape before finding the vertex.

If $a \gt 0$ the parabola opens upward and has a minimum (useful for a cost or average-cost curve). If $a \lt 0$ it opens downward and has a maximum (useful for revenue or profit).

Vertex form: finding the max or min without calculus

Vertex of a parabolaKey formulaFormula sheet →

x^* = -\frac{b}{2a} \qquad f(x^*) = c - \frac{b^2}{4a}

$x^*$: the input at the vertex
$f(x^*)$: the maximum (a<0) or minimum (a>0) value

Use it when you have a quadratic profit, revenue or cost function and need the optimal quantity and the optimal value, without derivatives.

Try it: shape the profit parabola

Raise the net price $b$ and the vertex moves up and right: a higher price raises both the profit-maximising quantity and the maximum profit. Raise the fixed cost $c$ and the whole curve shifts down: the vertex height falls and the break-even points move apart, since more revenue is now needed just to cover costs.

The quadratic formula: break-even for a curved profit

When profit is quadratic, break-even points are the roots: the $x$ values where $P(x) = 0$ . Factoring is not always possible, so use the quadratic formula.

Quadratic formulaFormula sheet →

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

$a, b, c$: the coefficients of ax^2 + bx + c
$b^2 - 4ac$: the discriminant

Use it when a quadratic profit or revenue function does not factor easily, or a question directly asks for its roots (break-even points).

The discriminant $b^2 - 4ac$ tells you how many real break-even points exist: positive gives two, zero gives exactly one (the vertex touches the axis), and negative means the curve never crosses zero — the firm never breaks even (if $a \lt 0$ and the vertex is below zero) or never loses money (if $a \gt 0$ and the vertex is above zero).

Exam practice

Summary and review

Linear function: $y = mx + b$ ; $m$ is the constant rate of change, $b$ is the value at $x = 0$ .
Break-even for linear cost and revenue: solve $R(x) = C(x)$ .
Quadratic function: $f(x) = ax^2 + bx + c$ ; $a \gt 0$ opens upward (minimum), $a \lt 0$ opens downward (maximum).
Vertex: $x^* = -b/(2a)$ , then substitute back in for the max or min value.
Quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ , for break-even points on a curved profit function.
Discriminant sign tells you how many real break-even points exist: two, one, or none.