**GOAL:** Understand the concept of slope for lines and
linear function s and learn how to visualize

quadratic functions by completing the square.

► A linear function is a function of the form

where
m and b are given numbers

• Slope = m =

**Exercise 1** Find the slope of the line passing through (−1, 1) and

(2, 7).

**• Equation of line passing through a point (x**_{1}, y_{1}) and with a given slope
m:

If (x, y) is another point on the line then
So we have

point-slope form :

**Exercise 2** Find the equation of the line through
(−1, 1) and with slope 2.

**Exercise 3** A small surf shop has fixed expenses of
$850 per month. Each surfboard costs $100 to

make and sells for $550.

(a) Write the monthly cost, revenue, and profit as
functions of the number of surfboards made.

Cost function = C(x)

Revenue function = R(x)

Profit function = P(x)

(b) Find the break-even point.

Ans. x ≈ 2

**Exercise 4** The demand curve of bread in a bakery
shop is q = D(p) = −50(p − 5) and its supply

curve is q = S(p) = 50(p − 1), where the price p is in dollars and the quantity
q is in loaves. Find the

equilibrium price p_{e} and equilibrium quantity q_{e}.

Ans. p_{e} = 3, q_{e} = 100

► A quadratic function is a function of the form f(x) =
ax^{2} + bx + c, where a ≠ 0, b and c are

given numbers. It always can be written in the informative form f(x) = a(x −
h)^{2} + k, which is a

horizontal translation by h and a vertical translation by k of the simple
parabola f (x) = ax^{2}.

**Exercise 5** Consider the quadratic function f(x) =
−x^{2} + 6x − 5.

(i) Complete the square to write it in the form f(x) = a(x − h)^{2} + k.

(ii) Use (i) to decide whether f(x) has a minimum value or a maximum value and
where it is taken.

(iii) Use (i) to find the roots of f (x).

(iv) De termine the axis of symmetry and the y- intercept and sketch the graph of
f (x).

**Exercise 6 **A furniture company making oak desks has
a fixed cost of $5, 000 per month and a cost

per desk of $500. Find how many desks per month it should produce to maximize
its profit if the price

is given by p = 1000 − 2.5x, where x denotes the number of oak desks produced by
the company.

Ans. x = 100

**Exercise 7** Consider the quadtratic f(x) = x^{2} − 5x
+ 4.

(a) Find its zeros using the quadratic formula :

(b) Factor it .

(c) Determine its sign .