School

Kent State UniversityDepartment

Business Administration InterdisciplinaryCourse Code

BUS 10123Professor

Eric Von HendrixLecture

13This

**preview**shows pages 1-2. to view the full**6 pages of the document.**• Chapter 2

• Mathematical and Statistical Foundations

• Functions

• A function is a mapping or relationship between an input or set of inputs and an output

• We write that y, the output, is a function f of x, the input, or y = f(x)

• y could be a linear function of x where the relationship can be expressed on a straight

line

• Or it could be non-linear where it would be expressed graphically as a curve

• If the equation is linear, we would write the relationship as

y = a + bx

where y and x are called variables and a and b are parameters

• a is the intercept and b is the slope or gradient

• Straight Lines

• The intercept is the point at which the line crosses the y-axis

• Example: suppose that we were modelling the relationship between a student’s average

mark, y (in percent), and the number of hours studied per year, x

• Suppose that the relationship can be written as a linear function

y = 25 + 0.05x

• The intercept, a, is 25 and the slope, b, is 0.05

• This means that with no study (x=0), the student could expect to earn a mark of 25%

• For every hour of study, the grade would on average improve by 0.05%, so another 100

hours of study would lead to a 5% increase in the mark

• Plot of Hours Studied Against Mark Obtained

• Straight Lines

• In the graph above, the slope is positive

– i.e. the line slopes upwards from left to right

• But in other examples the gradient could be zero or negative

• For a straight line the slope is constant – i.e. the same along the whole line

• In general, we can calculate the slope of a straight line by taking any two points on the

line and dividing the change in y by the change in x

• (Delta) denotes the change in a variable

• For example, take two points x=100, y=30 and x=1000, y=75

• We can write these using coordinate notation (x,y) as (100,30) and (1000,75)

• We would calculate the slope as

• Roots

• The point at which a line crosses the x-axis is known as the root

• A straight line will have one root (except for a horizontal line such as y=4 which has no

roots)

• To find the root of an equation set y to zero and rearrange

0 = 25 + 0.05x

• So the root is x = −500

• In this case it does not have a sensible interpretation: the number of hours of study

required to obtain a mark of zero!

• Quadratic Functions

Only pages 1-2 are available for preview. Some parts have been intentionally blurred.

• A linear function is often not sufficiently flexible to accurately describe the relationship

between two series

• We could use a quadratic function instead. We would write it as

y = a + bx + cx2

where a, b, c are the parameters that describe the shape of the function

• Quadratics have an additional parameter compared with linear functions

• The linear function is a special case of a quadratic where c=0

• a still represents where the function crosses the y-axis

• As x becomes very large, the x2 term will come to dominate

• Thus if c is positive, the function will be -shaped, while if c is negative it will be -

shaped.

• The Roots of Quadratic Functions

• A quadratic equation has two roots

• The roots may be distinct (i.e., different from one another), or they may be the same

(repeated roots); they may be real numbers (e.g., 1.7, -2.357, 4, etc.) or what are known

as complex numbers

• The roots can be obtained either by factorising the equation (contracting it into

parentheses), by ‘completing the square’, or by using the formula:

• The Roots of Quadratic Functions (Cont’d)

• If b2 > 4ac, the function will have two unique roots and it will cross the x-axis in two

separate places

• If b2 = 4ac, the function will have two equal roots and it will only cross the x-axis in one

place

• If b2 < 4ac, the function will have no real roots (only complex roots), it will not cross the

x-axis at all and thus the function will always be above the x-axis.

• Calculating the Roots of Quadratics - Examples

Determine the roots of the following quadratic equations:

1. y = x2 + x − 6

2. y = 9x2 + 6x + 1

3. y = x2 − 3x + 1

4. y = x2 − 4x

• Calculating the Roots of Quadratics - Solutions

• We solve these equations by setting them in turn to zero

• We could use the quadratic formula in each case, although it is usually quicker to

determine first whether they factorise

1. x2 + x − 6 = 0 factorises to (x − 2)(x + 3) = 0 and thus the roots are 2 and −3, which are

the values of x that set the function to zero. In other words, the function will cross the x-

axis at x = 2 and x = −3

2. 9x2 + 6x + 1 = 0 factorises to (3x + 1)(3x + 1) = 0 and thus the roots are −1/3 and −1/3.

This is known as repeated roots – since this is a quadratic equation there will always be

two roots but in this case they are both the same.

• Calculating the Roots of Quadratics – Solutions Cont’d

3. x2 − 3x + 1 = 0 does not factorise and so the formula must be used

###### You're Reading a Preview

Unlock to view full version