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Lecture 13

BUS 10123 Lecture Notes - Lecture 13: Quadratic Equation, The Intercept, The Roots


Department
Business Administration Interdisciplinary
Course Code
BUS 10123
Professor
Eric Von Hendrix
Lecture
13

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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

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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
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