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1. Topic 4: Lines, Parabolas and Systems
MATH6102 – Business Mathematics
Week 4

2. 4.1. Lines
4.2. Linear Function
4.3. Quadratic Functions
4.4. Systems of Linear Equations
4.5. Nonlinear System
4.6. Exponential Functions
Chapter Outline

3. 4.1. LINES

4.  Let (x1, y1) and (x2, y2) be to different points on a nonvertical line. The
slope of the line is
𝑚 =
𝑦2−𝑦1
𝑥2−𝑥1
 (slope formula)
𝑦 − 𝑦1 = 𝑚(𝑥 − 𝑥1)  (point-slope form)
𝑦 = 𝑚𝑥 + 𝑏  (slope-intercept form)
𝑥 = constant  (vertical line)
𝑦 = constant  (horizontal line)
 In summary, we can characterize the orientation of a line by its slope:
Zero slope  horizontal line
Undefined slope  vertical line
Positive slope  line rises from left to right
Negative slope  line falls from left to right
Slope of a Line

5. Slope of a Line

6. Example: Point-Slope
Form
Find an equation of the line that has slope 2 and
passes through (1, -3).
Solution:
Using a point-slope m = 2 and (x1, y1) = (1, -3) gives
y – y1 = m (x – x1)
y – (-3) = 2 𝑥 − 1
y + 3 = 2x – 2
2x – y – 5 = 0

7. Example: Determining a Line
from Two Points
Find an equation of the line passing through (-3, 8) and (4, -2).
Solution:
The line has slope:
m =
𝑦2−𝑦1
𝑥2−𝑥1
=
−2 −8
4 −(−3)
= −
10
7
Using a point-slope from with (-3, 8) as (x1, y1) gives
y – y1 = m (x – x1)
y – 8 = −
10
7
[x – (-3)]
y – 8 = −
10
7
(x + 3)
7(y – 8) = −10 (x + 3)
7y – 56 = −10x + 30
10x + 7y – 26 = 0

8. Example: Slope-Intercept
Form
Find an equation of the line with slope 3 and y-intercept – 4
Solution:
Using a slope-intercept form y = mx + b with m = 3 and b = - 4
gives
y = mx + b
y = 3x – 4

9. Example: Find the Slope
and y-Intercept of a Line
Find the slope and and y-intercept with equation y = 5(3 – 2x)
Solution:
y = mx + b
y = 5(3 – 2x)
y = 15 – 10x
Thus, m = -10 and b = 15, so the slope is -10 and and y-intercept is 15

10. 4.2. LINEAR FUNCTION

11.  A linear function was defined as a polynomial function of
degree 1.
 A function f is a linear function which can be written as f(x) =
ax + b where a ≠ 0
 Example, f(x) = 2x + 1
y = 2x + 1
Linear Function

12. Graphing a General Linear
Equation
Sketch the graph of 2x – 3y + 6 = 0
Solution:
• If x = 0, then – 3y + 6 = 0, so the y-intercept is 2,
line passing through (0, 2).
• If y = 0, then 2x + 6 = 0, so the x-intercept is -3,
line passing through (-3, 0).

13. 4.3. QUADRATIC FUNCTIONS

14.  A quadratic function was defined as a polynomial function
of degree 2
 Quadratic function is written as

where a, b and c are constants and a≠0
 Example,
 , is not quadratic
  c
bx
ax
x
f 

 2
  2
3
2


 x
x
x
f
2
1
)
(
x
x
f 
Quadratic Functions

15. The graph of quadratic function
is a parabola
1. If a > 0, the parabola opens upward. If a < 0, the parabola
opens downward.
2. The vertex is
3. The y-intercept is c
4. Axis of symmetry, factorial or
  c
bx
ax
x
f
y 


 2







 


a
ac
b
a
b
4
4
,
2
2
Graphing a Quadratic Function

16. Graph the quadratic function
Solution:
1. a = -1, b= -4, c = 12, so a < 0, it opens downward
2. The vertex is (-2, 16)
3. Y = 12
4. Axis of symmetry - factorial
  12
4
2




 X
X
x
f
Y
2
)
1
(
2
4
2








a
b
X
  
2
2
and
6
1
2
6
0
12
4
0 2










X
X
X
X
X
X
16
)
1
(
4
)
12
)(
1
(
4
)
4
(
4
4 2
2










a
ac
b
Y
Example: Graphing a Quadratic Function

17. 4.4. SYSTEMS OF LINEAR
EQUATIONS

18. Systems of Linear
Equations
• To solve systems of linear equations in both
two and three variables by using the technique
of elimination by addition or by substitution.

19. Example:
Solving a Two-Variable Linear System
Use elimination by addition to solve the system.
3𝑥 − 4𝑦 = 13
3𝑦 + 2𝑥 = 3
Solution:
3𝑥 − 4𝑦 = 13 | x 3  9𝑥 − 12𝑦 = 39
3𝑦 + 2𝑥 = 3 | x 4  8𝑥 + 12𝑦 = 12
17𝑥 = 51
𝑥 = 51/17
𝒙 = 𝟑
3𝑦 + 2𝑥 = 3
3𝑦 + 2(3) = 3
3𝑦 = 3 – 6
𝒚 = −𝟏

21. 4.5. NONLINEAR SYSTEM

22. • A system of equations with at least one nonlinear equation is
called a nonlinear system.
• if a nonlinear system contains a linear equation, usually solve
the linear equation for one variable and substitute for that
variable in the other equation
• Example,
Solution:










0
1
3
0
7
2
2
y
x
y
x
x
 
  
7
or
8
2
or
3
0
2
3
0
6
0
7
1
3
2
2
2

















y
y
x
x
x
x
x
x
x
x
x
Nonlinear System

24. 4.6. EXPONENTIAL FUNCTIONS

25. • The function f defined by
where b > 0, b  1, and the exponent x is any real number, is
called an exponential function.
• Rules for Exponents:
  x
b
x
f 

26. Example: Exponent
1. 32 = 3 x 3 = 9
2. 24 x 23= 24+3 = 27= 2 x 2 x 2 x 2 x 2 x 2 x 2
3. 55 : 53 = 55-3 = 52= 5 x 5
4. (4 x 2)3 = 43 x 23 = (4 x 4 x 4) x (2 x 2 x 2)

27. Graph the Exponential
Function
Example:
f(x) = (1/2)x
Solution:

28. Properties of Exponential
Functions

29. References
• Ernest F. Haeussler,Richard S. Paul,Richard J. Wood. (2019).
Introductory Mathematical Analysis for Business,
Economics, and the Life and Social Sciences. 14. Pearson
Canada Inc. Ontario. ISBN: 9780134141107.
• Chapter 3, 4.

30. Thank You