University Maths: Module 4 Linear Graphs Summary Assignment

This assignment is a comprehensive summary of Module 4's linear graphs, covering key concepts such as graphing equations by generating coordinate pairs and plotting them on the Cartesian Plane. It explains the process of finding an equation from a graph using the slope-intercept form (y = mx + c) and the rise-over-run method to determine the gradient and y-intercept. The summary also delves into finding equations using coordinates, including the gradient formula and the two-point formula. Furthermore, it explores the properties of parallel and perpendicular lines, detailing their gradient relationships. The document provides step-by-step examples and clear explanations to reinforce understanding of these fundamental concepts in linear algebra, with references to external resources for further study.

Summary Template – Module 4 Linear Graphs
Use this guide to help you create your own summary of the topics covered in Module 4.
1. Graphing an Equation:
a. To graph an equation, you need use the equation to generate pairs of coordinates
that can then be plotted on the Cartesian Plane.
b. Using the equation 𝑦 = 3𝑥 - 4 complete the table below so that you
have three pairs of coordinates that describe the equation.
𝒙 𝒚
0 -4
2 -2
2 2
Remember:
𝑥 and 𝑦 can take any values, so you
need to choose values for 𝑥 to
substitute into the equation to find 𝑦
(𝑥 = 0 is often a good place to start!)
c. Plot your coordinates on the grid below:
-4 -3 -2 -1 0 1 2 3 4 5 6
-15
-10
-5
0
5
10
15
x
y
Remember:
x Label the axes (𝑥 and 𝑦)

x Choose a suitable scale for each
axis based on the range of your 𝑥
and 𝑦 coordinates – you must go
up in even steps!
x Plot your coordinates (𝑥, 𝑦)
x Join your coordinates with a
straight line and extend in both directions
d. Why should you extend your straight line on the graph in both directions?
Extending the straight line on the graph allows for other points that lie within the line to be
observed. This is vita especially in using the line graph to model future predictions in areas such
sales growth in firms.
2. Finding an Equation from a Graph
a. For a linear equation in the general (slope-intercept) form:
𝑦 = 𝑚𝑥 + 𝑐
𝑥 and 𝑦 are independent and dependent variables respectively.
𝑚 is the slope of the equation
𝑐 is the y intercept, that is the value of y when x is zero (MathPlanet, 2019).
b. Using the Rise-over-Run Method:
i) Draw a suitable triangle onto the graph
and label the ‘rise’ and the ‘run’

-4 -3 -2 -1 0 1 2 3
-6
-4
-2
0
2
4
6
x
y
Rise
y intercept
Write down the values for rise and run:
Rise =( 3−−1 ) =4
Run =( 1−−1 )=2
ii) gradient=m= Δ y
Δ x =rise
run = 4
2 =2
iii) The letter Δ is used to refer a change in the variables
c. Mark the location of the y-intercept (′𝑐′) on the graph.
Marked in the graph
d. Using your answers to b) (iii) and c) write down the equation of the line shown
on the graph (StackExchange, 2019):
Run

y=2 x +1
3. Finding an Equation using Coordinates
a. Write down the Gradient Formula:
y= y2− y1
x2−x1
b. For the coordinates (-6, 9) and (1, -5) identify:
i. 𝑥1 =-6
ii. 𝑦1 =9
iii. 𝑥2 =1
iv. 𝑦2 =-5
d. Substitute your coordinates into the Gradient Formula to find the gradient of
the line between these points (be careful with negatives!)
y=(−5)−9
1−(−6) =−2
e. Using the gradient and one pair of coordinates, find the equation of the line.
(You can sub into 𝑦 = 𝑚𝑥 + 𝑐 or use the Two-Point Formula to do this)
Point (−6 , 9 )∧( x , y)
2= y−9
x+6
2∗ ( x+6 ) =( y−9)
−2 x−12+9= y
y=−2 x−3
4. Parallel and Perpendicular Lines:
a. Parallel lines have the same gradient.
b. Perpendicular lines cross at an angle of 90 °.
c. In your own words, explain the relationship between the gradients of
perpendicular lines.
The gradients0 of perpendicular lines are related by the formula
m1∗m2=−1
Where m1 is the gradient of line 1 and m2 the gradient of line 2.
Answers:

2b) (iii) 𝑚 = 2 2d) 𝑦 = 2𝑥 + 1
3d) 𝑚 = -2 3e) 𝑦 = -2𝑥 – 3
References
MathPlanet. (2019, March 27). How to solve linear equations. Retrieved from MathPlanet:
https://www.mathplanet.com/education/algebra-1/how-to-solve-linear-equations
StackExchange. (2019, March 27). Mathematics. Retrieved from StackExchange:
https://math.stackexchange.com/questions/385810/linear-equations-how-to-write-an-
equation-from-given-coordinates