Materi 3 Graph of an Equation

Graph of an Equation

MK Matematika I

Dosen Pengampu: Ir. Anis Siti Nurrohkayati, S.T., M.T

Introduction

What is a straight line equation?
What are the properties of straight line
equations?

A straight line graph is a visual
representation of a linear function.

Gradient

Before going into straight line
equations, you should first
understand about gradients.
Why?

Gradient

The gradient of a line can also be
positive or negative. If the line rises
from left to right then the gradient is
positive. Conversely, if the line goes
down from left to right then the
gradient is negative.

https://www.zenius.net/blog/rumus-grafik-persamaan-garis-lurus

Gradient

What does it stand for?

𝑦 = 𝑚𝑥 + 𝑏

y = how far up
x = how far along
m = slope or gradient (how steep the line is)
b = value of y when x = 0

How Do You Find “m” and “b”?

• b is easy: just see where the line crosses the Y axis
• m (the slope) needs some calculation:

• 𝑚 =!"#$%& ($ )

!"#$%& ($ *

Knowing this we can work out the
equation of a straight line:

• 𝑚 =

!

"= 2

• 𝑏 = 1 (value of y when x = 0)
• Putting that into 𝑦 = 𝑚𝑥 + 𝑏, gets us: 𝑦 = 2𝑥 + 1

Cont.

• With that equation we can now

• Choose any value for x and find the matching value for y

For example, when x is 1:

𝑦 = 2×1 + 1 = 3

Check for yourself that x = 1 and y = 3 is actually on the line.

Or we could choose another value for x, such as 7:

𝑦 = 2×7 + 1 = 15

And so when x = 7 you will have y = 15

Definition and Properties of
Straight Line Equations

A straight line equation is a two-variable linear equation with two
unknown variables. The properties of straight line equations are:

• Equations of straight lines that are parallel to each other
• Equations of straight lines that are mutually perpendicular
• Equations of straight lines that coincide with each other
• Equations of intersecting straight lines

Straight Line Equation Formula

Basically, straight line equations have two forms. The first is the implicit
form. Second, explicit form.

Implicit form
2x – y + 1 = 0

Explicit form
y = mx + b

Straight Line Equation Formula

There are two ways to find the equation of a straight line:
First, if the gradient and one of the intersection points are known.
Second, if two or more points are known.

1) Determine the equation of a straight line if the gradient m and one

of the points on the line are known: 𝒚 − 𝒚𝟏 = 𝒎 𝒙 − 𝒙𝟏

2) Determine the equation of a straight line if two points on the line

are known:

𝒚%𝒚𝟏
𝒚𝟐%𝒚𝟏=

𝒙%𝒙𝟏
𝒙𝟐%𝒙𝟏

How to Draw a Function Graph

First, start with a blank graph like
this. It has x-values going left-to-
right, and y-values going bottom-
to-top:

The x-axis and y-axis cross over
where x and y are both zero.

Plotting Points

• A simple (but not perfect) approach is to calculate the

function at some points and then plot them.

• A function graph is the set of points of the values taken

by the function.

Example: 𝑦 = 𝑥!− 5

Let us calculate some
points:

Then, plot them like this:

x

𝑦 = 𝑥!− 5

-3

-2

1

-4

0

-2

5

7

-3; 4

1; -4
0; -5

5; 20

-10

-5

0

5

10

15

20

25

-4

-3

-2

-1

0

1

2

3

4

5

6

Cont.

• Not very helpful yet. Let us add

some more points:

-10

-5

0

5

10

15

20

25

-4

-3

-2

-1

0

1

2

3

4

5

6

• Looking better. We can now guess

that plotting all the points will look
like a nice parabola.

Tips

• We should try to plot enough points to be confident in

what is going on!

• So "plotting some points" is useful, but can lead to

mistakes.

Positive and Negative Slope

• Going from left-to-right, the cyclist has to push on a positive slope:

Example

• 𝑚 =

%'

"= −3

• 𝑏 = 0
• Then gives us:

• 𝑦 = −3𝑥 + 0

We do not need the zero. So: 𝑦 = −3𝑥

Example 3: Vertical Line

• What is the equation for a vertical line?
• The slope is undefined, and where does it cross

the Y-axis?

• In fact, thus is a special case, and we use a different

equation, not “y=…”, but instead we use “x=…”.

• Like this:

• 𝑥 = 1.5
• Every point on the line has x coordinates 1.5, that is why its

equation is x = 1.5

Rise and Run

• Sometimes the words “rise” and “run” are used

• Rise is how far up
• Run is how far along

And so the slope “m” is:

𝑚 =+(,&

+-$

Terimakasih

• https://www.mathsisfun.com/equation_of_line.html