Linear functions
The equation of a line can be written in the form:
y = mx + b
where m = gradient
rise
m
run
and
b = y-intercept
Recall that the y-value is called the dependent variable as it depends
upon the x-value.
The x-value is called the independent variable as it can take on any
value we wish and then the y-value is calculated.
This is called a linear function because when it is
graphed it forms a straight line.
y
Other functions
6
4
We will look at several other types of functions.
Quadratics: y = ax2+ bx + c
the highest power is squared.
2
-6
-4
-2
2
4
x
6
-2
-4
y
6
-6
4
Cubic
ax3+
2
bx2
y=
+ cx + d
the highest power is cubed.
-6
-4
-2
2
4
x
6
-2
y
-4
6
-6
4
Exponential
y = ax
the x is in the power.
2
y
-6
-4
-2
2
y = a/x
or
xy = a
6
x
6
-2
4
-4
2
-6
Hyperbolic
4
-6
-4
-2
2
-2
-4
-6
4
6
x
Finding the equation from a table of values
Sometimes we are given a table of values and asked to find the
equation of the line that it forms.
1
x
y
1
5
2
2
8
3
5
4 9
14 29
6
15
To find the equation of a linear function such as this, we first work
out the gradient, then the y-intercept.
To find the gradient we need to know the rise or change in y-values
and the run or change in x-values.
rise 3 6 15
   3
m
5
run 1 2
If
y = mx + b
Then y = 3x + b
Sub in ANY point to find b.
so

5=3×1+b
b=2
y = 3x + 2
Example 1
Sian sells used cars and is paid $200 per week plus 5% commission
on her sales.
a) Form an equation for her pay
P
b) Complete the table of values.
1200
c) Graph the results.
d) From the graph:
1000
i) if Sian sells $16 000 what is
her pay?
800
ii) if Sian is paid $700, how
much did she sell?
600
a) P = 0·05S + 200
S 0 4000 12000 20000 400
P 200 400 800 1200
200
d i) $1000
d ii) $10 000
4000
8000 12000 16000 20000S
Example 2
State what type of function (linear, quadratic, cubic, exponential or
hyperbolic) each of the following are.
a) y = 8  5x
Linear
b) y = x3
Cubic
c) xy = 7
Hyperbolic
d) y = 3x
Exponential
e) y = x2 +7
Quadratic
f) y = 5/x
Hyperbolic
g) y = 10  5x + x2
Quadratic
h) x + y = 8
Linear
i) y = x3 + x2  7x + 9
Cubic
y
y
6
j)
k)
4
-4
-2
4
Quadratic
2
-6
y
6
2
4
6
x
2
-6
-4
-2
6
l)
Hyperbolic
2
-2
-2
-4
-4
4
6
4
x
2
Exponential
-6
-6
-6
-4
-2
2
4
6
x
Today’s work
Exercise 12A pg 361
2efgh, 3, 5, 9, 10