graphs_and_equations.jpgPlease note that this topic is now assessed at NCEA, Level 2 (Year 12).

There are several other types of graphs. These include hyperbolas, circles, growth curves and cubics.
These are shown below.

 

Hyperbolas
Circles
Exponential Functions
Cubics
Summary of Graphs

 

Hyperbolas

The general equation of a hyperbola is xy = c or Y10_Other_Graphs_01.gif

 

If c is positive.

e.g. xy = 4

If c is negative.

e.g. xy = - 4

Y10_Other_Graphs_02.gif
Y10_Other_Graphs_03.gif
Graph is in first and third quadrants.
Graph is in second and fourth quadrants.

 


 

Circles

The general equation of a circle is x 2 + y 2 = a 2. , where a is the radius of the circle.

This relation is not a function, because some of the x-values have two corresponding y-values.

e.g. x 2 + y 2 = 9

Y10_Other_Graphs_04.gif

 


 

Exponential Functions or Growth Curves

The general equation is y = ax

The greater the value of a, the steeper is the curve.

All growth curves of this type pass through the point (0, 1)

e.g. y = 3Y10_Other_Graphs_05.gif

 


 

Cubics

The general equation of a cubic function is y = (x − a)(x − b)(x − c)

The simplest cubic is y = x3.

e.g.

 

y = x3

y = (x − 1)(x − 3)(x + 2)

Y10_Other_Graphs_06.gif
Y10_Other_Graphs_07.gif


Summary of Graphs

 

Type of graph
General equation
Characteristics
Example

Straight line

ax + by + c = 0

y = mx + c

x and y terms both have exponents of 1

x + 2y = 6

y = 3x + 4

Parabola

y = ax 2 + bx + c

y = a(x − h)2 + k

y term has exponent of 1

x term has exponent of 2

y = x 2 + 4x + 3

y = 2(x − 3)2 + 4

Hyperbola

xy = c

x and y terms have exponents of 1

xy = 6

Circle

x 2 + y 2 = a2

x and y terms both have exponents of 2

x 2 + y 2 = 16

Growth curve
(exponential)

y = a x

x is the exponent

y = 5x

Cubic

y = (x − a)(x − b)(x − c)

x term has exponent of 3

y = (x − a)(x − b)(x − c)