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Nonlinear Functions


Linear functions are functions where x is raised only to the first power. On graphs, linear functions are always straight lines.
y = mx + b     3x + 5y - 10 = 0     y = 88x     are all examples of linear equations.
The graphs of nonlinear functions are not straight lines.


In this topic, we will be working with nonlinear functions with the form y = ax2 + b and y = ax3 b where a and b are integers.

Quadratic functions:   y = ax2 + b

The graph of the function y = ax2 + b will look like a "U". This "U" shape graph is called a parabola. When a is positive, then the parabola opens up. When a is negative, then the parabola opens down.

The highest or lowest point of parabolas is called the vertex. b determines where the vertex is on the graph. When b=0, the vertex is on the origin (0,0). When b = h where h is an integer, the vertex is on the point (0, h).
In this graph, the vertex is the lowest point.
b = 0 because the vertex is on the origin.

Use the point (2,12) to find a.
y = ax2
12 = a(2)2
12 = 4a
3 = a

The equation is y = 3x.


In this graph, the vertex is the highest point.
b = -5 because the vertex is on (0, -5).

Use the point (1, -7) to find a.
y = ax2 - 5
-7 = a(1)2 - 5
-7 = a - 5
-2 = a

The equation is y = -2x - 5.


Cubic functions:   y = ax3 + b

The graph of a cubic function has this shape
b = 0 when the point of transition (from an upwards curve to a downwards curve) is on the origin (0,0).
This is an example of y = ax3 where a is negative.

Use the point (1, -2) to find a.
y = ax3
-2 = a(1)3
-2 = a

The equation is y = -2x3.


b = -5 because the point of transition is on (0, -5).

Use the point (1, -2) to find a.
y = ax3 - 5
-2 = a(1)3 - 5
-2 = a - 5
3 = a

The equation is y = 3x3 - 5.


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