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1. Graphing polynomials
The graphs of polynomial functions of the form
| \(p(x)=a_nx^n+a_{n_1x^{n-1}}+\text…a_1x^1+a_0x^0\) |
depend on the degree n and the leading coefficient \(a_n\).
If n is an even number (i.e. n=2, 4, 6, 8,…), the ends of the graphs move towards the same direction. This means either both ends are moving upwards or both ends are moving downwards.
If n is an odd number (i.e. n=1, 3, 5, 7,…), the ends of the graphs move in opposite directions. This means if the left end is moving upwards, the right end is moving downwards and if the left end is moving downwards, then the right end is moving upwards.
- If n is an even number and \(a_n\) is a positive number, both ends move upwards.
- If n is an even number and an is a negative number, both ends move downwards.
- If n is an odd number and \(a_n\) has a positive value, the right end moves upwards and the left end moves downwards.
- If n is an odd number and \(a_n\) has a negative value, the left end moves upwards and the right end moves downwards.
2. Graph of linear function
A linear function also known as a first degree polynomial has the form:
The graph of a linear function is a straight line. Since n is an odd number that is n=1, the ends move in opposite directions.
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Diagram 1
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3. Graph of quadratic function
A quadratic function is a second degree polynomial of the form:
The graph of a quadratic function is called a parabola. A parabola either opens upwards or opens downwards, depending on the value of a.
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Diagram 2
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4. Graph of qiubic function
A qiubic function is a third degree polynomial of the form:
Since n is an odd number that is n=3, the ends move in opposite directions.
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Diagram 3
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5. Graph of quadruple function
A quadruple function is a fourth degree polynomial of the form:
| \(y=ax^4+bx^3+cx^2+dx+e\) |
Since n is an even number that is n=4, the ends move in the same directions.
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Diagram 4
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6. Graphing polynomials using factored form
The graph of a polynomial function can be easily drawn by factoring the polynomials. The solutions to p(x)=0, which are the x-intercepts, divide the horizontal axis into intervals. The sign on each interval can be determined by choosing a test point.
Example
The factored form for the polynomial \(y=x^3+2x^2-3x\) is:
\(y=x^3+2x^2-3x\)
\(=x(x^2+2x-3)\)
\(=x(x+3)(x-1)\)
The x-intercepts are x=0, x=-3 and x=1.
Choosing the following test points, the position of the graph can be determnined.
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Interval
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Test point
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Value of y
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Position of graph
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x<-3
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x=-5
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y=-60
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Below y-axis
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-3
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x=-1
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y=+4
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Above y-axis
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0
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x=0.5
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y=-78
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Below y-axis
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x>1
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x=2
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y=+10
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Above y-axis
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The graph is then as shown in Diagram 5.
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Diagram 5
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