Polynomials

Generally, polynomials are functions of

f (x) = a_nxⁿ + a_n-1x^n-1 + ... + a₁x + a₀,

where
n is the degree of the polynomials and it must be positive integer,
a_n is the leading coefficient of the polynomials and
a_i are coefficients.

Table below shows some examples of polynomials:

Function	Degree	Leading Coefficient	Type of Polynomial
4	0	4	Constant
5x + 1	1	5	Linear
25 - x2	2	-1	Quadratic
(x-1)2(1-2x)	3	-2	Cubic
(x-1)2(3x-1)2	4	9	Quartic

The domain for polynomial is always (-∞,∞); while the range is also (-∞,∞) for odd degree but for even degree we have to find it according to the function given. The graph shown below are the basic form of polynomials: f (x) = xⁿ and g(x) = -xⁿ. Try to slide the value of n to look at different types of the polynomials.

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Quartic Function

A quartic function is given as f (x) = ax⁴ + bx³ + cx² + dx + e where a, b, c, d and e are coefficients. a controls whether the graph ends upwards or downwards, while e tells us the y-intercept.


Slide the values of a, b, c, d and e to observe the changes of the graph f (x) = ax⁴ + bx³ + cx² + dx + e.

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Cubic Function

A cubic function is given as f (x) = ax³ + bx² + cx + d. The value of a decides the graph is mainly increasing or decreasing, while the value of d tells the y-intercept of the graph.


Slide the values of a, b, c and d to observe the changes of the graph f (x) = ax³ + bx² + cx + d.

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Quadratic Function

A quadratic function is given as f (x) = ax² + bx + c where a, b and c are coefficients. Sometimes, it is also written in the form of f (x) = a(x - h)² + k where (h, k) is the coordinates of the vertex. The graph below shows the quadratic curves in the first form (blue) and the second form (purple). The red dotted curve is the basic form of quadratic equation y = x².


Slide the values of a, b, c, h and k to observe the change of the graph f(x) = ax² + bx + c and f(x) = a(x - h)² + k. Can you get the formulae relating the values of a, b and c with the values of h and k?

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Linear Function

A linear function is given as f (x) = mx + c where m is the slope of the graph and c is the y-intercept.

Slide the values of m and c to observe the changes of graph f (x) = mx + c. You may realize that for

1. m > 0, the graph is increasing;
2. m = 0, the graph is constant;
3. m < 0, the graph is decreasing;

while for

1. c > 0, the graph has positive y-intercept;
2. c = 0, the graph passes through the origin;
3. c < 0, the graph has negative y-intercept.

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Constant Function

Constant function is given as f (x) = c where c is any real number.


Slide the value of c to observe the change of the graph f (x) = c. You may find out that for

1. c > 0, the line falls above the x-axis;
2. c = 0, the line falls on the x-axis;
3. c < 0, the line falls below the x-axis.

Betty, Created with GeoGebra