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This topic is part of the TCS FREE high school mathematics 'How-to Library'. It introduces polynomial functions, and shows you how to draw graphs of polynomials, as well as find zeros and turning points. (See the index page for a list of all available topics in the library.) To make best use of this topic, you need to download the Maths Helper Plus software. Click here for instructions.

Theory:

A polynomial is an expression having terms with decreasing powers of ‘x’, like this:  

2x³ + 3x² - x + 6

If:  axⁿ  is a term of a polynomial, then ‘a’ is the coefficient of the term, and ‘n’ is the index of the term. Polynomials are usually written so that there is only one term with a given index.

Technically, there must be more than one term in a polynomial (poly = many), but missing terms can be included by using zero coefficients, like this:

x³Â  è  x³ + 0x² + 0x + 0

The maximum index in a polynomial is called the degree of the polynomial, thus 5x² - 3x + 1  is a second degree polynomial, and 3x⁴ + 7x³ - 2x² - 9x - is a fourth degree polynomial.

Second degree polynomials are also called quadratic polynomials, and third degree polynomials are also called cubic polynomials.

Polynomial functions and their graphs:

This is an example of a polynomial function: 

y = x⁴ + 3x³ - 9x² - 23x - 12

 The graph of this polynomial function is as follows:

 

In general, polynomial function graphs consist of a smooth line with a series of hills and valleys.  The hills and valleys are called turning points. The maximum possible number of turning points is one less than the degree of the polynomial.

The polynomial above has degree 4 and has three turning points. This is the maximum possible number of turning points for a polynomial of this degree.

Zeros of polynomial functions:

A zero of a polynomial function is an ‘x’ value for which ‘y’ = 0. At these ‘x’ values, its graph cuts or touches the ‘x’ axis. The maximum number of zeros of a polynomial is the same as its degree.

The polynomial function y = x⁴ + 3x³ - 9x² - 23x - 12 graphed above, has only three zeros, at 'x' = -4, -1and 3. This is one less than the maximum of four zeros that a polynomial of degree four can have.

This polynomial intersects the 'x' axis at -4 and 3, but only touches the 'x' axis at 'x' = -1.

Usually you cannot be exactly sure of values you read from a printed graph, but Maths Helper Plus can find zeros and turning points of polynomial functions with very high accuracy.

The 'Method' section below shows you how to graph polynomial functions in Maths Helper plus, then use the software to find the zeros and turning points.

Method

To plot a polynomial function in Maths Helper Plus...

Step 1  Start with an empty Maths Helper Plus document

If you have just launched the software then you already have an empty document, otherwise hold down ‘Ctrl’ while you briefly press the ‘N’ key.

 

Step 2  Enter your polynomial function

1. Press the F3 key to activate the 'input box' for typing (see below):

2. Type your polynomial function into the input box, like this:

             

NOTE: To enter the index of a power, you can always use the '^' character followed by the index. For example, to enter x⁴, you type: x^4. For a power of 2 or 3, you can also use the 'squared' and 'cubed' operators. They do the same thing but look better. See the Maths Helper Plus 'help' to find out how.

 

3. Press Enter to complete the entry

 

Step 3  Adjust the scale of the graph

Most polynomial graph plots are far too big to fit the graphing area.

 

If you see no plot at all, you can try momentarily pressing the F10 key. This doubles the graph scale each time it is pressed. Repeat if necessary. To reduce the scale again,  hold down 'Shift' while you press F10.

 

Most polynomial graphs, including our example, are very large in the vertical ('y') direction and so need compressing vertically.

 

To compress the graph vertically, hold down a 'Ctrl' key, and briefly press the down arrow key. Repeat until all of the hills and valleys of the graph are visible. To undo this operation, use the up arrow key instead of the down arrow key.

 

If necessary, you can also compress the graph horizontally in the 'x' direction. To compress the graph horizontally, hold down 'Ctrl' and briefly press the left arrow key. Use the right arrow key instead of the left arrow key to undo this operation.

 

You can move the whole graph in any direction with a 'panning' operation. To pan, hold down a 'Ctrl' and a 'Shift' key while you briefly press any of the arrow keys.

To locate turning points and zeros with high accuracy in Maths Helper Plus...

Step 1  Graph the polynomial as described above

Step 2  To locate zeros of the function, select the 'intersection tool'.

The 'intersection tool' in Maths Helper Plus will locate most zeros of polynomials with high accuracy. This diagram illustrates the procedure:

At 'x' = -1, the graph does not intersect the 'x' axis, it only touches it, so the intersection tool will not work at 'x' = -1.

The steps are:

1. Click to select the intersection tool.

2. Move the centre of the of the intersection tool cursor over a point where the graph intersects an 'x' or 'y' axis.

3. Click the left mouse button. If the intersection point is found, a small dot will appear at the intersection point on the graph, and the intersection tool dialog box will display the coordinates. (See below.)

4. Repeat for each intersection point required, recording the coordinates in each case.

5. Press the 'Esc' key on your keyboard to cancel the intersection tool.

Step 3  To locate turning points of the function, select the 'turning point tool'.

The 'turning point tool' in Maths Helper Plus will locate turning points of polynomials with high accuracy. This diagram illustrates the procedure:

If the 'y' coordinate of a turning point is zero, then the 'x' value of the turning point is a zero of the function. For this example, the turning point at 'x' = -1 has a zero 'y' coordinate, so x = -1 is a zero of the function.

The steps are:

1. Click to select the turning point tool.

2. Move the centre of the box of the turning point tool cursor over a local maximum or local minimum on the y = f(x) graph.

3. Click the left mouse button. If the turning point is found, a small dot will appear at the turning point on the graph, and the turning point tool dialog box will display the coordinates. (See below.)

4. Repeat for each local maximum or local minimum on the graph, recording the coordinates in each case.

5. Press the 'Esc' key on your keyboard to cancel the turning point tool.