The graph of a polynomial or function reveals many features that would not be clear without a visual representation of the graph. One of these features is the axis of symmetry: a vertical line that divides the graph into two mirror and symmetrical images. Finding the axis of symmetry for a given polynomial is quite simple. Here are the two basic methods.
Steps
Method 1 of 2: Finding the Axis of Symmetry for Second Degree Polynomials
Step 1. Check the degree of the polynomial
The degree (or "order") of a polynomial is simply the highest exponent of the expression. If the degree of the polynomial is 2 (i.e. there is no exponent higher than x2), you can find the axis of symmetry using this method. If the degree of the polynomial is greater than two, use Method 2.
To illustrate this method, let's take the 2x polynomial as an example2 + 3x - 1. The highest exponent present is x2, so it is a second degree polynomial and it is possible to use the first method to find the axis of symmetry.
Step 2. Enter the numbers into the formula to find the axis of symmetry
To calculate the axis of symmetry of a second degree polynomial in the form x2 + bx + c (a parabola), use the formula x = -b / 2a.
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In the given example, a = 2, b = 3, and c = -1. Enter these values into the formula and you will get:
x = -3 / 2 (2) = -3/4.
Step 3. Write the equation of the axis of symmetry
The value calculated with the symmetry axis formula is the intersection of the symmetry axis with the abscissa axis.
In the given example, the symmetry axis is -3/4
Method 2 of 2: Graphically Find the Axis of Symmetry
Step 1. Check the degree of the polynomial
The degree (or "order") of a polynomial is simply the highest exponent of the expression. If the degree of the polynomial is 2 (i.e. there is no exponent higher than x2), you can find the axis of symmetry using the method described above. If the degree of the polynomial is greater than two, use the graphical method below.
Step 2. Draw the x and y axes
Draw two lines to form a kind of "plus" sign or a cross. The horizontal line is the abscissa axis, or x axis; the vertical line is the ordinate axis, or y axis.
Step 3. Number the chart
Mark both axes with numbers ordered at regular intervals. The distance between the numbers must be uniform on both axes.
Step 4. Calculate y = f (x) for each x
Take the function or polynomial into account and calculate the values of f (x) by inserting the values of x.
Step 5. For each pair of coordinates locate the corresponding point in the graph
You now have pairs of y = f (x) for each x on the axis. For each pair of coordinates (x, y), locate a point on the graph - vertically on the x-axis and horizontally on the y-axis.
Step 6. Draw the graph of the polynomial
After identifying all the points on the graph, connect them with a regular and continuous line to highlight the trend of the polynomial graph.
Step 7. Look for the axis of symmetry
Look carefully at the graph. Look for a point on the axis such that, if a line crosses it, the graph splits into two equal and mirrored halves.
Step 8. Find the axis of symmetry
If you have found a point - let's call it "b" - on the x axis, such that the graph splits into two mirror halves, then that "b" point is the axis of symmetry.
Advice
- The length of the abscissa and ordinate axes should be such as to allow a clear view of the graph.
- Some polynomials are not symmetric. For example, y = 3x does not have an axis of symmetry.
- The symmetry of a polynomial can be classified into even or odd symmetry. Any graph that has an axis of symmetry on the y axis has "even" symmetry; any graph that has an axis of symmetry on the x axis has "odd" symmetry.
