The graph of a polynomial or function reveals many properties that would not be obvious without being visually depicted. One of these properties is the axis of symmetry: the vertical line on the graph that divides the graph into two symmetrical mirror images. Finding the axis of symmetry for a given polynomial is quite easy. There are two basic ways.
Step
Method 1 of 2: Finding the Axis of Symmetry for a Level 2 Polynomial
Step 1. Check the degree of your polynomial
The degree (or "power") of a polynomial is simply the value of the largest exponent or power in an expression. If the degree of your polynomial is 2 (no exponent is greater than x2), you can find the axis of symmetry using this method. If the degree of your polynomial is more than 2, use Method 2.
To illustrate, take the polynomial 2x2 + 3x – 1 for example. The highest exponent in a polynomial is x2, so this polynomial is a degree 2 polynomial, and you can use this first method to find the axis of symmetry.
Step 2. Plug your numbers into the axis of symmetry formula
To calculate the axis of symmetry of a second degree polynomial of the form ax2 + bx + c (parabola), use the basic formula x = -b / 2a.
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In the example above, a = 2, b = 3, and c = -1. Plug these values into your formula, and you'll get:
x = -3 / 2(2) = -3/4.
Step 3. Write down the equation for the axis of symmetry
The value that you have calculated with the axis of symmetry formula is the x-intercept of the axis of symmetry.
In the example above, the axis of symmetry is -3/4
Method 2 of 2: Finding the Axis of Symmetry Using the Graph
Step 1. Check the degree of your polynomial
The degree (or "power") of a polynomial is simply the value of the largest exponent or power in an expression. If the degree of your polynomial is 2 (no exponent is greater than x2), you can find the axis of symmetry using this method. If the degree of your polynomial is more than 2, use the graphical method.
Step 2. Draw the x and y axes
Make two lines with a plus sign shape. The horizontal line is your x-axis; the vertical line is your y-axis.
Step 3. Put a number on your graph
Mark both axes with numbers at equal intervals. The distance between the numbers must be uniform on both axes.
Step 4. Calculate y = f(x) for each x
Take your polynomial or function and calculate the value of f(x) by plugging all the x values into it.
Step 5. Draw a point graph for each pair
Now, you have a pair of y = f(x) for each x on the axis. For each pair (x, y), draw a point on the graph – vertically on the x-axis and horizontally on the y-axis.
Step 6. Draw a graph of the polynomial
Once you've marked all the points of the graph, you can seamlessly connect your dots to see a continuous graph of your polynomial.
Step 7. Find the axis of symmetry
Check your charts carefully. Find the point on the axis that divides the graph into two equal parts and is a reflection of when a line passes through that point.
Step 8. Record the axis of symmetry
If you can find a point – let's say “b” – on the x-axis, that divides the graph into two reflecting halves, then that point, b, is your axis of symmetry.
Tips
- The length of your x and y axes should allow the overall shape of the graph to be clearly visible.
- Some polynomials are not symmetrical. For example, y = 3x has no axis of symmetry.
- The symmetry of a polynomial can be classified as odd or even 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 is "odd" symmetry.
