The graph of a polynomial function has certain features that are not so clear without a visual representation. One of these features is the symmetry axis, a vertical line that divides the graph into two symmetrical mirrored sides. Finding the axis of symmetry of a polynomial function is relatively easy, as there are two simple methods for this.
Steps
Method 1 of 2: Finding the axis of symmetry of degree 2 polynomials
Step 1. Check the degree of the polynomial in question
The degree (or order) of a polynomial is basically the largest exponent present in the expression. If the degree of a polynomial is 2 (which means that no exponent is greater than x2), you can find the axis of symmetry using this method. If the degree of the polynomial is greater than 2, use Method 2.
To illustrate, let's take the 2x polynomial as an example2 + 3x – 1. The largest exponent of the expression is x2, so it is a second order polynomial, which allows you to use this method to find the axis of symmetry.
Step 2. Substitute your numbers into the symmetry axis formula
To calculate the axis of symmetry of a second-order polynomial of the form ax2 + bx +c (a parabola), use the formula x = -b / 2a.
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In the example above, a = 2 b = 3 and c = -1. Replace the values and you will find:
x = -3 / 2(2) = -3/4.
Step 3. Write the axis of symmetry equation
The value you calculated using the formula above represents the point where the symmetry axis cuts the x axis.
In the example above, the axis of symmetry is the line x = -3/4
Method 2 of 2: Finding the Symmetry Axis Graphically
Step 1. Check the degree of the polynomial in question
The degree (or order) of a polynomial is basically the largest exponent present in the expression. If the degree of a polynomial is 2 (which means that no exponent is greater than x2), you can find the axis of symmetry using the method above, which uses a formula. However, if the degree is greater than 2, use this graphical method.
Step 2. Draw the x and y axes
Make two lines in the form of a "+" sign. The horizontal line will be the x axis while the vertical line will be the y axis.
Step 3. Number the chart
Mark the two axes with numbers leaving equal gaps between them.
Step 4. Calculate y = f(x) for each value of x
Use the polynomial function to calculate the values of f(x) by substituting the values of x into it.
Step 5. Make a dot on the graph for each pair
You will now have a value of y = f(x) for each value of x. For each (x, y) pair, make a point on the graph marking the meeting place by going vertically on the x-axis and horizontally on the y-axis.
Step 6. Draw the polynomial graph
After marking all points on the graph, you can connect them all together to reveal the continuous graph of the polynomial.
Step 7. Look for the axis of symmetry
Watch the chart carefully. Look for a point on it where, if a line were drawn, the graph would be split into two equal mirrored parts.
Step 8. Realize the axis of symmetry
If you can find a point, let's call it “b”, on the x-axis, where a line across it would split the graph into two equal mirror halves, then the line x = b is the symmetry axis you're looking for.
Tips
- The size of the x and y axes must be large enough to allow the graph format to be seen clearly.
- Some polynomials are not symmetric. The polynomial y = 3x, for example, has no axis of symmetry.
- The symmetry of a polynomial can be classified as even or odd. A graph that has an axis of symmetry on the y-axis has an even symmetry. If symmetry is on the x-axis, it will be odd.