How to Find an Axis of Symmetry: 11 Steps

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How to Find an Axis of Symmetry: 11 Steps
How to Find an Axis of Symmetry: 11 Steps

Video: How to Find an Axis of Symmetry: 11 Steps

Video: How to Find an Axis of Symmetry: 11 Steps
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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

Find an Axis of Symmetry Step 1
Find an Axis of Symmetry Step 1

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.

Find an Axis of Symmetry Step 2
Find an Axis of Symmetry Step 2

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.

  • In the example above, a = 2 b = 3 and c = -1. Replace the values and you will find:

    x = -3 / 2(2) = -3/4.

Find an Axis of Symmetry Step 3
Find an Axis of Symmetry Step 3

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

Find an Axis of Symmetry Step 4
Find an Axis of Symmetry Step 4

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.

Find an Axis of Symmetry Step 5
Find an Axis of Symmetry Step 5

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.

Find an Axis of Symmetry Step 6
Find an Axis of Symmetry Step 6

Step 3. Number the chart

Mark the two axes with numbers leaving equal gaps between them.

Find an Axis of Symmetry Step 7
Find an Axis of Symmetry Step 7

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.

Find an Axis of Symmetry Step 8
Find an Axis of Symmetry Step 8

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.

Find an Axis of Symmetry Step 9
Find an Axis of Symmetry Step 9

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.

Find an Axis of Symmetry Step 10
Find an Axis of Symmetry Step 10

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.

Find an Axis of Symmetry Step 11
Find an Axis of Symmetry Step 11

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.

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