# Axis of Symmetry

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Instructions:
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Use this calculator to find the axis of symmetry of a parabola, showing all the steps. Please provide a valid quadratic function
in the form box below.

## Axis of Symmetry Equation

This calculator will allow you to find the axis of symmetry equation for a given quadratic function, showing all the steps of the process.

You need to provide a valid expression quadratic function. For example, a valid quadratic function is something like 2x² - 5x + 1, but you can also provide a quadratic function that is not fully simplified like 2x² + 5x +3/4 x - x² , as the calculator will conduct any necessary quadratic simplification.

Once you provide a valid quadratic function, you need to click the "Calculate" button, and the solutions with all the steps will be provided.

The axis of symmetry has a strong geometric meaning, and it is the axis that serves as a "mirror" for the graph of a quadratic function, which is a parabola, and it is tightly linked to the roots of the quadratic function.

## Axis of symmetry formula

The graph of a quadratic function ax² +b x + c is a parabola, and this parabola will be symmetric around its axis of symmetry. The axis of symmetry equation is:

\[x = \displaystyle -\frac{b}{2a} \]## What are the steps for finding the axis of symmetry equation?

- Step 1: Identify the quadratic function and simplify it into its form ax² +b x + c
- Step 2: Once you have the quadratic function simplified, make sure that a ≠ 0, otherwise you cannot continue
- Step 3: The axis of symmetry equation is \(x = \displaystyle -\frac{b}{2a} \)
- Step 4: This means that the axis of symmetry is a vertical line, that passes through the point \(\left(\displaystyle -\frac{b}{2a}, 0\right) \)

Observe that this is the case for regular parabolas, without any axes rotation, which goes beyond the scope of this tutorial.

## Axis of symmetry calculator

This parabola calculator will receive the quadratic function provided, it will simplify it into its ax² +b x + c form, and it will plug the values a and b into the formula:

\[x = \displaystyle -\frac{b}{2a} \]But there are also other ways of finding the *axis of symmetry* of a parabola. Suppose that you solve the quadratic equation ax² +b x + c = 0, and you find
the roots u and v. How do you find the axis of symmetry when you know the roots of the quadratic equation?

- Step 1: Identify the given roots of the quadratic equations
- Step 2: You will have two roots u and v. If there is only one root, you define u and v as the same value
- Step 3: The axis of symmetry is found by computing the midpoint of the roots u and v: This is, we have the axis of symmetry formula \(x = \displaystyle \frac{u+v}{2}\). This works for either real or complex roots

When you have complex roots, they will be conjugate complex numbers, so then the average of them will yield a real number.

## Why would care about the axis of symmetry?

The axis of symmetry corresponds to the symmetric line for the graph of quadratic function, which is a parabola. So, having a reference for the symmetry gives a lot of information about the parabola.

For example, the roots of the equation will be disposed symmetrically with respect to this axis of symmetry.

### Example: Axis of symmetry

Consider the following quadratic equation: \(f(x) = 3x^2 + 2x + 1\). Find its axis of symmetry.

Solution:

which concludes the calculation.

### Example: Axis of Symmetry Equation

Assume you have the following quadratic expression: \(f(x) = x^2 + \frac{2}{3}x + \frac{5}{4}\). Use the formula to compute its axis of symmetry.

Solution:

which concludes the calculation.

### Example: Axis of symmetry formula from the roots

Assume that the roots of a quadratic equation are \(r_1 = 3\) and \(r_2 = 5\). Find the axis of symmetry equation of the parabola.

Solution: We know that when the roots are provided, we need to average the roots. Hence, the axis of symmetry equation of the parabola is

\[x = \displaystyle \frac{u+v}{2} = \displaystyle \frac{3+5}{2} = 4\]which concludes the calculation.

## More quadratic calculators

Finding the axis of symmetry of a parabola is just one many things you can do with quadratic functions. You can solve quadratic equations and compute the vertex.

Also, and as you probably have noticed, there is a strong link between the vertex formula and the axis of symmetry: Indeed, the axis of symmetry is a vertical line that passes through the vertex.