Axis of Symmetry

What is Axis of Symmetry?

The axis of symmetry in mathematics refers to a line that divides an object into two mirror-image halves. In the context of a quadratic function or the graph of a quadratic equation, which is a parabola, the axis of symmetry is a vertical line that passes through the vertex of the parabola and splits it into two congruent halves. Each point on one side of the axis has a corresponding point with the same distance from the axis on the other side, and the y-values of these corresponding points are equal.

For a quadratic function in its standard form:

$f(x) = ax^2 + bx + c, $

the axis of symmetry can be found using the formula:

$x = -\frac{b}{2a}, $

where $ a $ and $ b $ are the coefficients of the $ x^2 $ and $ x $ terms, respectively.

This axis of symmetry is important because:

• It helps in graphing the parabola, as once you know where the axis is, you can plot points on one side and reflect them across the axis to find corresponding points on the other side.
• The vertex of the parabola lies on the axis of symmetry. The x-coordinate of the vertex is given by $ x = -\frac{b}{2a} $, while the y-coordinate can be found by substituting this x-value back into the function to get $ f(-\frac{b}{2a}) $.
• For a parabola that opens upward (where $ a > 0 $), the vertex represents the minimum point on the graph; for one that opens downward (where $ a < 0 $), the vertex represents the maximum point.
• The axis of symmetry can be used in optimization problems to find the maximum or minimum value of a quadratic function.

In geometric shapes other than parabolas, such as circles, ellipses, squares, rectangles, and many other figures, the axis of symmetry also divides the shape into mirror-image halves, and the shape can have more than one axis of symmetry. However, in the specific case of a parabola described by a quadratic equation, there is only one unique axis of symmetry.

Axis of Symmetry of a Parabola

The axis of symmetry of a parabola is a straight line that vertically splits the parabola into two mirror-image halves. In the Cartesian coordinate system, where the parabola's equation is given in the form $ y = ax^2 + bx + c $, the axis of symmetry is vertical and its equation can be found using the formula:

$x = -\frac{b}{2a} $

Here, $ a $ and $ b $ are coefficients from the quadratic equation, with $ a $ corresponding to the quadratic term and $ b $ corresponding to the linear term. This equation yields the x-coordinate of the vertex of the parabola, which lies on the axis of symmetry.

Key Features of the Axis of Symmetry of a Parabola:

1. Perpendicular to the x-axis: For parabolas represented by the standard quadratic function, the axis of symmetry is always vertical.

2. Passes through the vertex: The axis of symmetry passes through the highest point on the parabola (for parabolas that open downward) or the lowest point on the parabola (for parabolas that open upward). This point is referred to as the vertex.

3. Reflective property: The axis of symmetry has the property that if you reflect a point from one side of the parabola to the other across this axis, the reflected point also lies on the parabola.

4. Equidistant: Every point on the parabola is equidistant from the axis of symmetry and the parabola's corresponding focus point.

5. Uniqueness: Each parabola has exactly one unique axis of symmetry.

6. Optimization: The vertex of the parabola represents the maximum or minimum value of the quadratic function, which can be critical in optimization problems.

Example:

For the quadratic equation $ y = 2x^2 - 8x + 3 $:

1. The coefficients are $ a = 2 $ and $ b = -8 $.

2. Calculate the axis of symmetry using the formula $ x = -\frac{b}{2a} $:

$x = -\frac{-8}{2 \cdot 2} $

$x = \frac{8}{4} $

$x = 2 $

3. The equation for the axis of symmetry is therefore $ x = 2 $, which means the line $ x = 2 $ divides the parabola into two symmetrical halves.

It's important to note that the axis of symmetry's concept extends beyond just quadratic equations and parabolas; it's a fundamental characteristic in various mathematical objects that exhibit symmetry, including geometrical figures and other graphs.

Axis of Symmetry Equation

The equation of the axis of symmetry for a parabola represented by a quadratic function is derived from the standard form of the quadratic equation. For a quadratic function given by:

$y = ax^2 + bx + c $

The axis of symmetry is a vertical line, and its equation is given by:

$x = -\frac{b}{2a} $

Here's a breakdown of the terms:

• $ a $: The coefficient of the $ x^2 $ term in the quadratic equation.
• $ b $: The coefficient of the $ x $ term in the quadratic equation.
• $ c $: The constant term in the quadratic equation. It's worth noting that $ c $ does not affect the axis of symmetry; it only affects the vertical translation of the parabola.

The axis of symmetry is the vertical line that goes through the vertex of the parabola. The x-coordinate of every point on this line is the same, and it's equal to $ -\frac{b}{2a} $, which is also the x-coordinate of the vertex of the parabola. This line divides the parabola into two mirror-image sections.

The formula $ x = -\frac{b}{2a} $ ensures that if you pick a point on one side of the axis and move horizontally to the other side, the corresponding y-value will be the same due to the symmetry. This is why the formula does not include the y-coordinate; it applies to all points along the vertical line regardless of y.

Example:

Consider the quadratic function: $y = 3x^2 - 12x + 7 $

To find its axis of symmetry:

• Identify $ a = 3 $ and $ b = -12 $.
• Plug into the axis of symmetry formula:

$x = -\frac{-12}{2 \cdot 3} $ $x = \frac{12}{6} $ $x = 2 $

So, the equation of the axis of symmetry for this parabola is $ x = 2 $.

Axis of Symmetry Formula

The axis of symmetry formula is used to find the line that divides a parabola into two mirror-image halves for a quadratic function of the form $ y = ax^2 + bx + c $. The equation of the axis of symmetry for such a parabola is given by:

$x = -\frac{b}{2a} $

Here's what each symbol represents:

• $ x $: The x-coordinate of any point on the axis of symmetry, which is also the x-coordinate of the vertex of the parabola.
• $ a $: The coefficient of the quadratic term ($ x^2 $) in the quadratic equation.
• $ b $: The coefficient of the linear term ($ x $) in the quadratic equation.
• $ c $: The constant term in the quadratic equation (not needed for calculating the axis of symmetry).

The axis of symmetry formula is derived from the method of completing the square or by taking the derivative of the quadratic function to find the x-value where the slope is zero, which corresponds to the vertex of the parabola. Since parabolas are symmetric, the vertical line that passes through the vertex represents the axis of symmetry.

Example:

For the quadratic function $ y = 4x^2 + 8x + 3 $, determine the axis of symmetry.

Solution:

• Identify the coefficients: $ a = 4 $ and $ b = 8 $.
• Apply the axis of symmetry formula:

$x = -\frac{b}{2a} $ $x = -\frac{8}{2 \cdot 4} $ $x = -\frac{8}{8} $ $x = -1 $

So, the axis of symmetry for the parabola described by the equation $ y = 4x^2 + 8x + 3 $ is the vertical line given by $ x = -1 $.

Derivation of the Axis of Symmetry for Parabola

The derivation of the axis of symmetry for a parabola can be understood by considering the quadratic function in standard form and using the method of completing the square, or through calculus by finding the vertex of the parabola.

Method 1: Completing the Square

Take the standard form of a quadratic equation:

$y = ax^2 + bx + c $

To find the vertex form, we complete the square:

1. Separate the constant term $ c $:

$y - c = ax^2 + bx $

2. Factor out the coefficient $ a $ from the first two terms:

$y - c = a(x^2 + \frac{b}{a}x) $

3. Add and subtract the square of half the coefficient of $ x $ within the parentheses (completing the square):

$y - c + a\left(\frac{b}{2a}\right)^2 = a\left(x^2 + \frac{b}{a}x + \left(\frac{b}{2a}\right)^2\right) - a\left(\frac{b}{2a}\right)^2 $

4. Rewrite the completed square as a binomial square and simplify:

$y = a\left(x + \frac{b}{2a}\right)^2 - a\left(\frac{b}{2a}\right)^2 + c $

5. This gives us the vertex form of the quadratic function:

$y = a\left(x - h\right)^2 + k $

where $ h = -\frac{b}{2a} $ and $ k $ is the new constant simplified from the right side. The vertex of the parabola is at the point $ (h, k) $.

Since the vertex lies on the axis of symmetry, the equation of the axis of symmetry is:

$x = h = -\frac{b}{2a} $

Method 2: Calculus

Another way to derive the axis of symmetry is using calculus:

1. Start with the standard form:

$y = ax^2 + bx + c $

2. To find the vertex, we need the x-coordinate where the slope of the tangent is zero (where the derivative of the function with respect to $ x $ equals zero):

$\frac{dy}{dx} = 2ax + b $

3. Set the derivative equal to zero to find the critical point:

$0 = 2ax + b $

$-b = 2ax $

$x = -\frac{b}{2a} $

4. The x-coordinate at the vertex is $ -\frac{b}{2a} $, and this is also the equation for the axis of symmetry.

Both methods yield the same result, showing that the axis of symmetry for a parabola described by a quadratic function in standard form $ y = ax^2 + bx + c $ is a vertical line through the x-coordinate of the vertex:

$x = -\frac{b}{2a} $

How to Find Axis of Symmetry

To find the axis of symmetry of a parabola defined by a quadratic equation in standard form, you can use the following process:

Step 1: Identify the Coefficients

Begin with the quadratic function in standard form:

$y = ax^2 + bx + c $

Here, $ a $, $ b $, and $ c $ are constants, with $ a \neq 0 $. Identify the coefficients $ a $ and $ b $ from the equation.

Step 2: Use the Axis of Symmetry Formula

The axis of symmetry can be found using the formula that gives the x-coordinate of the vertex of the parabola:

$x = -\frac{b}{2a} $

Step 3: Calculate

Substitute the identified coefficients $ a $ and $ b $ into the formula and calculate the x-coordinate:

$x_{vertex} = -\frac{b}{2a} $

This value gives you the x-coordinate of the vertex of the parabola.

Step 4: Write the Equation

The equation of the axis of symmetry is a vertical line that passes through the vertex. It can be written as:

$x = x_{vertex} $

Example:

Given the quadratic equation $ y = 2x^2 - 8x + 5 $, find its axis of symmetry.

Step 1: Identify the coefficients: $ a = 2 $ and $ b = -8 $.

Step 2: Use the axis of symmetry formula:

$x = -\frac{b}{2a} $

Step 3: Calculate the x-coordinate of the vertex:

$x_{vertex} = -\frac{-8}{2 \cdot 2} $ $x_{vertex} = -\frac{-8}{4} $ $x_{vertex} = \frac{8}{4} $ $x_{vertex} = 2 $

Step 4: Write the equation for the axis of symmetry:

$x = 2 $

So, the axis of symmetry for the given parabola is the vertical line $ x = 2 $.

The axis of symmetry is a fundamental property of parabolas that helps with graphing and understanding their geometric features. It always passes through the vertex, which is the parabola's point with either the maximum or minimum y-coordinate, depending on whether the parabola opens upward or downward.

Frequently Asked Questions and Answers about Axis of Symmetry.

What is an axis of symmetry?

Answer: An axis of symmetry is a line that divides a figure or a graph into two parts that are mirror images of each other. For a parabola represented by a quadratic function, it is a vertical line that passes through the vertex of the parabola.

How do you find the axis of symmetry in a parabola?

Answer: For a quadratic function written in standard form, $ y = ax^2 + bx + c $, the axis of symmetry can be found using the formula $ x = -\frac{b}{2a} $, where $ a $ and $ b $ are the coefficients of the quadratic and linear terms, respectively.

Does the axis of symmetry always pass through the vertex?

Answer: Yes, for a parabola, the axis of symmetry always passes through the vertex, which is either the highest or lowest point on the graph, depending on the direction the parabola opens.

Can there be more than one axis of symmetry for a parabola?

Answer: No, a parabola has only one axis of symmetry. It is the unique line that divides the parabola into two symmetrical halves.

How does the coefficient $ a $ in the quadratic equation affect the axis of symmetry?

Answer: The coefficient $ a $ does not affect the location of the axis of symmetry; it only determines the width and the direction (up or down) in which the parabola opens. However, $ a $ is part of the formula used to calculate the axis of symmetry $ x = -\frac{b}{2a} $.

Is the axis of symmetry relevant only for parabolas?

Answer: While the axis of symmetry is often associated with parabolas in the context of quadratic functions, many geometric shapes also have axes of symmetry, such as circles, squares, rectangles, and triangles, depending on their specific properties.

What if the quadratic has no linear term? Where is the axis of symmetry?

Answer: If the quadratic function is in the form $ y = ax^2 + c $ (without a linear $ bx $ term), then the axis of symmetry is the y-axis itself, i.e., $ x = 0 $. This is because the $ b $ coefficient is zero, and according to the formula $ x = -\frac{b}{2a} $, the axis of symmetry is at $ x = 0 $.

How can you find the axis of symmetry from the quadratic formula?

Answer: The vertex form of the quadratic formula, which is derived from completing the square, inherently contains the axis of symmetry as part of its structure. When you rewrite $ y = ax^2 + bx + c $ into the vertex form $ y = a(x - h)^2 + k $, the value $ h $ represents the x-coordinate of the vertex, and thus the axis of symmetry is $ x = h $.

If a quadratic equation is graphed, how can you identify the axis of symmetry?

Answer: On a graph, the axis of symmetry can be identified as the vertical line that passes through the vertex of the parabola. If you fold the graph along this line, the two halves of the parabola will match up perfectly.