Axis of Symmetry
- Introduction
- What Is The Axis of Symmetry?
- Axis of Symmetry of a Parabola
- Axis of Symmetry Equation
- Axis of Symmetry Formula in Standard Form and Vertex Form
- Deriving the Axis of Symmetry for a Parabola
- Solved Examples
- Practice Problems
- Frequently Asked Questions
Introduction
The axis of symmetry is a line that divides a shape or function into two identical halves as if folding it along that line. It's crucial in quadratic functions, where it's a vertical line passing through the vertex of the parabola. “Axis of symmetry” is also called “line of symmetry”. This line helps us understand symmetry and the behavior of mathematical functions and shapes.
What Is The Axis of Symmetry?
The axis of symmetry is popularly used in geometry to define a straight line that creates symmetry within a shape. This line serves as a mirror, reflecting the exact image of one side onto the other. Whether it's horizontal, vertical, or diagonal, the axis of symmetry ensures that if an object is folded along this line, both halves match perfectly.
Various shapes have varying numbers of axes of symmetry. A square shape has four lines of symmetry, while a rectangle possesses two. On the other hand, a parallelogram lacks any line of symmetry. However, a circle has infinite lines of symmetry, meaning it can be folded along any diameter to achieve symmetry. Additionally, note that the number of sides in a regular polygon determines the count of its axes of symmetry. A regular polygon with ‘`n`’ sides has ‘`n`’ axes of symmetry. For example, a regular hexagon has `6` axes of symmetry.
Axis of Symmetry of a Parabola
The “U-shaped” graph of any quadratic function is called a “parabola”. Understanding the axis of symmetry of a parabola is essential for analyzing the symmetrical properties of this geometric shape. A parabola possesses a single line of symmetry, also known as the axis of symmetry. It is important to note that the axis of symmetry of a parabola always passes through the vertex of the parabola. This straight line divides the parabola into two identical halves.
A parabola can exhibit four distinct orientations: vertical (facing up or down) and horizontal (facing left or right). Its axis of symmetry depends on the orientation of the parabola.
- When a parabola opens either upwards or downward, the axis of symmetry is a vertical line.
- When a parabola opens either leftwards or rightwards, the axis of symmetry is a horizontal line.
It's noteworthy that the axis of symmetry with a horizontal orientation exhibits a slope of zero, while the axis of symmetry with a vertical orientation has an undefined slope.
Axis of Symmetry Equation
To write the equation for the axis of symmetry of a parabola, the vertex plays a crucial role. Vertex is where the axis of symmetry intersects the parabola. This vertex serves as the focal point for determining the equation of the axis of symmetry.
When the parabola opens either upwards or downwards, indicating a vertical axis of symmetry, the equation of the axis is a simple vertical line passing through the vertex. For a parabola with a vertex at `(h, k)` that opens upwards or downwards, the axis of symmetry equation is \( x = h \).
Likewise, if the parabola opens either rightwards or leftwards, signifying a horizontal axis of symmetry, the equation of the axis becomes a horizontal line passing through the vertex. For a parabola with a vertex at `(h, k)` that opens leftwards or rightwards, the axis of symmetry equation is \( y = k \).
Axis of Symmetry Formula in Standard Form and Vertex Form
The axis of symmetry formula helps determine the line that divides a parabola into two symmetrical halves. In this section, we are focusing on the parabolas opening upwards or downwards that have vertical lines of symmetry. This formula varies slightly depending on whether the parabola is expressed in standard form or vertex form.
Standard Form
In the standard form of a quadratic equation, which is \( y = ax^2 + bx + c \), the axis of symmetry formula is given by:
\( x = \frac{-b}{2a} \)
This formula calculates the `x`-coordinate of the vertex, determining the line of symmetry for the parabola.
Example: Consider the quadratic function \( y = 2x^2 + 4x - 3 \). Write the equation for its axis of symmetry.
Solution:
To find the axis of symmetry using the standard form formula \( x = \frac{-b}{2a} \), where \( a = 2 \) and \( b = 4 \), plug these values into the formula:
\( x = \frac{-4}{2(2)} = \frac{-4}{4} = -1 \)
So, the axis of symmetry for the given quadratic function in standard form is \( x = -1 \).
Vertex Form
The vertex form of a quadratic equation is \( y = a(x - h)^2 + k \), where `(h, k)` represents the vertex of the parabola. The axis of symmetry formula is simply given by:
\( x = h \)
In this equation, '`h`' represents the `x`-coordinate of the vertex, directly providing the line of symmetry for the parabola.
Example: Consider the quadratic function \( y = 4(x - 3)^2 + 2 \). Write the equation for its axis of symmetry.
Solution:
By comparing the given equation with the vertex form \( y = a(x - h)^2 + k \), we can see that:
\( a = 4 \) , \( h = 3 \) and \( k = 2 \)
Meaning, the vertex of the parabola is at \( (3,2) \).
We can use the `x`-coordinate of the vertex to write the equation for the axis of symmetry.
So, the axis of symmetry for the given quadratic function in vertex form is \( x = 3 \).
Deriving the Axis of Symmetry for a Parabola
The axis of symmetry of a parabola opening upwards or downwards is a vertical line that divides it into two congruent halves, acting like a mirror image. Here's a step-by-step derivation of the formula for the `x`-coordinate of this line:
`1`. Start with the standard form: Begin with the standard form of the quadratic equation:
\( y = ax^2 + bx + c \)
`2`. Eliminate the constant term: Since the constant term only affects the vertical position of the parabola, we can ignore it for finding the axis of symmetry. Therefore, we focus on the equation:
\( y = ax^2 + bx \)
`3`. Find the `x`-intercepts: These are the points where the parabola intersects the `x`-axis, implying \( y = 0 \). Set the equation equal to zero and solve for `x`:
\( 0 = ax^2 + bx \)
This quadratic equation can be factored (assuming \( a \neq 0 \)):
\( 0 = x(ax + b) \)
Using the zero product property, we obtain two possible solutions:
\( x = 0 \) (first factor equals zero)
\( ax + b = 0 \) (second factor equals zero)
Solve for `x` in each case:
\( x_1 = 0 \)
\( x_2 = -\frac{b}{a} \)
`4`. Find the midpoint of the `x`-intercepts: The axis of symmetry passes exactly between the two `x`-intercepts. Therefore, the `x`-coordinate of the axis of symmetry is the average of \( x_1 \) and \( x_2 \):
\( x_{\text{axis}} = \frac{x_1 + x_2}{2} \)
Substitute the values of \( x_1 \) and \( x_2 \):
\( x_{\text{axis}} = \frac{0 + (-\frac{b}{a})}{2} \)
`5`. Conclusion: Therefore, the x-coordinate of the axis of symmetry for a parabola in standard form is given by:
\( x_{\text{axis}} = -\frac{b}{2a} \)
This formula applies to both upward and downward-facing parabolas, as long as the equation is in standard form (\( ax^2 + bx + c \)).
Solved Examples
Example `1`. Find the axis of symmetry of the parabola represented by the equation \( y = x^2 - 4x + 3 \).
Solution:
Identify the coefficients: \( a = 1 \), \( b = -4 \), \( c = 3 \)
Use the formula: \( x = -\frac{b}{2a} \)
Substitute the values: \( x = \frac{-4}{2 \times 1} = -2 \)
Therefore, the axis of symmetry is \( x = -2 \).
Example `2`. Determine the axis of symmetry of the parabola given by the equation \( y = 2(x - 1)^2 - 3 \).
Solution:
The vertex form already reveals the `x`-coordinate of the vertex: \( (1, -3) \)
Since the parabola is symmetrical around its vertex, the axis of symmetry is a vertical line passing through the vertex.
Therefore, the axis of symmetry is \( x = 1 \).
Example `3`. Write the equation for the axis of symmetry. \( y=x^{2}+8x+\ 11 \).
Solution:
The vertex of the parabola is at \( (-4, -5) \).
Since the parabola is symmetrical around its vertex, the axis of symmetry is a vertical line passing through the vertex.
Therefore, the axis of symmetry is \( x = -4 \).
Example `4`. What is the axis of symmetry for the function defined by \( f(x) = -2x^2 + 12x \)?
Solution:
Identify the coefficients: \( a = -2 \), \( b = 12 \), \( c = 0 \)
Use the formula: \( x = -\frac{b}{2a} \)
Substitute the values: \( x = \frac{-12}{2 \times (-2)} = 3 \)
Therefore, the axis of symmetry is \( x = 3 \).
Example `5`. The roots of a parabola are `(-7,0)` and `(3,0)`. Find the axis of symmetry for this parabola.
Solution:
We know the axis of symmetry passes exactly through the middle of its `2` `x`-intercepts.
We can calculate the average of the two `x`-intercepts to find the `x`-coordinate of the vertex.
\( x_{\text{axis}} = \frac{x_1 + x_2}{2} \)
Substitute \( x_1 = -7 \) and \( x_2 = 3 \) into the formula:
\( x_{\text{axis}} = \frac{-7 + 3}{2} \)
\( x_{\text{axis}} = 2 \)
Therefore, the axis of symmetry is \( x = 2 \).
Practice Problems
Q`1`. Which of the following equations represents a parabola with an axis of symmetry at \( x = -1 \)?
- \( y = x^2 - 2x + 1 \)
- \( y = (x - 1)^2 + 3 \)
- \( y = 3x^2 - 2x \)
- \( y = -2(x + 1)^2 \)
Answer: d
Q`2`. What is the axis of symmetry of the parabola \( y = 2x^2 - 12x + 18 \)?
- \( x = 2 \)
- \( x = 3 \)
- \( x = -3 \)
- \( x = 6 \)
Answer: b
Q`3`. Which of the following decides the equation for the axis of symmetry of a parabola opening left?
- `x`-coordinate of the vertex
- `y`-coordinate of the vertex
- none of the above
Answer: b
Q`4`. A square has ______.
- `1` axis of symmetry
- `2` axes of symmetry
- `4` axes of symmetry
- No axes of symmetry
Answer: c
Q`5`. If the zeroes of a parabola are at `(1,0)` and `(9,0)`, what is the equation for its axis symmetry?
- \( x = 4 \)
- \( x = 5 \)
- \( x = -5 \)
- \( x = 4.5 \)
Answer: b
Frequently Asked Questions
Q`1`. What is the axis of symmetry of a parabola?
Answer: The axis of symmetry of a parabola is a vertical or horizontal line that divides the parabola into two equal and symmetrical halves. It passes through the vertex of the parabola.
Q`2`. How do you find the axis of symmetry of a parabola opening upwards or downwards?
Answer: To find the axis of symmetry of a parabola, you can use the formula \(x = -\frac{b}{2a}\) for a quadratic equation in the standard form \(y = ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are coefficients of the quadratic equation.
Q`3`. Does every parabola have an axis of symmetry?
Answer: Yes, every parabola has an axis of symmetry. It may be vertical, horizontal, or even oblique, depending on the orientation and position of the parabola.
Q`4`. What is the relationship between the axis of symmetry and the vertex of a parabola?
Answer: The axis of symmetry of a parabola passes through its vertex. The vertex represents the highest or lowest point of the parabola, and it lies on the axis of symmetry.
Q`5`. Can the axis of symmetry for a parabola open upwards or downwards by a Horizontal Line?
Answer: No, the axis of symmetry of a parabola facing up or down is always a vertical line. This is because a horizontal line would not divide such a parabola into two symmetrical halves.