Axis of Symmetry of a Parabola

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Symmetry shows the balanced and proportional relationship between different parts of an object, shape, or function. It's an important concept to grasp in geometry, algebra, and calculus. In a quadratic function, the axis of symmetry is the line that divides the parabola into two halves. Let’s explore this further below.

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In this lesson, you’ll learn:

• What is an axis of symmetry

• The axis of symmetry formula

• How to find the axis of symmetry of a parabola

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What is the Axis of Symmetry?

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The axis of symmetry of a parabola is a vertical line that divides the parabola into two equal, mirror-image halves. 

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For a quadratic equation in standard form: y = ax² + bx +c

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The axis of symmetry formula is: x= -b/2a

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This line passes through the vertex and helps identify the parabola's highest or lowest point, depending on whether it opens upward or downward.

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Find the Axis of Symmetry of a Parabola

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1. Identify coefficients from the quadratic equation.

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For example, the given quadratic function in standard form is y = x² + 4x + 3.

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In this equation, the coefficients are: a = 1 , b = 4 , c = 3

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2. Plug the values of a and b into the formula x=-b/2a.

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x = - (4) / 2(1)

x = -4 / 2

x = -2

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3. The result is the x-value of the axis of symmetry.

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Therefore, x = –2 is the axis of symmetry of the parabola, indicating that the vertical line of symmetry passes through that point on the graph.

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Example 1:

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Find the axis of symmetry of the quadratic equation y = x² - 6x + 5.

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Solution:

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a = 1 , b = -6 , c = 5

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Substituting the given values into the axis of symmetry formula:

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x = - (-6) / 2(1)
x = 6 / 2
x = 3

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Therefore, the axis of symmetry is x = 3. 

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Example 2:

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Find the axis of symmetry of the graph y = 3x² - 12x + 5.

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Solution:

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a = 3 , b = -12 , c = 5

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Substituting the given values into the axis of symmetry formula:

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x = - (-12) / 2(3)

x = 12 / 6

x = 2

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Therefore, the axis of symmetry is x = 2.

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Find the Axis of Symmetry of a Parabola in Vertex Form

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When a quadratic function is written in vertex form,  y = a (x - h)²  + k, the values of h and k represent the coordinates of the vertex.

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To find the axis of symmetry, we simply use the formula: x = h. 

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1. Identify the coefficients from the given equation.

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For example, the given equation in vertex form is y = 2 (x - 3)² + 5.

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In this quadratic function, the coefficients are: a = 2 , h = 3 , k = 5‍

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2. Determine the vertex (h, k).

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Based on the data above, the vertex of the parabola is (3, 5).‍

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3. The value of h is the axis of symmetry.

Therefore, x = 3 is the axis of symmetry of the parabola, indicating that the vertical line of symmetry passes through that point on the graph.

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Example 3:

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Find the axis of symmetry of the parabola y = 3 (x - 2)² - 4.

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Solution:

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a = 3, h = 2, k = -4

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Substituting the given values, the vertex coordinates are (2, -4).

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Since the axis of symmetry always passes through the vertex, the value of h gives us the line of symmetry.

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Thus, the axis of symmetry is x = 2.

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