Vertex and axis of symmetry

Vertex and Axis of Symmetry

A quadratic function can be expressed in standard form as f(x) = ax² + bx + c, where a, b, and c are constants. The graph of a quadratic function is a parabola. Understanding the vertex and the axis of symmetry is crucial for graphing and interpreting these functions.

Vertex of a Quadratic Function

The vertex is the highest or lowest point on the graph of a quadratic function, depending on the direction of the parabola. If a > 0, the parabola opens upwards, and the vertex is the minimum point. If a < 0, the parabola opens downwards, and the vertex is the maximum point.

The x-coordinate of the vertex can be calculated using the formula:

x = -b / (2a)

Once you find the x-coordinate, you can substitute it back into the original function to find the y-coordinate of the vertex.

Example 1: Finding the Vertex

Consider the quadratic function f(x) = 2x² - 8x + 6.

1. Identify the coefficients: a = 2, b = -8, c = 6.
2. Calculate the x-coordinate of the vertex:

x = -(-8) / (2 × 2) = 8 / 4 = 2

3. Substitute x = 2 back into the function to find the y-coordinate:

f(2) = 2(2)² - 8(2) + 6 = 2(4) - 16 + 6 = 8 - 16 + 6 = -2

Thus, the vertex of the function f(x) = 2x² - 8x + 6 is (2, -2).

Axis of Symmetry

The axis of symmetry is a vertical line that divides the parabola into two mirror-image halves. The equation of the axis of symmetry can be expressed as:

x = -b / (2a)

From the previous example, we found the x-coordinate of the vertex to be 2. Therefore, the equation of the axis of symmetry for the function f(x) = 2x² - 8x + 6 is:

x = 2

Example 2: Finding the Axis of Symmetry

Using the same function, f(x) = 2x² - 8x + 6, we already calculated the axis of symmetry:

x = -(-8) / (2 × 2) = 2

Thus, the axis of symmetry is the line x = 2.

Graphing the Quadratic Function

Now that you have the vertex and the axis of symmetry, you can graph the quadratic function. Start by plotting the vertex. Then, use the axis of symmetry to find additional points on the graph. Since the parabola is symmetric, if you find a point on one side of the axis, you can find the corresponding point on the other side.

Example 3: Graphing the Function

Let’s graph the function f(x) = 2x² - 8x + 6.

1. Plot the vertex (2, -2) on the graph.
2. Determine another point by choosing a value for x. For example, let x = 1:

f(1) = 2(1)² - 8(1) + 6 = 2 - 8 + 6 = 0

3. Plot the point (1, 0).

Example 4: Finding the Vertex and Axis of Symmetry for Another Function

Consider the function f(x) = -x² + 4x - 3.

1. Identify the coefficients: a = -1, b = 4, c = -3.
2. Calculate the x-coordinate of the vertex:

x = -4 / (2 × -1) = -4 / -2 = 2

3. Substitute x = 2 back into the function to find the y-coordinate:

f(2) = -2² + 4(2) - 3 = -4 + 8 - 3 = 1

The vertex is (2, 1).

4. Calculate the axis of symmetry:

x = 2

The axis of symmetry is x = 2.

Summary

• The vertex is the highest or lowest point of a quadratic function.
• The axis of symmetry is a vertical line that passes through the vertex.
• The x-coordinate of the vertex can be found using the formula x = -b / (2a).
• Substituting the x-coordinate back into the function gives the y-coordinate of the vertex.

Check your understanding

1. Find the vertex and axis of symmetry for the function f(x) = 3x² + 6x + 2.
2. What is the vertex of the function f(x) = -2x² + 8x - 5?
3. How does the axis of symmetry relate to the vertex?
4. Explain how to graph a quadratic function using the vertex and axis of symmetry.