Standard form of a quadratic function

Standard Form of a Quadratic Function

A quadratic function is a polynomial function of degree two. Its standard form is expressed as:

f(x) = ax² + bx + c

In this equation, a, b, and c are constants, and a cannot be zero. The variable x represents the input value, while f(x) represents the output value of the function.

Components of the Standard Form

Each component of the quadratic function plays a crucial role in determining the shape and position of the parabola, which is the graph of a quadratic function.

• a: This coefficient determines the direction of the parabola. If a is positive, the parabola opens upwards. If a is negative, it opens downwards.
• b: This coefficient affects the position of the vertex (the highest or lowest point) of the parabola along the x-axis.
• c: This constant represents the y-intercept, which is the point where the graph intersects the y-axis.

Example of a Quadratic Function

Let's consider the quadratic function:

f(x) = 2x² - 4x + 1

In this function:

• a = 2: The parabola opens upwards.
• b = -4: This will affect the x-coordinate of the vertex.
• c = 1: The y-intercept is at (0, 1).

Finding the Vertex of the Parabola

The vertex of a quadratic function can be found using the formula for the x-coordinate:

x = -b / (2a)

Using our example, we can find the vertex:

x = -(-4) / (2 × 2) = 4 / 4 = 1

Now, substitute x = 1 back into the function to find the y-coordinate of the vertex:

f(1) = 2(1)² - 4(1) + 1 = 2 - 4 + 1 = -1

Thus, the vertex of the parabola is at the point (1, -1).

Graphing the Quadratic Function

To graph the quadratic function, identify the vertex and the y-intercept. From the vertex, you can plot additional points by choosing values for x and calculating corresponding f(x) values.

For example, let’s calculate f(x) for x = 0, x = 2, and x = 3:

f(0) = 2(0)² - 4(0) + 1 = 1

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

f(3) = 2(3)² - 4(3) + 1 = 2(9) - 12 + 1 = 18 - 12 + 1 = 7

Now we have the points:

• (0, 1)
• (2, 1)
• (3, 7)

Plot these points along with the vertex (1, -1) on a Cartesian plane:

• The vertex is at (1, -1).
• The y-intercept is (0, 1).
• At x = 2, f(x) = 1 gives another point.
• At x = 3, f(x) = 7 gives another point.

Axis of Symmetry

The axis of symmetry is a vertical line that divides the parabola into two mirror images. It passes through the vertex. The equation of the axis of symmetry can be given as:

x = -b / (2a)

In our example, the axis of symmetry is:

x = 1

Common Mistakes

Summary

• A quadratic function is in the form f(x) = ax² + bx + c.
• The coefficient a determines the direction of the parabola.
• The vertex can be found using x = -b / (2a).
• The axis of symmetry is the vertical line that goes through the vertex.

Check your understanding

1. What is the standard form of a quadratic function?
2. How do you determine if a parabola opens upwards or downwards?
3. Calculate the vertex for the quadratic function f(x) = -3x² + 6x + 2.
4. What is the equation of the axis of symmetry for the function f(x) = x² - 4x + 3?