Fungsi Kuadrat

Introduction

This tutorial explores the concept of quadratic functions, a fundamental topic in mathematics, particularly relevant for high school students. Understanding quadratic functions is essential for solving equations, graphing parabolas, and applying these concepts in real-world scenarios. This guide will break down the key aspects of quadratic functions and how to work with them effectively.

Step 1: Understanding the Quadratic Function

A quadratic function is typically expressed in the standard form:

[ f(x) = ax^2 + bx + c ]

• a, b, and c are constants.
• a cannot be zero, as that would make the function linear.

Key Characteristics of Quadratic Functions

• The graph of a quadratic function is a parabola.
• The direction of the parabola (opening upwards or downwards) is determined by the sign of a:
  • If a > 0, the parabola opens upwards.
  • If a < 0, the parabola opens downwards.

Step 2: Identifying the Vertex and Axis of Symmetry

To analyze the graph of a quadratic function, it’s essential to find its vertex and axis of symmetry.

Finding the Vertex

The vertex can be calculated using the formula:

[ x = -\frac{b}{2a} ]

• Substitute the values of b and a to find the x-coordinate of the vertex.
• To find the y-coordinate, substitute the x value back into the original equation.

Axis of Symmetry

The axis of symmetry is the vertical line that passes through the vertex:

[ x = -\frac{b}{2a} ]

Step 3: Finding the Roots of the Quadratic Function

Roots (or zeros) of the quadratic function are the x-values where the graph intersects the x-axis. There are several methods to find the roots:

1. Factoring

• Rewrite the quadratic equation in factored form, if possible.
• Set each factor equal to zero and solve for x.

2. Completing the Square

• Rearrange the equation to move the constant term to the other side.
• Create a perfect square trinomial on one side of the equation, then solve for x.

3. Using the Quadratic Formula

If factoring is difficult, use the quadratic formula:

[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} ]

• This formula provides the roots directly based on the coefficients of the quadratic equation.

Step 4: Graphing the Quadratic Function

Once you have the vertex, axis of symmetry, and roots, you can graph the quadratic function.

Steps to Graph

1. Plot the Vertex on the graph based on the coordinates found.
2. Draw the Axis of Symmetry as a dashed vertical line through the vertex.
3. Plot the Roots on the x-axis.
4. Sketch the Parabola by drawing a smooth curve through the vertex and roots, ensuring it opens in the correct direction.

Conclusion

Understanding quadratic functions involves recognizing their standard form, identifying key characteristics like the vertex and roots, and effectively graphing them. Mastering these concepts is crucial for success in higher-level mathematics. For further practice, try solving different quadratic equations using the methods outlined in this tutorial and graphing them to reinforce your understanding.