Menyusun fungsi kuadrat, Fungsi kuadrat

Introduction

This tutorial focuses on constructing quadratic functions, understanding their properties, and graphing them effectively. Quadratic functions are essential in algebra and have various applications in fields such as physics, engineering, and economics. By the end of this guide, you will be able to create quadratic functions, identify key characteristics, and graph them accurately.

Step 1: Understanding the Quadratic Function

A quadratic function is typically expressed in the standard form:

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

• a is the coefficient of ( x^2 )
• b is the coefficient of ( x )
• c is the constant term

Tips:

• The value of a determines the direction of the parabola (upward if a > 0, downward if a < 0).
• If you know the roots of the function, you can also express it in factored form:

[ f(x) = a(x - r_1)(x - r_2) ]

where ( r_1 ) and ( r_2 ) are the roots.

Step 2: Finding the Axis of Symmetry

The axis of symmetry is a vertical line that divides the parabola into two mirror-image halves. It can be calculated using the formula:

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

Practical Advice:

• Plug in the values of b and a from your quadratic function to find the axis of symmetry.
• This line will also help you find the vertex of the parabola.

Step 3: Determining the Vertex

The vertex of a quadratic function is the highest or lowest point on the graph, depending on the direction of the parabola. You can find the vertex coordinates using:

• ( x ) coordinate: ( x = -\frac{b}{2a} )
• ( y ) coordinate: Substitute the ( x ) value back into the function: ( y = f(x) )

Example:

1. If ( a = 1 ) and ( b = 2 ):
   • Calculate ( x = -\frac{2}{2 \cdot 1} = -1 )
   • Find ( y = f(-1) = 1(-1)^2 + 2(-1) + c )

Step 4: Finding the Discriminant

The discriminant helps determine the nature of the roots of the quadratic function. It is given by:

[ D = b^2 - 4ac ]

Interpret the Discriminant:

• If ( D > 0 ): Two distinct real roots
• If ( D = 0 ): One real root (the vertex is on the x-axis)
• If ( D < 0 ): No real roots (the parabola does not intersect the x-axis)

Step 5: Graphing the Quadratic Function

To graph a quadratic function accurately, follow these steps:

1. Plot the Vertex: Use the coordinates found in Step 3.
2. Draw the Axis of Symmetry: Draw a dashed line through the vertex.
3. Identify the Y-Intercept: This is the value of c in the function.
4. Calculate Additional Points: Choose x-values around the vertex to find more points on the graph.
5. Sketch the Parabola: Connect the points smoothly to form a U-shaped curve.

Common Pitfalls:

• Ensure the vertex and axis of symmetry are accurate as they guide the entire graph.
• Check your calculations for the y-coordinate of the vertex to avoid misrepresentation.

Conclusion

In this tutorial, you learned how to construct quadratic functions, find their axis of symmetry, determine their vertices, analyze the discriminant, and graph them effectively. Understanding these concepts is crucial for applications in various fields. Next, practice by creating your own quadratic functions and graphing them to reinforce your learning!