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Introduction

This tutorial will guide you through the process of forming a quadratic function. Understanding quadratic functions is essential in algebra, as they are used to model various real-world scenarios, including physics, engineering, and economics. By the end of this tutorial, you will be able to construct a quadratic function from its key components.

Step 1: Identify the Standard Form of a Quadratic Function

A quadratic function can be expressed in the standard form:

• ( f(x) = ax^2 + bx + c )
  • a: coefficient of ( x^2 ) (determines the direction and width of the parabola)
  • b: coefficient of ( x ) (affects the position of the vertex)
  • c: constant term (y-intercept of the graph)

Practical Tip

• Ensure that a is not zero; if it is, the function is not quadratic.

Step 2: Determine Key Features of the Quadratic Function

To construct a quadratic function, identify the following features:

• Vertex: The highest or lowest point of the parabola.
• Axis of Symmetry: A vertical line that divides the parabola into two mirror-image halves, given by the formula:
  • ( x = -\frac{b}{2a} )

Practical Advice

• Use the vertex coordinates to help plot the function accurately.

Step 3: Find the Roots of the Quadratic Function

The roots (or x-intercepts) of the quadratic function can be found using:

• Factoring: Set the equation to zero and factor.
• Quadratic Formula: Use the formula if factoring is difficult:
  • ( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} )

Common Pitfall

• Ensure the discriminant (( b^2 - 4ac )) is non-negative to confirm there are real roots.

Step 4: Sketch the Graph of the Quadratic Function

Once you have the key features:

• Plot the vertex on the graph.
• Draw the axis of symmetry.
• Mark the roots (if any) on the x-axis.
• Sketch the parabola, ensuring it opens upwards if ( a > 0 ) or downwards if ( a < 0 ).

Practical Tips

• Use additional points by substituting values into the function to ensure the graph accurately reflects the quadratic's behavior.

Conclusion

You have now learned how to construct a quadratic function step-by-step, from identifying its standard form to graphing it. Remember to find the vertex and roots, which are crucial for accurately plotting the function. As a next step, practice constructing and graphing quadratic functions using different sets of coefficients to strengthen your understanding.