Lesson: Converting Standard Parabola Equation to Turning Point Form

Objective

• Learn how to convert a standard parabola equation into its Turning Point form.

Introduction

• Example equation: f(x) = x² - 4x - 6
• Exams may ask to rewrite it as: f(x) = a(x - p)² + q
• Conversion is achieved through completing the square.

Steps to Convert Standard Form to Turning Point Form

Step 1: Ensure Coefficient of x² is 1

• Start with f(x) = x² - 4x (coefficient is already 1).

Step 2: Completing the Square

• Write the first two parts: x² - 4x
• Add and subtract the square of half the coefficient of x:
  • Add: ((\frac{-4}{2})^2 = 4)
  • Subtract the same: (-4)
• Adjust for constant term (-6): f(x) = x² - 4x + 4 - 4 - 6

Step 3: Simplify to Turning Point Form

• Combine into a bracket squared: (x - 2)²
• Calculate the remaining part using a calculator: -6 - 4 = -10
• Result: f(x) = (x - 2)² - 10

Explanation of Terms

• a = 1, p = 2, q = -10
• The turning point is ((p, q) = (2, -10))

Importance of Turning Point Form

• The formula provides the turning point directly.
• Useful in exams for scoring marks.

Additional Example

Convert Another Equation

• Start with another equation:
  • Example: f(x) = x² - 5x - 8
• Use the same process:
  • Add: ((\frac{-5}{2})^2 = \frac{25}{4})
  • Adjust equation: f(x) = x² - 5x + \frac{25}{4} - \frac{25}{4} - 8
• Result: f(x) = (x - \frac{5}{2})² - 4.25

Calculate Turning Point

• Turning Point: ((\frac{5}{2}, -4.25))
• Alternatively, use the formula: x = -b/2a for x value of the turning point.

Conclusion

• Ensure familiarity with completing the square and converting to turning point form for exams.
• Know the importance of identifying turning points as it is frequently examined.