MathIsimple

Advanced Quadratic Functions

Explore vertex form, transformations, maximum/minimum values, and real-world applications of quadratic functions. Master completing the square and analyzing parabolas!

10th Grade
Functions
~60 min
🎮 Interactive Activity: Vertex Form Converter

Convert to vertex form and find the vertex!

Standard Form:

y = x² + 4x + 3

Vertex form: y = a(x - h)² + k

🎮 Interactive Activity: Parabola Analyzer

Adjust parameters and see how the parabola changes!

Equation: y = (x - 0)² + 0

Opens: Upward ∪

Vertex: (0, 0)

Axis of Symmetry: x = 0

Minimum Value: 0

1. Review of Quadratic Functions

Quadratic Function Forms

Quadratic functions can be written in three main forms:

  • Standard Form: y = ax² + bx + c
  • Vertex Form: y = a(x - h)² + k (vertex is (h, k))
  • Factored Form: y = a(x - r₁)(x - r₂) (zeros are r₁ and r₂)
Example 1: Identifying Forms

Standard: y = 2x² - 8x + 6

Vertex: y = 2(x - 2)² - 2 (vertex at (2, -2))

Factored: y = 2(x - 1)(x - 3) (zeros at x = 1 and x = 3)

Example 2: Key Features

For y = -x² + 4x - 3:

a = -1: Opens downward

Vertex: Complete the square to find (2, 1)

Maximum value: 1 (at x = 2)

y-intercept: (0, -3)

2. Vertex Form and Its Applications
3. Completing the Square
4. Quadratic Transformations
5. Maximum and Minimum Values
6. Real-World Applications
7. Graphing Quadratic Functions
8. Problem-Solving Strategies
Frequently Asked Questions

Practice Time!

Practice Quiz
10
Questions
0
Correct
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Score
1
What is the vertex of the parabola y = (x - 3)² + 2?
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2
For y = -2(x + 1)² - 4, does the parabola open upward or downward?
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3
What is the axis of symmetry for y = (x - 5)² + 3?
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4
Which quadratic has a maximum value?
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5
What is the minimum value of y = (x - 2)² + 7?
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6
Convert y = x² + 6x + 5 to vertex form. What is the vertex?
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7
For y = a(x - h)² + k, what does the parameter 'a' control?
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8
What is the y-intercept of y = (x - 2)² - 3?
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9
Which transformation changes y = x² to y = (x - 4)²?
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10
For what value of k does y = x² - 4x + k have its vertex on the x-axis?
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