📊 Part 1: Identifying Quadratic Function Forms

1. Match each quadratic function with its correct form by drawing lines between the columns:

f(x) = 2x² - 8x + 6

f(x) = 2(x - 2)² - 2

f(x) = 2(x - 1)(x - 3)

A. Vertex Form

B. Standard Form

C. Factored Form

2. Circle the coefficient 'a' in each quadratic function:

For f(x) = -3x² + 5x - 2, the coefficient 'a' is:

-3

5

-2

3. What does the coefficient 'a' tell us about the parabola? Check all that apply:

If a > 0, the parabola opens upward

If a < 0, the parabola opens downward

The larger |a| is, the wider the parabola

The larger |a| is, the narrower the parabola

🔄 Part 2: Converting Between Forms

4. Convert f(x) = x² - 6x + 8 from standard form to factored form. Show your work:

5. Complete the square to convert f(x) = x² - 6x + 8 to vertex form:

6. Fill in the blanks for the vertex form f(x) = a(x - h)² + k:

In vertex form, the vertex is at the point (_____, _____), where h = _____ and k = _____.

7. From the factored form f(x) = (x - 2)(x - 4), the x-intercepts (zeros) are:

📈 Part 3: Graphing and Analysis

8. For the function f(x) = -(x + 1)² + 4, identify the following:

Vertex: (_____, _____)

Axis of symmetry: x = _____

Maximum or minimum value: ____________

Direction of opening: ____________

9. Sketch the graph of f(x) = (x - 1)(x - 3) on the coordinate plane below. Label the vertex and x-intercepts:

10. Circle the correct statement about the function f(x) = 2x² - 4x + 1:

The parabola has a maximum value

The parabola has a minimum value

The parabola opens to the right

The parabola has no vertex

🌍 Part 4: Real-World Applications

11. A ball is thrown upward from a height of 6 feet with an initial velocity of 32 feet per second. The height h(t) of the ball after t seconds is given by h(t) = -16t² + 32t + 6.

a) What is the initial height of the ball? _____ feet

b) When does the ball reach its maximum height? t = _____ seconds

c) What is the maximum height reached? _____ feet

12. Show your work for finding the maximum height in problem 11:

13. A farmer wants to create a rectangular pen with 100 feet of fencing. If the width is x feet, then the area A(x) = x(50 - x). What width gives the maximum area?

💭 Part 5: Reflection and Critical Thinking

14. Explain in your own words why the vertex form a(x - h)² + k makes it easy to identify the vertex of a parabola:

15. Compare the advantages of each form of quadratic functions:

Standard form (ax² + bx + c) is best for: ________________________________

Vertex form a(x - h)² + k is best for: ________________________________

Factored form a(x - r)(x - s) is best for: ________________________________

16. Describe one real-life situation where quadratic functions might be used and explain why:

Exploring Quadratic Functions

Exploring Quadratic Functions

📊 Part 1: Identifying Quadratic Function Forms

🔄 Part 2: Converting Between Forms

📈 Part 3: Graphing and Analysis

🌍 Part 4: Real-World Applications

💭 Part 5: Reflection and Critical Thinking

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