Lesson Plan: Vertex Form of a Quadratic Function

 

Lesson Summary

This 50-minute lesson focuses on the vertex form of a quadratic function, teaching students how to rewrite quadratic equations into vertex form and use this representation to identify the vertex and axis of symmetry. Multimedia resources from Media4Math.com are incorporated, including video definitions for key terms and individual clip art resources to enhance understanding. A 10-question quiz with an answer key concludes the lesson.

Lesson Objectives

  • Rewrite quadratic equations into vertex form using algebraic techniques.
  • Identify the vertex and axis of symmetry in vertex form equations.
  • Analyze how parameters h and k influence the graph of a quadratic function.

Common Core Standards

  • HSF.BF.3: Identify the effect on the graph of replacing f(x) by f(x) + k, kf(x), f(kx), and f(x + k); find the value of k given the graphs.
  • HSF.IF.8a: Use the process of factoring and completing the square to reveal zeros, extreme values, and symmetry, and interpret these in a context.

Prerequisite Skills

  • Proficiency in algebraic manipulation.
  • Familiarity with the standard form of quadratic equations.

Key Vocabulary

 

 

Warm Up Activities (10 minutes)

Choose from one or more of these activities.

Desmos Graphing Activity

Access the Desmos graphing calculator at https://www.desmos.com/calculator.

  • Plot y = x2 + 6x + 5. Observe the graph.
  • Then plot this graph: y = (x + 3)2 - 4. 
  • Compare the original and transformed graphs. Discuss the shift to the vertex (-3, -4).

 

Quadratics

 

  • Discuss with students why the graphs overlap. Ask them how they could convert a function in standard form to one in vertex form.

Clip Art Activity

Use the following slide show to learn the basics of quadratic functions. You can also incorporate this into the Teach section.

Completing the Square with Algebra Tiles

Objective: Students will visualize the process of completing the square using algebra tiles to better understand the transformation of a quadratic equation into vertex form.

Materials needed:

  • Digital or physical algebra tiles: x² tiles, x tiles, and unit tiles.
  • Whiteboard or interactive display (if using digital tiles).
  • Grid paper (for physical tiles).

Step 1: Represent the Expression

Write the quadratic expression on the board: x² + 6x + __

  • Display an x² tile at the top-left corner.
  • Show six x tiles laid horizontally to the right of the x² tile.

 

Completing the Square

 

Step 2: Form a Square

  • Rearrange the six x tiles into two equal groups of three tiles.
  • Place three x tiles vertically along the right of the x² tile and three horizontally along the top.

 

Completing the Square

 

Step 3: Identify the Missing Piece

  • Highlight the empty square in the top-right corner of the arrangement.
  • Add unit tile squares in this space to represent the missing area.
  • Explain that 3 × 3 = 9, so we add 9 to complete the square.

 

Completing the Square

 

Step 4: Write the Completed Expression

The completed square represents: (x + 3)²

Explain how adding 9 to the expression x² + 6x made it a perfect square trinomial.

Reflection Questions

  1. What does completing the square mean in terms of the algebra tiles?
  2. Why did we add 9 to complete the square?
  3. How would the process change if the middle term was 4x instead of 6x?

Extension

Invite students to explore other expressions, such as x² + 8x or x² + 4x, using algebra tiles.

 

 

Teach (25 minutes)

Understanding the Vertex Form of a Quadratic Function

In this lesson, we explore the vertex form of a quadratic function and the mathematical process of completing the square, both of which are foundational concepts for analyzing and solving quadratic equations.

The vertex form of a quadratic function is written as y = a(x-h)2 + k, where (h, k) represents the vertex of the parabola. This form is particularly useful because it provides an immediate visual understanding of the graph’s key characteristics: its vertex (the highest or lowest point of the graph, depending on the parabola's orientation) and its axis of symmetry, which is the vertical line x = h. The parameter a indicates the parabola’s direction of opening (upward for a > 0 and downward for a < 0) and its degree of vertical stretch or compression.

To convert a quadratic function in standard form, y = ax2 + bx + c, into vertex form, we use the technique of completing the square. This method involves rewriting the quadratic expression so that it becomes a perfect square trinomial, making it easier to express the function in its vertex form. Completing the square not only simplifies the process of identifying the vertex but also reveals critical properties of the quadratic function, such as its symmetry and transformations.

Incorporating technology tools such as graphing calculators or dynamic graphing software allows students to visually compare the standard and vertex forms. By completing the square and transitioning between these forms, students deepen their understanding of the quadratic function’s structure and graphical representation, laying the groundwork for more advanced algebraic concepts.

This section introduces students to deriving the vertex form of a quadratic equation by completing the square. Students will practice converting standard form to vertex form step by step. Examples illustrate the relationship between the quadratic equation and its graph, highlighting how the vertex and axis of symmetry are derived.

Example 1: Rewrite y = x2 + 4x + 3 in vertex form.

Given the quadratic equation:

y = x2 + 4x + 3

Step 1: Write the equation in standard form.

The equation is already in standard form:

y = x2 + 4x + 3

Step 2: Isolate the quadratic and linear terms.

Focus on the x2 + 4x terms, leaving the constant term (+3) outside:

y = (x2 + 4x) + 3

Step 3: Complete the square for x2 + 4x.

  • Take half the coefficient of x, which is 4. Half of 4 is 2.
  • Square 2: 22 = 4.
  • Add and subtract 4 inside the parentheses:

y = (x2 + 4x + 4 - 4) + 3

Simplify:

y = ((x + 2)2 - 4) + 3

Step 4: Simplify the equation.

Combine constants (-4 + 3):

y = (x + 2)2 - 1

Step 5: Write the equation in vertex form.

The vertex form of the equation is:

y = (x + 2)2 - 1

Vertex:

The vertex is at (-2, -1).

Summary: The equation was transformed into vertex form by completing the square, making it easier to identify the vertex (-2, -1).

Example 2: Identify the vertex and axis for y = 2(x - 1)2 + 3.

  1. The equation is already in vertex form.
  2. Identify h = 1 and k = 3.

Solution: Vertex: (1, 3), Axis: x = 1.

Summary: The equation in vertex form provides the vertex and axis of symmetry directly, (1, 3) and x = 1, respectively.

Example 3 (Real-world): A ball follows h(t) = -16t2 + 32t + 5. Find the maximum height.

  • Start by factoring out -16: h(t) = -16(t2 - 2t) + 5.
  • Complete the square for t2 - 2t

t2 - 2t = t2 - 2•1t + 1 - 1

= (t2 - 2•1t + 1) - 1

= (t - 1)2 - 1

  • Write the completed square term: 

h(t) = -16((t - 1)2 - 1) + 5.

  • Simplify: 

h(t) = -16(t - 1)2 + 21.

Solution: Maximum height: 21 feet at t = 1 second.

Summary: The vertex form reveals the maximum height (21 feet) and the time it occurs (1 second).

 

Quadratics

 

 

 

Review (10 minutes)

Reinforce the concept of the vertex form by summarizing the derivation process and reviewing key terms. Utilize the following video to show an application of quadratic functions.

https://www.media4math.com/library/39605/asset-preview

Vocabulary Review

  • Vertex Form: A way of writing a quadratic equation as y = a(x-h)2 + k, where (h, k) is the vertex of the parabola.
  • Completing the Square: A method used to rewrite a quadratic equation in vertex form by creating a perfect square trinomial.
  • Vertex: The highest or lowest point of a parabola, represented by the coordinates (h, k) in the vertex form of a quadratic equation.
  • Axis of Symmetry: A vertical line that divides the parabola into two symmetrical halves, given by the equation x = h, where h is the x-coordinate of the vertex.

Example 1: Rewrite y = 3x2 - 12x + 7 in vertex form.

  • Factor out 3: y = 3(x2 - 4x) + 7.
  • Complete the square (x2 - 4x): 

x2 - 2•2 + 4 - 4 = (x2 - 2•2 + 4) - 4

= (x - 2)2 - 4

  • Write the completed square equation: 

y = 3((x - 2)2 - 4) + 7

  • Simplify: 

y = 3(x - 2)2 - 5

Solution: Vertex: (2, -5).

Summary: By rewriting the equation into vertex form, the vertex (2, -5) is easily identified.

Example 2: Identify the vertex of y = -2(x + 1)2 + 4.

Recognize h = -1 and k = 4.

Solution: Vertex: (-1, 4).

Summary: The vertex form clearly identifies the vertex (-1, 4) without additional computation.

Additional Multimedia Resources

 

Quiz

Answer the following questions.

  1. Rewrite y = x2 + 2x + 5 in vertex form.

     
  2. Convert y = -x2 + 6x - 8 to vertex form.

     
  3. Find the vertex of y = 2(x - 3)2 - 7.

     
  4. Identify the axis of symmetry for y = 4(x + 2)2 + 1.

     
  5. Rewrite y = x2 + 10x + 16 in vertex form.

     
  6. Find the maximum value of y = -2x2 + 8x - 6.

     
  7. Identify the vertex of y = -3(x + 4)2 + 5.

     
  8. Rewrite y = 5x2 - 20x + 15 in vertex form.

     
  9. Find the vertex of y = (x - 2)2 - 3.

     
  10. Identify the axis of symmetry for y = x2 + 4x + 4.

Answer Key

  1. y = (x + 1)2 + 4
  2. y = -(x - 3)2 + 1
  3. (3, -7)
  4. x = -2
  5. y = (x + 5)2 - 9
  6. (2, 2)
  7. (-4, 5)
  8. y = 5(x - 2)2 - 5
  9. (2, -3)
  10. x = -2