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Lesson Plan: Comparing Forms of Quadratic Functions

 

Lesson Summary

In this lesson, students will explore the three primary forms of quadratic functions: standard, vertex, and factored forms. They will compare these forms to understand their unique advantages and applications in solving, graphing, and interpreting quadratic equations. Students will also complete a hands-on warm-up activity, followed by guided instruction with real-world examples. Multimedia resources from Media4Math.com are incorporated to enhance learning. The lesson concludes with a 10-question quiz and an answer key. The estimated time for this lesson is 50 minutes.

Lesson Objectives

  • Compare standard, vertex, and factored forms of quadratic functions.
  • Analyze the advantages of each form for solving, graphing, and interpretation.
  • Convert between the different forms of quadratic functions.

Common Core Standards

  • HSF.IF.8a Write a quadratic function in different but equivalent forms to reveal and explain different properties of the function.

Prerequisite Skills

  • Understanding the forms of quadratic functions.
  • Basic skills in graphing quadratic equations.
  • Ability to recognize key features of a parabola (vertex, axis of symmetry, and roots).

Key Vocabulary

 

 

Warm Up Activities

Choose from one or more of these activities.

Activity 1: Desmos

Students will use Desmos to graph the same quadratic equation in three forms: standard, vertex, and factored.

  1. Access the Desmos graphing calculator: https://www.desmos.com/calculator
  2. Enter the equation in standard form: y = x2 - 6x + 8.
  3. Rewrite and enter the equation in vertex form: y = (x - 3)2 - 1.
  4. Rewrite and enter the equation in factored form: y = (x - 2)(x - 4).
  5. Observe and discuss how the graphs are identical and how each form highlights different features of the parabola.

Activity 2: Interactive Game: Jeopardy

Now that your students are near the end of this unit on quadratic equations and functions, play this Jeopardy game to review key concepts. 

https://www.media4math.com/library/4814/asset-preview

Activity 3: Communication

Given verbal descriptions of a parabola, students sketch it. Example: "The parabola opens upwards, has a vertex at (2, -3), and passes through the point (0, 1)." Students create a rough sketch on graph paper or using graphing tools.

 

 

Teach

Quadratic functions are foundational in algebra and provide a gateway to understanding more advanced mathematical concepts. They describe parabolic shapes and are represented by a second-degree polynomial equation. Quadratic functions can be expressed in three common forms: standard form, vertex form, and factored form. Each has unique characteristics and applications, allowing students to analyze and solve problems in different ways.

Standard Form

The standard form of a quadratic equation is y = ax2 + bx + c, where a, b, and c are constants. This form is particularly useful for identifying the y-intercept, which occurs at (0, c). It also provides a starting point for many algebraic methods, such as completing the square or applying the quadratic formula, to find the roots of the equation.

Vertex Form

The vertex form, y = a(x - h)2 + k, highlights the vertex of the parabola at (h, k). This form is excellent for analyzing the graph of a quadratic function, as it directly reveals the maximum or minimum point (depending on the direction of the parabola). Additionally, the value of a determines the width and direction of the parabola, making this form highly visual and intuitive.

Factored Form

The factored form, y = a(x - p)(x -q ), is most helpful for identifying the roots or zeros of the function, located at x = p and x = q. This form simplifies solving equations by setting the quadratic equal to zero and solving for x. It also provides insight into the symmetry of the parabola and how it interacts with the x-axis.

By understanding these forms, students gain a versatile toolkit for solving and interpreting quadratic functions in diverse contexts.

Use the following examples to explore the different quadratic forms. You can also explore additional examples in these collections:

Example 1

Solving Using Factored Form
Solve x2 - 5x + 6 = 0 by factoring:

  • Find the factors of 6 whose product is 6 and whose sum is -5. These factors are -2 and -3:

-2 • (-3) = 6

-2 + (-3) = -5

  • Write in factored form: 

(x - 2)(x - 3) = 0

  • Solutions: x = 2 and x = 3. Factored form quickly reveals the roots.

Summary: Factored form is ideal for quickly identifying the roots of a quadratic equation. These roots also correspond to the x-intercepts of the graph:

 

Quadratics

 

Example 2

Identifying the Vertex Using Vertex Form
Graph y = 2(x - 1)2 - 3.

  • The vertex is (1, -3), which is easily identified from the vertex form.
  • Plot the vertex and analyze symmetry.

Summary: Vertex form provides a clear way to identify the vertex and symmetry of the parabola.

Example 3

Real-World Application Using Standard Form
A ball is thrown upward with the equation h(t) = -16t2 + 32t + 5.

  • Use standard form to determine initial height (5) and time at maximum height.
  • Convert to vertex form: h(t) = -16(t - 1)2 + 21. Maximum height: 21 feet at t = 1 second.

Summary: Standard form is useful for modeling real-world problems and provides an overview of the quadratic function.

Applications: Pyrotechnics

VIDEO: Algebra Applications: Quadratic Functions, 2

This segment explores quadratic equations in fireworks displays. Concepts include vertex form, roots, and parabola shape modifications via constants a, b, and c. Key vocabulary: vertex, roots, trajectory. Applications include predicting and modeling pyrotechnic paths.
Resource: https://www.media4math.com/library/1514/asset-preview

Interactive Math Game: Pyrotechnic Math

Use this math game to review graphs of Quadratic Functions in Vertex Form. Aim a rocket at a star-shaped target. Hit the target and get an explosion of algebra. This game is ideal for exploring graphs of Quadratic Functions in the context of curve fitting.
Resource: https://www.media4math.com/library/4864/asset-preview

 

 

Review

Vocabulary Review

  • Quadratic Function: A function that can be expressed in the form f(x) = ax^2 + bx + c, where a ≠ 0. Its graph is a parabola.
  • Standard Form: The representation of a quadratic function as y = ax^2 + bx + c.
  • Vertex Form: The representation of a quadratic function as y = a(x - h)^2 + k, where (h, k) is the vertex of the parabola.
  • Factored Form: The representation of a quadratic function as y = a(x - r1)(x - r2), where r1 and r2 are the roots of the equation.
  • Vertex: The highest or lowest point on the graph of a quadratic function, located at (h, k) in the vertex form.
  • Axis of Symmetry: The vertical line that passes through the vertex of the parabola, given by x = h.
  • Roots/Zeros: The values of x for which f(x) = 0. These are the points where the parabola intersects the x-axis.
  • Discriminant: The expression b^2 - 4ac in the quadratic formula, which determines the nature and number of roots.

Revisit key vocabulary and summarize the advantages of each form:

  • Standard Form: Overview of the parabola.
  • Vertex Form: Highlights the vertex.
  • Factored Form: Identifies roots.

Example 1: Solving Using Factored Form

Solve x2 - 9x + 20 = 0 using factored form.

  • Find factors of 20 whose product is 20 and whose sum is -9. This would be -5 and -4, as shown below:

-4 • (-5) = 20
-4 + (-5) = -9

  • Write in factored form:

(x - 4)(x - 5) = 0

  • Find the solutions:

x = 4
x = 5

Summary: The factored form quickly identifies the roots of the quadratic equation. These can be validated by graphing the function.

 

Quadratics

 

Example 2: Converting Standard Form to Vertex Form

Convert the quadratic function y = 2x2 + 8x + 5 into vertex form.

  1. Identify coefficients: Here, a = 2, b = 8, and c = 5.
  2. Complete the square:
    • Factor out the coefficient of x2 from the first two terms: y = 2(x2 + 4x) + 5.
    • Find the value that completes the square inside the parentheses. Take half of the coefficient of x (which is 4), square it to get 4, and add and subtract this value inside the parentheses: y = 2(x2 + 4x + 4 - 4) + 5.
    • Rewrite the expression by grouping the perfect square trinomial and the constant: y = 2[(x + 2)2 - 4] + 5.
    • Distribute the 2 and simplify: y = 2(x + 2)2 - 8 + 5, which simplifies to y = 2(x + 2)2 - 3.

Result: The vertex form is y = 2(x + 2)2 - 3, with the vertex at (-2, -3).

Example 3: Solving a Quadratic Equation Using the Quadratic Formula

Solve the quadratic equation 3x2 - 2x - 8 = 0 using the quadratic formula.

  1. Identify coefficients: Here, a = 3, b = -2, and c = -8.
  2. Apply the quadratic formula: x = (-b ± √(b2 - 4ac)) / (2a).
  3. Calculate the discriminant: b2 - 4ac = (-2)2 - 4(3)(-8) = 4 + 96 = 100.
  4. Compute the roots:
    • x = (2 ± √100) / 6, which simplifies to x = (2 ± 10) / 6.
    • This yields two solutions: x = (2 + 10) / 6 = 2 and x = (2 - 10) / 6 = -4/3.

Result: The solutions are x = 2 and x = -4/3.

Additional Multimedia Resources

 

 

Quiz

Directions: Answer all questions below. 

  1. Rewrite y = x2 + 6x + 8 in factored form. ______

     
  2. Convert y = (x - 3)2 - 4 to standard form. ______

     
  3. Identify the vertex of y = -2(x + 1)2 + 5. ______

     
  4. Solve y = x2 - 4x + 4 using factored form. ______

     
  5. Write the factored form of y = x2 - 9. ______

     
  6. Determine the maximum height of h(t) = -16t2 + 64t + 3. ______

     
  7. Identify the roots of y = (x - 1)(x - 7). ______

     
  8. Write y = x2 + 2x + 1 in vertex form. ______

     
  9. Explain the advantage of vertex form for graphing. ______

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

Answer Key:

  1. y = (x + 4)(x + 2)
  2. y = x2 - 6x + 5
  3. Vertex: (-1, 5)
  4. x = 2
  5. y = (x - 3)(x + 3)
  6. Maximum height: 67 feet
  7. Roots: x = 1, x = 7
  8. y = (x + 1)2
  9. Shows the vertex clearly.
  10. y = 2(x - 2)2 - 2