Lesson Plan: Graphing Quadratic Functions

Lesson Summary

In this 50-minute lesson, students will learn how to graph quadratic functions by creating tables of values and plotting parabolas. Key features such as the vertex and axis of symmetry will be identified. This lesson incorporates multimedia resources, including videos, clip art, and interactive games, from Media4Math.com. A 10-question quiz with an answer key will assess understanding at the end.

Lesson Objectives

• Create tables of values to graph quadratic functions.
• Identify and label the vertex and axis of symmetry on a graph.
• Develop fluency in recognizing the shape and orientation of parabolas.

Common Core Standards

• HSF.IF.7a Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.
  - Graph linear and quadratic functions and show intercepts, maxima, and minima.

Prerequisite Skills

• Plotting points on a coordinate plane.
• Basic understanding of symmetry.

Key Vocabulary

• Vertex: The highest or lowest point on a parabola. 
  • Multimedia Resource: https://www.media4math.com/library/74587/asset-preview
• Axis of Symmetry: A vertical line that divides the parabola into two symmetrical halves. 
  • Multimedia Resource: https://www.media4math.com/library/74586/asset-preview
• Parabola: The U-shaped curve formed by the graph of a quadratic function. 
  • Multimedia Resource: https://www.media4math.com/library/74573/asset-preview
• Table of Values: A method of organizing x- and y-values to graph quadratic functions.

Additional Multimedia Resources

• This is a collection of video definitions related to Quadratics: https://www.media4math.com/MathVideoCollection--QuadraticsDefinitions 
• This is a collection of definitions related to Quadratics: https://www.media4math.com/Definitions--Quadratics 

Warm-up Activities

Choose from one of these activities.

1. Desmos Activity

Open the Desmos graphing calculator by navigating to Desmos Graphing Calculator. Begin by discussing the standard form of a quadratic function: y = ax2 + bx + c. Create a quadratic formula template that uses sliders for the values of a, b, and c.

• Explain how the coefficients a, b, and c influence the shape and position of the parabola.
• Vertex Identification:
  • Use the sliders in Desmos to create variables for a, b, and c.
  • Observe how changing the values affects the vertex's position.
  • Note the vertex's coordinates by observing the graph or using the trace feature.
• Axis of Symmetry:
  • Identify the axis of symmetry, which is the vertical line passing through the vertex.
  • Verify this by noting the equation x = -b/(2a).
• Effect of Coefficient a:
  • Adjust a to see how it affects the parabola's width and direction (concave up for a > 0, concave down for a < 0).

2. Drag-N-Drop Activity

This game tests a student's understanding of evaluating a quadratic function. Use this activity for students who still need to work on the skill of evaluating quadratics.

https://www.media4math.com/library/4828/asset-preview

3. Slide Show

Use this slide show to provide a brief introduction to graphs of quadratics, focusing more on matching coordinates. This also includes a Desmos activity.

https://www.media4math.com/library/21559/asset-preview

Teach

In this lesson, students will delve into the graphical representations of quadratic functions, focusing on how their defining parameters shape their behavior. Quadratic functions, represented by equations like y = ax2 + bx + c in standard form, can also be expressed in vertex form: y = a(x - h)2 + k. This alternate representation highlights the role of the vertex and simplifies the process of graphing.

The vertex is a key feature of a parabola, representing its highest or lowest point depending on the parabola's orientation. In vertex form, the vertex is explicitly given as (h, k), making it easier to identify the parabola's maximum or minimum value. The axis of symmetry, a vertical line passing through the vertex at x = h, divides the parabola into two symmetric halves. Additionally, the coefficient a determines the orientation of the parabola: a positive a produces an upward-facing parabola, while a negative a results in a downward-facing one. The value of a also controls the width or steepness of the curve.

In this section, students will gain a step-by-step understanding of graphing quadratic functions. They will create tables of values, plot points, and identify key features such as the vertex and axis of symmetry. Each example builds on the previous one to deepen understanding and demonstrate real-world applications.

Example 1: Graphing a Basic Quadratic Function

Equation: $y = x^2$

• Start by selecting values for $x$ within the range $-3 \leq x \leq 3$. Compute the corresponding $y$-values:

• Plot the points on a coordinate plane: $(-3, 9), (-2, 4), (-1, 1), (0, 0), (1, 1), (2, 4), (3, 9)$.
• Connect the points to form a symmetric U-shaped parabola. The vertex is at $(0, 0)$, and the axis of symmetry is $x = 0$.
• Discuss how the parabola is symmetric about the axis of symmetry. 

Key Takeaway: The basic quadratic function $y = x^2$ produces a parabola with its vertex at the origin and opens upwards.

Example 2: Shifting and Stretching a Parabola

Equation: $y = 2(x - 1)^2 - 3$

•  Identify the vertex using the form $y = a(x - h)^2 + k$. Here, $(h, k) = (1, -3)$.
• The axis of symmetry is $x = 1$.
• Create a table of values for $x$ around the vertex:

• Plot these points: $(-1, 5), (0, -1), (1, -3), (2, -1), (3, 5)$.
• Connect the points to form the parabola. The graph is narrower than $y = x^2$ because $a = 2$, stretching the parabola vertically. 

Key Takeaway: Changing the vertex shifts the parabola, and the coefficient $a$ affects the width and direction of the graph.

Example 3: Real-world Application

Problem: A ball is thrown, following the equation $h(t) = -4.9t^2 + 9.8t + 2$, where $h(t)$ is the height (in meters) and $t$ is time (in seconds).

• Identify the vertex to find the maximum height. Use $t = -\frac{b}{2a}$:
  • $a = -4.9, b = 9.8$
  • $t = -\frac{9.8}{2(-4.9)} = 1$ second
  • Substitute $t = 1$ into $h(t)$: $h(1) = -4.9(1)^2 + 9.8(1) + 2 = 6.9$ meters.
• Create a table of values for $t$:

•  Plot the points $(0, 2), (0.5, 5.675), (1, 6.9), (1.5, 5.675), (2, 2)$.
• Sketch the parabola to show the trajectory of the ball. The vertex $(1, 7.9)$ represents the maximum height, and $t = 2.186$ seconds marks when the ball hits the ground. 

Key Takeaway: Quadratic equations can model real-world scenarios like projectile motion, and the vertex provides critical information about the maximum or minimum value.

Additional Multimedia Resources

• This is a collection of video definitions related to Quadratics: https://www.media4math.com/MathVideoCollection--QuadraticsDefinitions 
• This is a slide show of math examples of graphs of quadratic functions in standard form: https://www.media4math.com/library/slideshow/math-examples-graphs-quadratic-functions-standard-form
• This is a slide show of math examples of graphs of quadratic functions in vertex form: https://www.media4math.com/library/slideshow/math-examples-graphs-quadratic-functions-vertex-form

Review

In this review, students will consolidate their understanding of quadratic functions, focusing on graphing, identifying key features, and applying the vocabulary learned during the lesson. Emphasis will be placed on the terms Vertex, Axis of Symmetry, Parabola, and Table of Values.

Step 1: Recall that the Vertex represents the highest or lowest point on the graph of a Parabola. Use the equation $y = -x^2 + 4x - 3$. Identify the vertex using the formula $x = -\frac{b}{2a}$, calculate $x = 2$, and find $y = -1$. The vertex is $(2, -1)$. Discuss the role of the vertex as the turning point.

Step 2: Define the Axis of Symmetry. For $y = -x^2 + 4x - 3$, the axis of symmetry is $x = 2$, dividing the Parabola into two mirror images. Highlight how every point on one side has a corresponding point on the other side.

Step 3: Create a Table of Values for $x = 0, 1, 2, 3, 4$. Plot these points:

• $(0, -3)$
• $(1, 0)$
• $(2, -1)$
• $(3, 0)$
• $(4, -3)$

Use the table to plot and sketch the graph. Show how the vertex and axis of symmetry guide the graphing process.

Additional Practice Example: Graph $y = (x - 3)^2 - 4$. Identify:

• Vertex: $(3, -4)$
• Axis of Symmetry: $x = 3$
• Table of Values: $x = 1, 2, 3, 4, 5$

Highlight the role of symmetry and the U-shaped curve characteristic of a Parabola.

Additional Multimedia Resources

• This is a collection of video definitions related to Quadratics: https://www.media4math.com/MathVideoCollection--QuadraticsDefinitions 
• This is a collection of definitions related to Quadratics: https://www.media4math.com/Definitions--Quadratics 
• INSTRUCTIONAL RESOURCE: Tutorial: Graphs of Quadratic Functions: https://www.media4math.com/library/21524/asset-preview
• INSTRUCTIONAL RESOURCE: Tutorial: Analyzing Graphs of Quadratic Functions in Standard Form: https://www.media4math.com/library/36131/asset-preview

By the end of this review, students should confidently identify and graph the vertex and axis of symmetry, complete a table of values, and accurately sketch a quadratic function.

Quiz

Answer the following questions.

1. Define vertex.
   
   
    
2. What is the axis of symmetry for $y = (x - 2)^2 - 5$?
   
   
    
3. Sketch $y = x^2 + 2x + 1$.
   
   
    
4. Identify the vertex of $y = -x^2 + 4x - 3$.
   
   
    
5. Graph $y = 2x^2 - 8x + 6$.
   
   
    
6. How does $y = -2x^2$ differ from $y = x^2$?
   
   
    
7. Describe the shape of a parabola.
   
   
    
8. What is the effect of a negative coefficient on $x^2$?
   
   
    
9. Solve $y = (x - 3)^2$ when $y = 9$.
   
   
    
10. Find the axis of symmetry for $y = x^2 - 4x + 3$.
    
     

Answer Key:

1. The highest or lowest point of a parabola.
2. $x = 2$.
3. The graph is a parabola with vertex $(-1, 0)$ and opens upwards.
4. The vertex is $(2, 1)$.
5. The graph is a parabola with vertex $(2, -2)$ and opens upwards.
6. $y = -2x^2$ is narrower than $y = x^2$ and opens downward.
7. A U-shaped curve.
8. A negative coefficient on $x^2$ flips the parabola to open downward.
9. $x = 6$ and $x = 0$.
10. $x = 2$.