Lesson Plan: Introduction to Quadratic Functions
Lesson Summary
In this 50-minute lesson, students will learn what a quadratic function is. Key features such as the vertex and axis of symmetry will be identified. This lesson incorporates multimedia resources, including videos, clip art, and from Media4Math.com. A 10-question quiz with an answer key will assess understanding at the end.
Lesson Objectives
- Understand the definition and general form of a quadratic function.
- Identify quadratic functions in equations and graphs.
- Analyze key features of quadratic graphs, including vertex and axis of symmetry.
Common Core Standards
- HSF.IF.4: For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities.
- HSF.IF.5: Relate the domain of a function to its graph and the quantitative relationship it describes.
Prerequisite Skills
- Recognizing and plotting linear functions.
- Basic understanding of function notation.
Key Vocabulary
- Quadratic function: A function described by the equation \( y = ax^2 + bx + c \), where \( a \neq 0 \).
- Multimedia Resource: https://www.media4math.com/library/39563/asset-preview
- Parabola: The U-shaped graph of a quadratic function.
- Multimedia Resource: https://www.media4math.com/library/74573/asset-preview
- Multimedia Resource: https://www.media4math.com/library/22118/asset-preview
- Standard form: The equation \( y = ax^2 + bx + c \) representing a quadratic function.
- Multimedia Resource: https://www.media4math.com/library/74572/asset-preview
- Vertex: The point representing the maximum or minimum value of a parabola.
- Multimedia Resource: https://www.media4math.com/library/74587/asset-preview
- Multimedia Resource: https://www.media4math.com/library/43008/asset-preview
- Axis of symmetry: A vertical line that divides a parabola into two symmetrical halves. V
- Multimedia Resource: https://www.media4math.com/library/74586/asset-preview
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 one or more activities below.
Activity 1: Using Algebra Tiles to Model Quadratic Expressions
Objective: Use algebra tiles to visually model and simplify quadratic expressions.
Materials Needed: Algebra tiles (tiles representing x², x, and constants).
Introduce the Algebra Tiles: Review the representation of algebra tiles:
- Large square: x²
- Long rectangle: x
- Small square: constant term
Example Setup: Write the quadratic expression x² + 3x + 2 on the board.
Student Task:
- Model the given quadratic expression using algebra tiles.
- Arrange the tiles in a rectangular form to visualize the expression as an area model.
- Discuss the relationship between the tiles and the terms in the expression.
- Challenge Question: Model and factor x² + 5x + 6 using algebra tiles.
Debrief: Highlight how algebra tiles help visualize and factor quadratic expressions.
Activity 2: Finding the Areas of Rectangles with Variable Dimensions
Objective: Practice finding areas of rectangles with variable dimensions.
Present Examples:
- Rectangle A: Length = x + 2, Width = x + 3
- Rectangle B: Length = x - 1, Width = x + 4
Student Task:
- Calculate the area of each rectangle by multiplying length and width.
- Expand the product to form a quadratic expression.
Guided Practice:
- Rectangle A: Area = (x + 2)(x + 3) = x² + 5x + 6
- Rectangle B: Area = (x - 1)(x + 4) = x² + 3x - 4
Extension: Include examples with negative coefficients or fractions.
Debrief: Discuss the connection between multiplication of expressions and quadratic forms.
Activity 3: Graphing Functions Using Desmos.com
Activity:
- Go to Desmos.com.
- Graph \( y = 2x + 1 \) (linear) and \( y = x^2 - 4 \) (quadratic).
- Observe differences between a straight line and a parabola.
- Discuss with a partner how the graph of \( y = x^2 - 4 \) changes when the coefficient of \( x^2 \) is increased.
Teach
Introduction to Quadratics
Quadratic functions are a foundational concept in algebra, representing relationships where the highest power of the variable is two. These functions are written in standard form y = ax2 + bx + c, where a, b, and c are constants, and a ≠ 0. Quadratics are unique because they model parabolic shapes, which appear in various real-world contexts, such as the trajectory of a thrown ball or the design of suspension bridges.
The graph of a quadratic function, called a parabola, has several key features. The vertex is the highest or lowest point of the parabola, depending on whether it opens upwards (a > 0) or downwards (a < 0). The axis of symmetry is a vertical line that passes through the vertex, dividing the parabola into two mirror-image halves. Quadratic graphs may also have one, two, or no x-intercepts, which represent the points where the graph crosses the x-axis. Additionally, the y-intercept is the point where the graph intersects the y-axis, located at (0, c).
In this lesson, students will explore how changes in the coefficients a, b, and c affect the shape and position of the parabola, gaining insights into the essential features of quadratic graphs.
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:
x | y |
|---|---|
| -3 | 9 |
| -2 | 4 |
| -1 | 1 |
| 0 | 0 |
| 1 | 1 |
| 2 | 4 |
| 3 | 9 |
- 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: \( y = 2(x - 1)^2 - 3 \). This is an equation in vertex form.
- 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:
x | y |
|---|---|
| -1 | 5 |
| 0 | -1 |
| 1 | -3 |
| 2 | -1 |
| 3 | 5 |
- 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 \( t \):
t | h(t) |
|---|---|
| 0 | 2 |
| 0.5 | 5.675 |
| 1 | 6.9 |
| 1.5 | 5.675 |
| 2 | 2 |
- Plot the points \( (0, 2), (0.5, 6.025), (1, 7.9), (1.5, 6.625), (2, 2) \).
- Sketch the parabola to show the trajectory of the ball. The vertex \( (1, 6.9) \) represents the maximum height.
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
Review
Vocabulary Review
Before diving into the examples, let’s briefly review some key vocabulary terms related to quadratic functions:
- Quadratic Function: A function of the form y = ax2 + bx + c, where a, b, and c are constants, and a ≠ 0.
- Parabola: The U-shaped graph of a quadratic function.
- Vertex: The highest or lowest point of the parabola, representing its minimum or maximum value.
- Axis of Symmetry: A vertical line passing through the vertex that divides the parabola into two mirror-image halves.
- X-Intercepts: The points where the parabola crosses the x-axis, also called roots or solutions.
- Y-Intercept: The point where the parabola crosses the y-axis, located at (0, c).
Tutorial Video
Watch the video below to reinforce your understanding of quadratic graphs and their features.
Example 1: Analyzing a Function in Standard Form
Consider the quadratic function y = 2x2 - 4x + 1. This is a quadratic in standard form y = ax2 + bx + c. Let’s identify its features:
- We can conclude the following from just looking at the equation:
- The positive value for a means the parabola has a U-shape.
- Because a > 1, the parabola is narrow.
- The y-intercept is at (0, 1)
- This is what we can conclude by making calculations using the values of a, b, and c.
- The x-value for the vertex is x = -b/(2a) = -(-4)/(2•2) = 1
- The axis of symmetry is x = 1
- The y coordinate of the vertex is y = 2(1)2 - 4(1) + 1 = -1
- The coordinates of the vertex are (1, -1)
Graph this parabola using the vertex, intercepts, and axis of symmetry.
Example 2: Analyzing a Function in Vertex Form
Consider the quadratic function y = -(x - 3)2 + 1. This is a quadratic function in vertex form: y = a(x - h)2 + k. Let’s identify its features:
- We can conclude the following from just looking at the equation:
- The negative value for a means the parabola has an upside down U-shape.
- The coordinates of the vertex are (3, 1)
- The axis of symmetry is x = 3.
- This is what we can conclude by making calculations:
- Find the y-intercept by evaluating for x = 0: y = -(-3)2 + 1 = -9 + 1 = -8
Reflection
Practice identifying the features of quadratic graphs with different equations. Pay special attention to how the coefficients and other parameters influence the parabola's shape, direction, and position.
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
- Closed Captioned Video: Quadratics: What Is a Quadratic Function? https://www.media4math.com/library/39563/asset-preview
- Closed Captioned Video: Quadratics: What Is a Quadratic Equation? https://www.media4math.com/library/39562/asset-preview
Quiz
Answer the following questions about quadratic functions.
Part 1: Standard Form
Recall that the standard form of a quadratic function is y = ax2 + bx + c.
- Given:
y = 2x2 - 4x + 1
Coordinates of the Vertex: ___________
Equation for the Axis of Symmetry: ___________
Coordinates for the y-intercept: ___________ - Given:
y = -3x2 + 6x - 2
Coordinates of the Vertex: ___________
Equation for the Axis of Symmetry: ___________
Coordinates for the y-intercept: ___________ - Given:
y = x2 + 8x + 15
Coordinates of the Vertex: ___________
Equation for the Axis of Symmetry: ___________
Coordinates for the y-intercept: ___________ - Given:
y = -x2 + 2x + 5
Coordinates of the Vertex: ___________
Equation for the Axis of Symmetry: ___________
Coordinates for the y-intercept: ___________ - Given:
y = 4x2 - 16x + 7
Coordinates of the Vertex: ___________
Equation for the Axis of Symmetry: ___________
Coordinates for the y-intercept: ___________
Part 2: Vertex Form
Recall that the vertex form of a quadratic function is y = a(x - h)2 + k, where (h, k) is the vertex.
- Given:
y = 2(x - 3)2 + 4
Coordinates of the Vertex: ___________
Equation for the Axis of Symmetry: ___________
Coordinates for the y-intercept: ___________ - Given:
y = -1(x + 2)2 - 5
Coordinates of the Vertex: ___________
Equation for the Axis of Symmetry: ___________
Coordinates for the y-intercept: ___________ - Given:
y = 0.5(x - 1)2 + 3
Coordinates of the Vertex: ___________
Equation for the Axis of Symmetry: ___________
Coordinates for the y-intercept: ___________ - Given:
y = -2(x + 4)2 + 6
Coordinates of the Vertex: ___________
Equation for the Axis of Symmetry: ___________
Coordinates for the y-intercept: ___________ - Given:
y = 3(x - 5)2 - 2
Coordinates of the Vertex: ___________
Equation for the Axis of Symmetry: ___________
Coordinates for the y-intercept: ___________
Answer Key
- (1, -1); x = 1; (0, 1)
- (1, 1); x = 1; (0, -2)
- (-4, -1); x = -4; (0, 15)
- (1, 6); x = 1; (0, 5)
- (2, -9); x = 2; (0, 7)
- (3, 4); x = 3; (0, 22)
- (-2, -5); x = -2; (0, -9)
- (1, 3); x = 1; (0, 3.5)
- (-4, 6); x = -4; (0, -26)
- (5, -2); x = 5; (0, 73)
