Lesson 4: Graphs of Linear and Quadratic Functions

👨🏽‍🏫 Introduction

Have you ever seen a straight road or a curved rollercoaster? In mathematics, we describe those shapes using graphs of functions 📈.

This lesson will teach you how to draw and interpret linear (straight-line) and quadratic (parabola-shaped) graphs. These graphs help us model and solve real-life problems like motion, cost, and design.

By the end of this lesson, you’ll be able to:

• Recognize and sketch linear and quadratic graphs
• Generate table values and plot coordinates
• Interpret features like roots, turning point, and axis of symmetry

🧠 Key Concepts and Explanations

📏 Linear Functions

Linear functions have the form:

Where:

• m is the gradient (slope)
• c is the y-intercept (where the graph crosses the y-axis)

Graph is a straight line.

🟰 Quadratic Functions

Quadratic functions have the form:

The graph is a curve called a parabola:

• If a > 0, it opens upwards (U-shaped)
• If a < 0, it opens downwards (n-shaped)

📉 Key Features of a Quadratic Graph

• Roots or Zeros: x-values where y = 0 (i.e., where the graph crosses the x-axis)
• Vertex or Turning Point: The highest or lowest point on the curve
• Axis of Symmetry: Vertical line that divides the parabola in half

📊 Sample Problem Walkthroughs

1. 1. Sketch the graph of y = 2x + 1 for x = -2 to 2
      Create a table:

      x | -2 | -1 |  0 |  1 |  2
      y | -3 | -1 |  1 |  3 |  5
      

      Plot the points and draw a straight line through them.

1. Sketch the graph of y = x² – 2x – 3 for x = -1 to 4
   Calculate values:

   x | -1 |  0 |  1 |  2 |  3 | 4
   y |  6 | -3 | -4 | -3 |  0 | 5
   

   Plot the points and draw a smooth U-shaped curve.

   

   Roots: x = 3 and x = -1 (where y = 0)

   Turning Point: At x = 1, y = -4

   Axis of Symmetry: x = 1

✍🏽 Practice Exercises

🅰️ A. Linear Graphs

1. Draw the graph of y = -x + 4 for x = -2 to 3
2. Draw the graph of y = 3x – 2 for x = -1 to 2

🅱️ B. Quadratic Graphs

1. Sketch the graph of y = x² + x – 6 for x = -3 to 3
2. Sketch the graph of y = -x² + 4x for x = 0 to 5

🆎 C. Interpretation

1. From the graph of y = x² – 4x + 3, state the roots, turning point, and axis of symmetry.
2. What is the minimum value of y = x² – 6x + 8?

✅ Answers and Explanations

🅰️ A. Linear Graphs

• 1. y = -x + 4 gives points (-2, 6), (-1, 5), (0, 4), (1, 3), (2, 2), (3, 1)
• 2. y = 3x – 2 gives points (-1, -5), (0, -2), (1, 1), (2, 4)

🅱️ B. Quadratic Graphs

• 1. y = x² + x – 6 has roots at x = -3 and x = 2; vertex at x = -0.5
• 2. y = -x² + 4x has roots at x = 0 and x = 4; vertex at x = 2, y = 4

🆎 C. Interpretation

• 1. For y = x² – 4x + 3:
  Roots: x = 1 and x = 3
  Axis of symmetry: x = 2
  Turning point: (2, -1)
• 2. For y = x² – 6x + 8:
  Turning point at x = 3, so minimum y =

📌 Recap

• ✅ Linear graphs form straight lines; quadratic graphs form curves (parabolas)
• ✅ Key parts of quadratic graphs include roots, vertex (turning point), and axis of symmetry
• ✅ Tables help generate values for accurate plotting

💭 Reflection Prompt

If a business’s profit graph is a parabola, what might the turning point represent? Why would it be important to find it?