IIT Madras
Mathematics 1
Quadratic functions

Quadratic functions

A well-defined collection of distinct objects called elements or members.

https://youtu.be/U4gr4zMosMk

Learning Outcomes

  1. Compare quadratic functions and linear functions.
  2. Define a quadratic function and represent it using a parabola.
  3. Identify different terms in a quadratic function and comprehend the importance of their coefficients.
  4. Represent the ‘axis of symmetry’ and ‘vertex’ used to represent parabola.

Exercise Questions 🤯

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Hello! On this Wednesday evening here in India, let’s go through these questions about quadratic functions. These problems involve understanding the shape and key features of parabolas from their equations.

Core Concepts: Understanding Quadratic Functions ($y = ax^2 + bx + c$)

Before we solve the problems, let’s review the basics. The graph of a quadratic function is a U-shaped curve called a parabola.

  1. Direction of Opening: The sign of the leading coefficient, ‘$a$’, tells you everything about the parabola’s direction and whether it has a minimum or maximum point.

    • If $a > 0$ (positive), the parabola opens upwards ($\cup$) and has a minimum value at its lowest point.
    • If $a < 0$ (negative), the parabola opens downwards ($\cap$) and has a maximum value at its highest point.

  2. The Vertex: The vertex is the turning point of the parabola—either the very bottom (minimum) or the very top (maximum). We can find its coordinates $(x_v, y_v)$ with a simple formula:

    • x-coordinate of the vertex: $x_v = -\frac{b}{2a}$
    • y-coordinate of the vertex: Plug the x-coordinate you just found back into the equation. This y-value is the minimum or maximum value of the function.

Question 1: Identifying the Graph of a Parabola (from file image_00850a.png)

The Question: Which of the graphs in Figure 1 represents the following function: $y = x^2 - x + 1$?

Detailed Solution:

  1. Analyze the Equation: The function is $y = x^2 - x + 1$. Let’s identify the coefficients:

    • $a = 1$
    • $b = -1$
    • $c = 1$

  2. Determine the Direction:

    • Since $a = 1$ (which is positive), the parabola must open upwards.
    • Looking at the four graphs, only Graph D shows a parabola opening upwards. Graphs B and C are not parabolas, and Graph A opens downwards.

  3. (Optional) Confirm by Finding the Vertex:

    • Let’s find the vertex to be absolutely sure.
    • x-coordinate: $x_v = -\frac{b}{2a} = -\frac{-1}{2(1)} = \frac{1}{2} = 0.5$.
    • y-coordinate: $y_v = (0.5)^2 - (0.5) + 1 = 0.25 - 0.5 + 1 = 0.75$.
    • The vertex is at $(0.5, 0.75)$.
    • Graph D shows a parabola opening upwards with its vertex slightly to the right of the y-axis and above the x-axis, which perfectly matches our calculated vertex.

Final Answer: Option D.

Question 2: Identifying the Graph of a Parabola (from file image_0081ff.png)

The Question: Which of the graphs in Figure 1 represents the following function: $y = -x^2 + 2x + 1$?

Detailed Solution:

  1. Analyze the Equation: The function is $y = -x^2 + 2x + 1$. Let’s identify the coefficients:

    • $a = -1$
    • $b = 2$
    • $c = 1$

  2. Determine the Direction:

    • Since $a = -1$ (which is negative), the parabola must open downwards.
    • Looking at the four graphs, only Graph A shows a parabola opening downwards.

  3. (Optional) Confirm by Finding the Vertex:

    • x-coordinate: $x_v = -\frac{b}{2a} = -\frac{2}{2(-1)} = -\frac{2}{-2} = 1$.
    • y-coordinate: $y_v = -(1)^2 + 2(1) + 1 = -1 + 2 + 1 = 2$.
    • The vertex is at $(1, 2)$.
    • Graph A shows a parabola opening downwards with its peak (vertex) at the point $(1, 2)$, which is a perfect match.

Final Answer: Option A.

Question 3: Finding the Minimum/Maximum Value (from file image_0081a4.png)

The Question: For the function $y = x^2 + 10x + 1$, determine whether $y$ has a minimum or maximum value and identify it from the options given below.

  • Maximum value -24
  • Minimum value -24
  • Maximum value 76
  • Neither maximum nor minimum value exists

Detailed Solution:

  1. Analyze the Equation: The function is $y = x^2 + 10x + 1$. The coefficients are:

    • $a = 1$
    • $b = 10$
    • $c = 1$

  2. Determine if it’s a Minimum or Maximum:

    • Since $a = 1$ (positive), the parabola opens upwards.
    • An upward-opening parabola has a lowest point, which is a minimum value.
    • This immediately eliminates the “Maximum value” options.

  3. Find the Minimum Value (the y-coordinate of the vertex):

    • First, find the x-coordinate of the vertex: $$x_v = -\frac{b}{2a} = -\frac{10}{2(1)} = -5$$
    • Now, substitute this x-value back into the original equation to find the y-coordinate, which is our minimum value: $$y_{min} = (-5)^2 + 10(-5) + 1$$ $$y_{min} = 25 - 50 + 1$$ $$y_{min} = -24$$

Final Answer: The function has a Minimum value -24.

Graded Assignment 2Examples of Quadratic =