IIT Madras
Mathematics 1
Examples of Quadratic functions

Examples of Quadratic functions

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

https://youtu.be/dvJKbgIPG8Q

Learning Outcomes

  1. Determine the minimum and maximum value of a quadratic function.
  2. Explain the concept of range and domain of a quadratic function.
  3. Demonstrate the ability to apply these concepts in real life scenarios.

Exercise Questions 🤯

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Hello! On this Wednesday evening here in India, I’d be glad to help you with this set of problems. They are excellent examples of how quadratic functions are used to model real-world situations, from the path of an ant to maximizing business profits.

Core Concepts: The Parabola ($y = ax^2 + bx + c$)

All these problems revolve around the properties of parabolas. Let’s quickly review the key formula.

  • The Vertex: The vertex is the highest or lowest point of the parabola. Its coordinates $(x_v, y_v)$ are found using:
    • x-coordinate: $x_v = -\frac{b}{2a}$
    • y-coordinate: Plug $x_v$ back into the function to find $y_v$.
  • Maximum vs. Minimum:
    • If ‘$a$’ is positive, the parabola opens upwards ($\cup$), and the vertex is a minimum.
    • If ‘$a$’ is negative, the parabola opens downwards ($\cap$), and the vertex is a maximum.
  • Axis of Symmetry: This is the vertical line that cuts the parabola in half. Its equation is simply $x = x_v$.

Question 1: Ant on a Banana (from file image_0079c2.png)

The Question: The curve on the surface of the banana as shown in Figure 2 can be described using the equation $y = x^2 + 2x + 4$. An ant (shown in blue color in Figure 2) is walking from one end of the banana to the other end. What will be the x-coordinate of the ant’s location once it reaches the vertex of its path?

Core Concept: The question asks for the x-coordinate of the vertex of the parabola.

Detailed Solution:

  1. Identify the coefficients from the equation $y = x^2 + 2x + 4$:
    • $a = 1$
    • $b = 2$
    • $c = 4$
  2. Apply the formula for the x-coordinate of the vertex: $$x_v = -\frac{b}{2a}$$ $$x_v = -\frac{2}{2(1)} = -1$$

Final Answer: The x-coordinate of the ant at the vertex is -1.

Question 2: Deb and Ananya’s Toys (from file image_0076dc.png)

The Question: Deb and Ananya bought 20 toys together. From these 20 toys, Deb lost 3 toys and Ananya lost 4 toys. Product of the current number of their toys is 42. Can you form an equation for Deb to know how many toys did he have initially? [Let us assume Deb initially had x number of toys.]

Core Concept: Translating a word problem into a mathematical equation by defining expressions for each quantity.

Detailed Solution:

  1. Define initial quantities:

    • Number of toys Deb had initially = $x$.
    • Total toys = 20.
    • Number of toys Ananya had initially = $20 - x$.

  2. Define current quantities (after losing some):

    • Number of toys Deb has now = $x - 3$.
    • Number of toys Ananya has now = $(20 - x) - 4 = 16 - x$.

  3. Form the equation based on the product:

    • The product of their current number of toys is 42.
    • (Deb’s current toys) $\times$ (Ananya’s current toys) = 42
    • $(x - 3)(16 - x) = 42$

  4. Expand the equation to match the options:

    • Use the FOIL method: $x(16) + x(-x) - 3(16) - 3(-x) = 42$
    • $16x - x^2 - 48 + 3x = 42$
    • Combine like terms: $-x^2 + 19x - 48 = 42$

Final Answer: The correct equation is $-x^2 + 19x - 48 = 42$.

Question 3: Vaccine Manufacturing Cost (from file image_0076dc.png)

The Question: Represent the following problem in the form of an equation: a medicine manufacturer produces y number of vaccines every day. The manufacturing cost of each vaccine is ₹100 plus the number of vaccines manufactured on that day. On a particular day, the total manufacturing cost was ₹10,000. How many vaccines were manufactured on that day?

Core Concept: Total Cost = (Number of Units) $\times$ (Cost per Unit).

Detailed Solution:

  1. Define the quantities from the problem:

    • Number of vaccines produced = $y$.
    • Cost per vaccine = “₹100 plus the number of vaccines”, which translates to $100 + y$.
    • Total manufacturing cost = ₹10,000.

  2. Set up the equation:

    • Total Cost = (Number of Units) $\times$ (Cost per Unit)
    • $10000 = y \times (100 + y)$

  3. Expand and rearrange the equation to match the options:

    • $10000 = 100y + y^2$
    • This is commonly written with the highest power first: $y^2 + 100y = 10000$.

Final Answer: The correct equation is $y^2 + 100y = 10000$.

Question 4: Vertex and Axis of Symmetry (from file image_007662.png)

The Question: Find the vertex and axis of symmetry of the graph of the quadratic function: $f(x) = x^2 + 4x + 5$.

Core Concept: The vertex is the point $(x_v, y_v)$ and the axis of symmetry is the vertical line $x = x_v$.

Detailed Solution:

  1. Identify the coefficients from $f(x) = x^2 + 4x + 5$:

    • $a = 1$
    • $b = 4$
    • $c = 5$

  2. Find the x-coordinate of the vertex:

    $$x_v = -\frac{b}{2a} = -\frac{4}{2(1)} = -2$$

  3. Determine the axis of symmetry:

    • The axis of symmetry is the line $x = x_v$, so it is $x = -2$.

  4. Find the y-coordinate of the vertex:

    • Substitute $x = -2$ back into the function:
    • $y_v = f(-2) = (-2)^2 + 4(-2) + 5$
    • $y_v = 4 - 8 + 5 = 1$

  5. State the vertex:

    • The vertex is the point $(x_v, y_v)$, which is $(-2, 1)$.

Final Answer: The correct option is vertex= (-2,1), axis of symmetry x=-2.

Question 5: Maximizing Profit (Finding the Input) (from file image_007662.png)

The Question: Ananya runs a shop selling books. The profit she makes from her shop is given by the function $P(u) = 100 + 40u - 2u^2$, where u is the amount that she spends on bookbinding. Find the value of u in order to maximize the profit $P(u)$.

Core Concept: A downward-opening parabola (negative ‘a’ value) has a maximum value at its vertex. This question asks for the input value ($u$) that gives this maximum profit, which is the u-coordinate of the vertex.

Detailed Solution:

  1. Rearrange the function and identify coefficients:

    • $P(u) = -2u^2 + 40u + 100$
    • $a = -2$
    • $b = 40$
    • $c = 100$
    • Since ‘a’ is negative, this parabola opens downwards and has a maximum.

  2. Find the u-coordinate of the vertex:

    • $u_{vertex} = -\frac{b}{2a} = -\frac{40}{2(-2)} = -\frac{40}{-4} = 10$

Final Answer: The value of u that maximizes the profit is 10.

Question 6: Finding the Maximum Profit (from file image_007622.png)

The Question: Find the maximum profit obtained by Ananya’s shop.

Core Concept: This is the follow-up to the previous question. The maximum profit is the actual maximum value of the function, which is the y-coordinate (or in this case, the P-coordinate) of the vertex.

Detailed Solution:

  1. From the previous question, we know that profit is maximized when $u = 10$.

  2. Calculate the profit $P(u)$ for $u = 10$:

    • $P(10) = 100 + 40(10) - 2(10)^2$
    • $P(10) = 100 + 400 - 2(100)$
    • $P(10) = 500 - 200$
    • $P(10) = 300$

Final Answer: The maximum profit is 300.

Examples of Quadratic =Quadratic