Using the Maximum or Minimum Value to Sketch the Graph

Introduction

Key Concepts

1. Quadratic Functions Overview

A quadratic function is a second-degree polynomial of the form: $$ f(x) = ax^2 + bx + c $$ where \( a \), \( b \), and \( c \) are constants, and \( a \neq 0 \). The graph of a quadratic function is a parabola, which can open upwards or downwards depending on the sign of \( a \).

2. Vertex of a Parabola

The vertex of a parabola is its highest or lowest point, representing the maximum or minimum value of the function. For the quadratic function \( f(x) = ax^2 + bx + c \), the x-coordinate of the vertex is found using: $$ x = -\frac{b}{2a} $$ Substituting this back into the function gives the y-coordinate: $$ f\left(-\frac{b}{2a}\right) = a\left(-\frac{b}{2a}\right)^2 + b\left(-\frac{b}{2a}\right) + c $$ Simplifying, we find: $$ f\left(-\frac{b}{2a}\right) = c - \frac{b^2}{4a} $$ Thus, the vertex is \( \left(-\frac{b}{2a}, c - \frac{b^2}{4a}\right) \).

3. Maximum and Minimum Values

The maximum or minimum value of a quadratic function depends on the coefficient \( a \):

• If \( a > 0 \), the parabola opens upwards, and the vertex represents the minimum value.
• If \( a < 0 \), the parabola opens downwards, and the vertex represents the maximum value.

• If \( a > 0 \): \( f(x) \geq c - \frac{b^2}{4a} \)
• If \( a < 0 \): \( f(x) \leq c - \frac{b^2}{4a} \)

4. Axis of Symmetry

The axis of symmetry of a parabola is a vertical line that passes through the vertex, dividing the parabola into two mirror images. The equation of the axis of symmetry is: $$ x = -\frac{b}{2a} $$ This line is critical in sketching the graph as it helps determine the symmetry and accurate plotting of points.

5. Graphing the Parabola Using Vertex and Axis of Symmetry

To sketch the graph of a quadratic function using its maximum or minimum value:

1. Determine the coefficients \( a \), \( b \), and \( c \) from the quadratic equation.
2. Calculate the x-coordinate of the vertex using \( x = -\frac{b}{2a} \).
3. Find the y-coordinate by substituting the x-coordinate back into the function.
4. Identify the axis of symmetry with the equation \( x = -\frac{b}{2a} \).
5. Plot the vertex on the graph.
6. Use the axis of symmetry to plot additional points on either side of the vertex.
7. Draw the parabola opening upwards or downwards based on the sign of \( a \).

6. Examples

Example 1: Graph \( f(x) = 2x^2 - 4x + 1 \)

1. Identify \( a = 2 \), \( b = -4 \), \( c = 1 \).
2. Calculate vertex's x-coordinate: $$ x = -\frac{-4}{2 \times 2} = 1 $$
3. Find y-coordinate: $$ f(1) = 2(1)^2 - 4(1) + 1 = -1 $$
4. Vertex: \( (1, -1) \), Axis of symmetry: \( x = 1 \).
5. Plot additional points, e.g., \( x = 0 \), \( f(0) = 1 \); \( x = 2 \), \( f(2) = 1 \).
6. Draw the parabola opening upwards.

Example 2: Graph \( f(x) = -x^2 + 6x - 8 \)

1. Identify \( a = -1 \), \( b = 6 \), \( c = -8 \).
2. Calculate vertex's x-coordinate: $$ x = -\frac{6}{2 \times -1} = 3 $$
3. Find y-coordinate: $$ f(3) = -3^2 + 6(3) - 8 = 1 $$
4. Vertex: \( (3, 1) \), Axis of symmetry: \( x = 3 \).
5. Plot additional points, e.g., \( x = 2 \), \( f(2) = 0 \); \( x = 4 \), \( f(4) = 0 \).
6. Draw the parabola opening downwards.

7. Determining Range

The range of a quadratic function is the set of possible y-values. It depends on the parabola's direction:

• If \( a > 0 \): Range is \( [k, \infty) \), where \( k \) is the y-coordinate of the vertex.
• If \( a < 0 \): Range is \( (-\infty, k] \), where \( k \) is the y-coordinate of the vertex.

8. Real-World Applications

Quadratic functions model various real-life scenarios such as projectile motion, optimization problems, and areas. Understanding the maximum or minimum values helps in determining optimal solutions, such as maximizing profit or minimizing cost.

Advanced Concepts

1. Derivation of the Vertex Formula

The vertex form of a quadratic function provides a straightforward way to identify its vertex. Starting with the standard form: $$ f(x) = ax^2 + bx + c $$ We complete the square to convert it to vertex form: \begin{align*} f(x) &= a\left(x^2 + \frac{b}{a}x\right) + c \\ &= a\left(x^2 + \frac{b}{a}x + \left(\frac{b}{2a}\right)^2 - \left(\frac{b}{2a}\right)^2\right) + c \\ &= a\left(\left(x + \frac{b}{2a}\right)^2 - \frac{b^2}{4a^2}\right) + c \\ &= a\left(x + \frac{b}{2a}\right)^2 - \frac{b^2}{4a} + c \\ &= a\left(x + \frac{b}{2a}\right)^2 + \left(c - \frac{b^2}{4a}\right) \end{align*} Thus, the vertex form is: $$ f(x) = a\left(x + \frac{b}{2a}\right)^2 + \left(c - \frac{b^2}{4a}\right) $$ From this, the vertex is clearly \( \left(-\frac{b}{2a}, c - \frac{b^2}{4a}\right) \).

2. Discriminant and Its Role

The discriminant of a quadratic equation \( ax^2 + bx + c = 0 \) is given by: $$ D = b^2 - 4ac $$ It determines the nature of the roots:

• If \( D > 0 \): Two distinct real roots.
• If \( D = 0 \): One real root (repeated).
• If \( D < 0 \): No real roots, complex roots.

3. Optimization Problems

Quadratic functions are pivotal in optimization problems where the goal is to find the maximum or minimum value of a certain quantity. For example, determining the maximum area of a rectangular enclosure with a fixed perimeter involves quadratic functions.

4. Integration with Calculus

While the Cambridge IGCSE curriculum does not delve deeply into calculus, the concept of maximum and minimum values lays the groundwork for understanding derivatives. The derivative of a quadratic function reveals the slope of the tangent at any point, and setting it to zero identifies critical points corresponding to maxima or minima.

5. Applications in Physics and Engineering

Quadratic functions model projectile motion, where the height of an object is a quadratic function of time. The maximum height achieved corresponds to the vertex of the parabola. In engineering, optimizing material usage or stress distribution often involves quadratic relationships.

6. Complex Problem-Solving Strategies

Advanced problems may require combining multiple concepts:

• Finding the maximum area of a rectangle inscribed under a parabola.
• Determining the optimal pricing strategy to maximize revenue based on quadratic models.

7. Symmetry and Parabola Properties

Exploring the symmetrical properties of parabolas enhances graphing skills. Recognizing that the parabola is symmetric about its axis allows for efficient plotting of points and a more accurate sketch.

8. Interdisciplinary Connections

Quadratic functions intersect with disciplines such as economics, where cost and revenue functions are often quadratic, and biology, where population models may exhibit quadratic growth patterns. Understanding maxima and minima in these contexts provides valuable insights into optimization and equilibrium states.

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Summary and Key Takeaways