Quadratic Function Examples With Answers

Understanding and Solving Quadratic Function Examples: A Comprehensive Guide

Quadratic functions are fundamental to algebra and have wide-ranging applications in various fields, from physics and engineering to economics and computer science. Understanding how to identify, graph, and solve quadratic functions is crucial for anyone pursuing further studies in mathematics or related disciplines. This comprehensive guide provides numerous examples of quadratic functions with detailed solutions, designed to enhance your understanding and problem-solving skills. We'll explore various methods for solving quadratic equations and delve into real-world applications.

What is a Quadratic Function?

A quadratic function is a polynomial function of degree two, meaning the highest power of the variable (usually x) is 2. It can be expressed in the general form:

f(x) = ax² + bx + c

where a, b, and c are constants, and a ≠ 0 (if a were 0, it would be a linear function). The graph of a quadratic function is a parabola, a U-shaped curve that opens either upwards (if a > 0) or downwards (if a < 0). The value of a also affects the parabola's width; a larger absolute value of a results in a narrower parabola, while a smaller absolute value results in a wider one.

Key Features of a Quadratic Function:

Before we dive into examples, let's review the important features of a quadratic function:

• Vertex: The highest or lowest point on the parabola. Its coordinates are given by (-b/2a, f(-b/2a)).
• Axis of Symmetry: A vertical line that passes through the vertex, dividing the parabola into two symmetrical halves. Its equation is x = -b/2a.
• x-intercepts (Roots or Zeros): The points where the parabola intersects the x-axis (where y = 0). These are found by solving the quadratic equation ax² + bx + c = 0.
• y-intercept: The point where the parabola intersects the y-axis (where x = 0). This is simply the value of c.

Methods for Solving Quadratic Equations:

There are several methods to solve quadratic equations (ax² + bx + c = 0):

1. Factoring: This method involves rewriting the quadratic expression as a product of two linear expressions. It's the easiest method when applicable.

2. Quadratic Formula: This formula provides the solutions for any quadratic equation:

   x = [-b ± √(b² - 4ac)] / 2a

   The discriminant (b² - 4ac) determines the nature of the roots:

   • If b² - 4ac > 0, there are two distinct real roots.
   • If b² - 4ac = 0, there is one real root (a repeated root).
   • If b² - 4ac < 0, there are two complex roots (involving imaginary numbers).

3. Completing the Square: This method involves manipulating the quadratic expression to form a perfect square trinomial, making it easier to solve.

Examples of Quadratic Functions with Solutions:

Let's work through several examples, illustrating the different methods and highlighting key concepts.

Example 1: Factoring

Solve the quadratic equation x² + 5x + 6 = 0.

• Solution: We look for two numbers that add up to 5 and multiply to 6. These numbers are 2 and 3. Therefore, we can factor the equation as (x + 2)(x + 3) = 0. This means either x + 2 = 0 or x + 3 = 0. Solving these gives us x = -2 and x = -3. The x-intercepts are (-2, 0) and (-3, 0).

Example 2: Quadratic Formula

Solve the quadratic equation 2x² - 7x + 3 = 0.

• Solution: Here, a = 2, b = -7, and c = 3. Applying the quadratic formula:

  x = [7 ± √((-7)² - 4 * 2 * 3)] / (2 * 2) x = [7 ± √(49 - 24)] / 4 x = [7 ± √25] / 4 x = [7 ± 5] / 4

  This gives us two solutions: x = (7 + 5) / 4 = 3 and x = (7 - 5) / 4 = 0.5. The x-intercepts are (3, 0) and (0.5, 0).

Example 3: Completing the Square

Solve the quadratic equation x² + 6x + 5 = 0.

• Solution: To complete the square, we take half of the coefficient of x (which is 6), square it (3² = 9), and add and subtract it to the equation:

  x² + 6x + 9 - 9 + 5 = 0 (x + 3)² - 4 = 0 (x + 3)² = 4 x + 3 = ±√4 x + 3 = ±2 x = -3 ± 2

  This gives us two solutions: x = -1 and x = -5. The x-intercepts are (-1, 0) and (-5, 0).

Example 4: Finding the Vertex and Axis of Symmetry

Find the vertex and axis of symmetry for the quadratic function f(x) = x² - 4x + 3.

• Solution: Here, a = 1, b = -4, and c = 3.

  The x-coordinate of the vertex is -b/2a = -(-4) / (2 * 1) = 2. The y-coordinate of the vertex is f(2) = (2)² - 4(2) + 3 = -1. Therefore, the vertex is (2, -1). The axis of symmetry is x = 2.

Example 5: Real-world Application – Projectile Motion

A ball is thrown upwards with an initial velocity of 20 m/s from a height of 1.5 meters. Its height (h) in meters after t seconds is given by the equation h(t) = -4.9t² + 20t + 1.5. Find the maximum height reached by the ball.

• Solution: This is a quadratic function where height is a function of time. The maximum height occurs at the vertex of the parabola. The t-coordinate of the vertex is -b/2a = -20 / (2 * -4.9) ≈ 2.04 seconds. Substituting this into the equation gives the maximum height:

  h(2.04) = -4.9(2.04)² + 20(2.04) + 1.5 ≈ 21.4 meters.

Example 6: Quadratic Inequalities

Solve the inequality x² - x - 6 > 0.

• Solution: First, we solve the corresponding quadratic equation x² - x - 6 = 0. This factors as (x - 3)(x + 2) = 0, giving solutions x = 3 and x = -2. These values divide the number line into three intervals: (-∞, -2), (-2, 3), and (3, ∞). Testing a value from each interval reveals that the inequality is satisfied when x < -2 or x > 3.

Frequently Asked Questions (FAQ):

• Q: What if the quadratic equation has no real solutions?

  • A: This happens when the discriminant (b² - 4ac) is negative. The solutions are then complex numbers involving the imaginary unit i (where i² = -1).

• Q: Can I use a graphing calculator to solve quadratic equations?

  • A: Yes, graphing calculators can graph the quadratic function and you can find the x-intercepts (roots) visually from the graph. Many calculators also have built-in functions to solve quadratic equations directly.

• Q: What are some other applications of quadratic functions?

  • A: Quadratic functions are used to model various phenomena, including the trajectory of projectiles, the shape of a suspension bridge cable, optimization problems in business, and many more.

Conclusion:

Quadratic functions are a cornerstone of algebra and have numerous practical applications. Mastering the techniques for solving quadratic equations—factoring, using the quadratic formula, and completing the square—is essential for success in mathematics and related fields. By understanding the key features of quadratic functions, such as the vertex, axis of symmetry, and intercepts, you can effectively analyze and interpret their graphs and solve real-world problems. Remember to practice regularly with different examples to solidify your understanding and build your problem-solving skills. The more you practice, the more confident and proficient you will become in working with quadratic functions.