Quadratic Function Examples With Answers

Understanding Quadratic Functions: Examples and Solutions

Quadratic functions are fundamental to algebra and have widespread applications in various fields, from physics and engineering to economics and computer graphics. Understanding quadratic functions, their graphs (parabolas), and how to solve equations involving them is crucial for anyone pursuing further studies in mathematics or related disciplines. This comprehensive guide will delve into the world of quadratic functions, providing numerous examples with detailed solutions and explanations to solidify your understanding. We'll cover various methods for solving quadratic equations and explore 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 written in the general form:

f(x) = ax² + bx + c

where a, b, and c are constants, and a is not equal to zero (if a were zero, it would no longer be a quadratic function). The graph of a quadratic function is a parabola, a symmetrical U-shaped curve. The value of a determines the parabola's direction (opening upwards if a > 0 and downwards if a < 0), while b and c influence its position on the coordinate plane.

Key Features of a Quadratic Function and its Graph

Before diving into examples, let's review some essential features of quadratic functions and their parabolic graphs:

• Vertex: The vertex is the lowest (minimum) or highest (maximum) point on the parabola. Its coordinates are given by: x = -b / 2a and substituting this x value back into the function to find the corresponding y value.

• Axis of Symmetry: This is a vertical line passing through the vertex, dividing the parabola into two symmetrical halves. Its equation is x = -b / 2a.

• x-intercepts (Roots or Zeros): These are the points where the parabola intersects the x-axis (where y = 0). They are the solutions to the quadratic equation ax² + bx + c = 0.

• y-intercept: This is the point where the parabola intersects the y-axis (where x = 0). It is simply the value of c.

Methods for Solving Quadratic Equations

Several methods exist for solving quadratic equations (finding the x-intercepts):

• Factoring: This involves expressing the quadratic expression as a product of two linear factors. For example, x² + 5x + 6 = (x + 2)(x + 3) = 0, which gives solutions x = -2 and x = -3. This method is only effective for easily factorable quadratics.

• Quadratic Formula: This formula provides the solutions for any quadratic equation, regardless of its factorability:

x = [-b ± √(b² - 4ac)] / 2a

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

* **b² - 4ac > 0:** Two distinct real roots (two x-intercepts)
* **b² - 4ac = 0:** One real root (repeated root; the vertex touches the x-axis)
* **b² - 4ac < 0:** No real roots (the parabola doesn't intersect the x-axis)

• Completing the Square: This method involves manipulating the quadratic expression to create a perfect square trinomial, allowing for easy solution.

Examples of Quadratic Functions with Detailed Solutions

Let's work through several examples, applying different solution methods:

Example 1: Finding the Vertex, Axis of Symmetry, and Intercepts

Consider the quadratic function: f(x) = x² - 4x + 3

1. Vertex: a = 1, b = -4, c = 3. x = -b / 2a = -(-4) / 2(1) = 2. f(2) = 2² - 4(2) + 3 = -1. The vertex is (2, -1).

2. Axis of Symmetry: x = 2

3. x-intercepts: x² - 4x + 3 = 0. This factors to (x - 1)(x - 3) = 0, giving x-intercepts at x = 1 and x = 3.

4. y-intercept: When x = 0, f(0) = 3. The y-intercept is (0, 3).

Example 2: Solving using the Quadratic Formula

Solve the quadratic equation: 2x² + 5x - 3 = 0

Using the quadratic formula with a = 2, b = 5, c = -3:

x = [-5 ± √(5² - 4 * 2 * -3)] / (2 * 2) = [-5 ± √49] / 4 = [-5 ± 7] / 4

This gives two solutions: x = 1/2 and x = -3

Example 3: Completing the Square

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

1. Move the constant term to the right side: x² + 6x = -5

2. Take half of the coefficient of x (6/2 = 3), square it (3² = 9), and add it to both sides: x² + 6x + 9 = -5 + 9

3. This creates a perfect square trinomial: (x + 3)² = 4

4. Take the square root of both sides: x + 3 = ±2

5. Solve for x: x = -3 ± 2, giving x = -1 and x = -5

Example 4: A Quadratic with No Real Roots

Consider the quadratic function: f(x) = x² + 2x + 5

Let's find the discriminant: b² - 4ac = 2² - 4 * 1 * 5 = -16. Since the discriminant is negative, there are no real roots; the parabola does not intersect the x-axis.

Example 5: Real-world Application - Projectile Motion

The height (h) of a projectile launched vertically upwards with an initial velocity (v₀) and initial height (h₀) is given by the quadratic equation:

h(t) = -gt²/2 + v₀t + h₀

where g is the acceleration due to gravity (approximately 9.8 m/s²), and t is time in seconds.

Let's say a ball is thrown upwards with an initial velocity of 20 m/s from a height of 1.5 meters. The equation becomes:

h(t) = -4.9t² + 20t + 1.5

To find the maximum height, we find the vertex: t = -b / 2a = -20 / (2 * -4.9) ≈ 2.04 seconds.

Substituting this value back into the equation gives the maximum height: h(2.04) ≈ 21.4 meters.

Frequently Asked Questions (FAQ)

Q1: How do I determine if a parabola opens upwards or downwards?

The parabola opens upwards if the coefficient of x² (a) is positive and downwards if a is negative.

Q2: What is the significance of the discriminant?

The discriminant (b² - 4ac) tells us about the nature of the roots (solutions) of a quadratic equation. A positive discriminant means two distinct real roots, a zero discriminant means one real root (repeated), and a negative discriminant means no real roots.

Q3: Can a quadratic function have only one x-intercept?

Yes, this occurs when the discriminant is zero, and the vertex of the parabola lies on the x-axis.

Q4: What are some real-world applications of quadratic functions besides projectile motion?

Quadratic functions are used to model various phenomena, including the trajectory of a ball, the shape of a satellite dish, the path of a water jet, and optimization problems in business and economics.

Conclusion

Quadratic functions are a cornerstone of algebra, offering a rich landscape of mathematical concepts and practical applications. Mastering the ability to solve quadratic equations using factoring, the quadratic formula, or completing the square is essential. Understanding the characteristics of the parabola, including its vertex, axis of symmetry, and intercepts, allows for a deeper comprehension of the function's behavior. The examples provided here, coupled with further practice, will build a solid foundation for tackling more complex mathematical challenges and appreciating the multifaceted nature of quadratic functions in various fields. Remember to always check your answers and practice regularly to enhance your understanding and problem-solving skills.