Quadratic Function Examples With Answers

Understanding Quadratic Functions: Examples and Solutions

Quadratic functions are a fundamental concept in algebra, describing relationships between variables that result in a parabola when graphed. They are expressed in the general form: f(x) = ax² + bx + c, where 'a', 'b', and 'c' are constants and 'a' is not equal to zero. Understanding quadratic functions is crucial for various applications, from physics (projectile motion) to economics (maximizing profit). This article will explore various examples of quadratic functions, providing detailed solutions to help solidify your understanding. We’ll delve into different methods for solving these functions, including factoring, the quadratic formula, and completing the square. Let’s dive in!

I. Introduction to Quadratic Functions

The defining characteristic of a quadratic function is the x² term. This term dictates the parabolic shape of its graph. The value of 'a' determines the parabola's orientation (opens upwards if a > 0, downwards if a < 0) and its "steepness." The 'b' coefficient influences the parabola's horizontal position, and 'c' represents the y-intercept (where the graph crosses the y-axis).

Understanding the different forms of a quadratic function is also key. Besides the standard form (ax² + bx + c), we have:

• Vertex Form: f(x) = a(x - h)² + k, where (h, k) represents the vertex (the highest or lowest point) of the parabola.
• Factored Form (Intercept Form): f(x) = a(x - r₁)(x - r₂), where r₁ and r₂ are the x-intercepts (where the graph crosses the x-axis).

II. Examples of Quadratic Functions and Solutions

Let's explore various examples, demonstrating different solution techniques.

Example 1: Solving by Factoring

Find the roots (x-intercepts) of the quadratic function: f(x) = x² + 5x + 6

Solution:

1. Factor the quadratic expression: We need to find two numbers that add up to 5 (the coefficient of x) and multiply to 6 (the constant term). These numbers are 2 and 3. Therefore, we can factor the equation as: (x + 2)(x + 3) = 0

2. Set each factor to zero: This gives us two equations: x + 2 = 0 and x + 3 = 0

3. Solve for x: Solving each equation yields x = -2 and x = -3. These are the roots (or zeros) of the quadratic function. They represent the x-coordinates where the parabola intersects the x-axis.

Example 2: Solving using the Quadratic Formula

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

Solution:

The quadratic formula is a powerful tool for solving any quadratic equation, even those that are difficult or impossible to factor:

x = [-b ± √(b² - 4ac)] / 2a

In this equation: a = 2, b = -7, and c = 3. Substituting these values into 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 x = (7 - 5) / 4 = 1/2

Therefore, the roots of the quadratic equation are x = 3 and x = 1/2.

Example 3: Completing the Square

Find the vertex of the parabola represented by the quadratic function: f(x) = x² - 6x + 5

Solution:

Completing the square is a method used to rewrite a quadratic function in vertex form, f(x) = a(x - h)² + k. The vertex (h, k) can then be easily identified.

1. Group the x terms: Rewrite the equation as: f(x) = (x² - 6x) + 5

2. Complete the square: To complete the square for the expression inside the parentheses, take half of the coefficient of x (-6/2 = -3), square it (-3)² = 9, and add and subtract this value inside the parentheses:

f(x) = (x² - 6x + 9 - 9) + 5

3. Factor the perfect square trinomial: The first three terms inside the parentheses form a perfect square trinomial: (x - 3)².

f(x) = (x - 3)² - 9 + 5

4. Simplify:

f(x) = (x - 3)² - 4

Now the equation is in vertex form. The vertex is (h, k) = (3, -4).

Example 4: Word Problem

A ball is thrown upwards from the ground with an initial velocity of 40 m/s. Its height (h) after t seconds is given by the equation h(t) = -5t² + 40t. When does the ball reach its maximum height, and what is the maximum height?

Solution:

This is a classic example of a quadratic function applied to projectile motion. The equation h(t) = -5t² + 40t is a parabola that opens downwards (because the coefficient of t² is negative). The maximum height occurs at the vertex of the parabola.

We can find the vertex using the formula for the x-coordinate of the vertex: t = -b / 2a. In this case, a = -5 and b = 40.

t = -40 / (2 * -5) = 4 seconds

To find the maximum height, substitute t = 4 seconds back into the equation:

h(4) = -5(4)² + 40(4) = 80 meters

Therefore, the ball reaches its maximum height of 80 meters after 4 seconds.

III. Different Methods for Solving Quadratic Equations

We've already touched upon factoring, the quadratic formula, and completing the square. Let's briefly summarize their strengths and weaknesses:

• Factoring: Simple and elegant when the quadratic expression can be easily factored. However, not all quadratic equations are easily factorable.

• Quadratic Formula: A universal method that works for all quadratic equations, regardless of whether they are factorable or not. It's a reliable tool, but can be slightly more time-consuming than factoring.

• Completing the Square: Useful for rewriting the quadratic equation in vertex form, which readily reveals the vertex of the parabola. It can also be used to solve quadratic equations, but it can be less intuitive than the quadratic formula for some.

IV. Understanding the Discriminant

The discriminant (b² - 4ac) within the quadratic formula provides valuable information about the nature of the roots:

• Discriminant > 0: Two distinct real roots (the parabola intersects the x-axis at two points).

• Discriminant = 0: One real root (a repeated root; the parabola touches the x-axis at only one point – the vertex).

• Discriminant < 0: No real roots (the parabola does not intersect the x-axis). The roots are complex numbers (involving the imaginary unit 'i').

V. Applications of Quadratic Functions

Quadratic functions find applications in diverse fields:

• Physics: Projectile motion, describing the trajectory of objects under gravity.

• Engineering: Designing parabolic antennas and reflectors.

• Economics: Modeling cost, revenue, and profit functions.

• Computer Graphics: Creating curved shapes and animations.

• Statistics: Analyzing data distributions and regression models.

VI. Frequently Asked Questions (FAQ)

Q1: What is a parabola?

A parabola is a symmetrical, U-shaped curve that is the graph of a quadratic function.

Q2: Can a quadratic function have more than two roots?

No, a quadratic function can have at most two real roots.

Q3: What if 'a' is equal to zero in the quadratic equation?

If a = 0, the equation is no longer quadratic; it becomes a linear equation.

Q4: How do I graph a quadratic function?

You can graph a quadratic function by plotting several points and connecting them to form a parabola. Alternatively, you can use the vertex and x-intercepts to sketch the graph. Using graphing software or calculators can be very useful.

Q5: What is the axis of symmetry?

The axis of symmetry is a vertical line that divides the parabola into two symmetrical halves. Its equation is x = -b / 2a.

VII. Conclusion

Quadratic functions are a cornerstone of algebra and have widespread applications in various fields. Mastering the techniques for solving quadratic equations – factoring, the quadratic formula, and completing the square – is essential for understanding and solving problems involving quadratic relationships. Remember the discriminant, which provides insights into the nature of the roots. Through practice and a clear understanding of the concepts, you can confidently tackle quadratic function problems and appreciate their significance in mathematics and beyond. Keep practicing, and you’ll become proficient in handling these important mathematical tools!