Quadratic Functions Intercepts, Vertex, And Solving Equations

In the realm of mathematics, quadratic functions hold a significant position, serving as fundamental tools for modeling a wide array of real-world phenomena. From the trajectory of a projectile to the shape of a suspension bridge, quadratic functions provide a versatile framework for understanding and analyzing parabolic relationships. In this comprehensive guide, we delve into the intricacies of quadratic functions, focusing on key aspects such as finding intercepts, determining the vertex, and sketching the graph. We will also explore practical techniques for solving quadratic equations, such as completing the square. Mastering these concepts is crucial for anyone seeking a deeper understanding of mathematical modeling and its applications.

A quadratic function is a polynomial function of degree two, generally expressed in the form f(x) = ax^2 + bx + c, where a, b, and c are constants and a ≠ 0. The graph of a quadratic function is a parabola, a U-shaped curve that opens upwards if a > 0 and downwards if a < 0. The key features of a parabola, such as its intercepts and vertex, provide valuable insights into the behavior of the function. In the following sections, we will explore how to determine these features and use them to sketch the graph of a quadratic function.

Part a: Analyzing the Quadratic Function f(x) = x^2 + 4x - 5

In this section, we will dissect the quadratic function f(x) = x^2 + 4x - 5 to uncover its key characteristics. Our exploration will involve finding the x-intercepts, y-intercept, vertex coordinates, and ultimately, sketching the graph of the function. By meticulously analyzing these elements, we will gain a comprehensive understanding of the function's behavior and its graphical representation. Let's embark on this journey of mathematical discovery.

i. Determining the x-intercept(s)

The x-intercepts of a function are the points where the graph intersects the x-axis. At these points, the value of the function, f(x), is zero. To find the x-intercepts of the quadratic function f(x) = x^2 + 4x - 5, we need to solve the equation x^2 + 4x - 5 = 0. This can be achieved through various methods, including factoring, completing the square, or using the quadratic formula. In this case, we will employ factoring, a technique that involves expressing the quadratic expression as a product of two linear factors. Factoring is often the quickest method when the quadratic expression can be easily factored.

The given quadratic equation is x^2 + 4x - 5 = 0. We seek two numbers that multiply to -5 and add up to 4. These numbers are 5 and -1. Therefore, we can factor the quadratic expression as follows:

x^2 + 4x - 5 = (x + 5)(x - 1) = 0

Setting each factor equal to zero, we get:

x + 5 = 0 or x - 1 = 0

Solving for x, we find:

x = -5 or x = 1

Thus, the x-intercepts of the function are (-5, 0) and (1, 0). These points are where the parabola crosses the x-axis.

ii. Finding the y-intercept

The y-intercept of a function is the point where the graph intersects the y-axis. This occurs when x = 0. To find the y-intercept of the function f(x) = x^2 + 4x - 5, we substitute x = 0 into the equation:

f(0) = (0)^2 + 4(0) - 5 = -5

Therefore, the y-intercept is (0, -5). This is the point where the parabola crosses the y-axis.

iii. Calculating the Vertex Coordinates

The vertex of a parabola is the point where the parabola changes direction. It is either the minimum point (if the parabola opens upwards) or the maximum point (if the parabola opens downwards). For a quadratic function in the form f(x) = ax^2 + bx + c, the x-coordinate of the vertex, denoted as h, can be found using the formula:

h = -b / 2a

In our case, f(x) = x^2 + 4x - 5, so a = 1 and b = 4. Plugging these values into the formula, we get:

h = -4 / (2 * 1) = -2

To find the y-coordinate of the vertex, denoted as k, we substitute h = -2 into the function:

k = f(-2) = (-2)^2 + 4(-2) - 5 = 4 - 8 - 5 = -9

Therefore, the vertex of the parabola is (-2, -9). Since the coefficient of the x^2 term is positive (a = 1), the parabola opens upwards, and the vertex represents the minimum point of the function.

iv. Sketching the Graph

Now that we have determined the x-intercepts (-5, 0) and (1, 0), the y-intercept (0, -5), and the vertex (-2, -9), we can sketch the graph of the quadratic function f(x) = x^2 + 4x - 5. To sketch the graph, we plot these key points on a coordinate plane and then draw a smooth, U-shaped curve (parabola) that passes through these points. The vertex will be the lowest point on the graph, and the parabola will be symmetric about the vertical line passing through the vertex (the axis of symmetry).

The graph will be a parabola opening upwards with its vertex at (-2, -9). The x-intercepts are (-5, 0) and (1, 0), and the y-intercept is (0, -5). These points provide a clear outline for sketching the curve. The parabola will be symmetric about the vertical line x = -2.

Part b: Solving the Quadratic Equation 2x + 1 = 5 / (x + 3) by Completing the Square

In this section, we will tackle the quadratic equation 2x + 1 = 5 / (x + 3) using the technique of completing the square. Completing the square is a powerful algebraic method that allows us to rewrite a quadratic equation in a form that can be easily solved. This technique involves manipulating the equation to create a perfect square trinomial on one side, enabling us to isolate the variable and find the solutions. Let's embark on this process and unravel the solutions to the given equation.

To solve the quadratic equation 2x + 1 = 5 / (x + 3) by completing the square, we first need to eliminate the fraction and rearrange the equation into the standard quadratic form, ax^2 + bx + c = 0. Then, we can apply the completing the square method to find the solutions for x.

Step 1: Eliminate the Fraction

Multiply both sides of the equation by (x + 3) to get rid of the fraction:

(2x + 1)(x + 3) = 5

Step 2: Expand and Simplify

Expand the left side of the equation:

2x^2 + 6x + x + 3 = 5

Combine like terms:

2x^2 + 7x + 3 = 5

Step 3: Rearrange into Standard Quadratic Form

Subtract 5 from both sides to set the equation to zero:

2x^2 + 7x - 2 = 0

Step 4: Divide by the Leading Coefficient

To make the coefficient of x^2 equal to 1, divide the entire equation by 2:

x^2 + (7/2)x - 1 = 0

Step 5: Complete the Square

To complete the square, we need to add and subtract the square of half the coefficient of the x term. The coefficient of the x term is 7/2, so half of it is 7/4, and its square is (7/4)^2 = 49/16. Add and subtract this value from the left side of the equation:

x^2 + (7/2)x + 49/16 - 49/16 - 1 = 0

Step 6: Rewrite as a Perfect Square

Rewrite the first three terms as a perfect square:

(x + 7/4)^2 - 49/16 - 1 = 0

Step 7: Combine Constants

Combine the constant terms:

(x + 7/4)^2 - 49/16 - 16/16 = 0

(x + 7/4)^2 - 65/16 = 0

Step 8: Isolate the Squared Term

Add 65/16 to both sides:

(x + 7/4)^2 = 65/16

Step 9: Take the Square Root

Take the square root of both sides:

x + 7/4 = ±√(65/16)

x + 7/4 = ±√65 / 4

Step 10: Solve for x

Subtract 7/4 from both sides:

x = -7/4 ± √65 / 4

So, the solutions for x are:

x = (-7 + √65) / 4 and x = (-7 - √65) / 4

These are the two solutions to the quadratic equation obtained by completing the square.

In this comprehensive exploration of quadratic functions, we have delved into the essential techniques for analyzing and solving these mathematical expressions. We began by dissecting the quadratic function f(x) = x^2 + 4x - 5, meticulously determining its x-intercepts, y-intercept, and vertex. This process not only allowed us to sketch the graph of the function but also provided valuable insights into its behavior and characteristics. Furthermore, we tackled the quadratic equation 2x + 1 = 5 / (x + 3), employing the powerful method of completing the square to arrive at its solutions. By mastering these techniques, we have equipped ourselves with the tools necessary to confidently navigate the world of quadratic functions and their applications. This knowledge serves as a solid foundation for further mathematical explorations and problem-solving endeavors. The ability to analyze quadratic functions and solve quadratic equations is a valuable asset in various fields, including physics, engineering, and economics, where these mathematical concepts are frequently employed to model real-world phenomena. Therefore, a thorough understanding of these concepts is crucial for anyone seeking to excel in these domains.