Finding The Quadratic Function With Zeros At X=-2 And X=5

In mathematics, identifying a function with specific zeros is a fundamental concept, particularly when dealing with quadratic functions. A zero of a function is an input value that makes the function's output equal to zero. In graphical terms, these are the points where the function's graph intersects the x-axis. This article delves into the process of finding a quadratic function that has zeros at x = -2 and x = 5. We will explore the underlying principles, the step-by-step methodology, and practical applications of this concept. Understanding how to construct a function with predetermined zeros is crucial in various mathematical contexts, including solving equations, modeling real-world phenomena, and analyzing graphs.

Understanding Zeros of a Function

Before we dive into the specifics, let's clarify what it means for a function to have zeros at certain points. A zero of a function, also known as a root or x-intercept, is a value of x for which the function f(x) equals zero. In other words, if f(a) = 0, then a is a zero of the function f. For a quadratic function, which is a polynomial of degree two, there can be at most two real zeros. These zeros correspond to the points where the parabola, the graph of the quadratic function, intersects the x-axis.

When we know the zeros of a quadratic function, we can construct the function in factored form. If a quadratic function has zeros at x = a and x = b, then the function can be written in the form f(x) = k(x - a) (x - b), where k is a non-zero constant. This form is particularly useful because it directly incorporates the zeros into the function's expression. The constant k determines the vertical stretch or compression of the parabola. By expanding this factored form, we can obtain the standard form of the quadratic function, which is f(x) = ax² + bx + c, where a, b, and c are constants.

The relationship between the zeros and the coefficients of the quadratic function is a cornerstone of algebra. Knowing this relationship allows us to work backward from the zeros to find the function itself, which is the core objective of this article. Understanding the nature and behavior of quadratic functions is essential for a wide range of applications, from solving algebraic problems to modeling physical phenomena. The concept of zeros extends beyond quadratic functions and is fundamental in the study of polynomials and other types of functions as well.

Methodology: Constructing the Quadratic Function

To construct a quadratic function with zeros at x = -2 and x = 5, we will follow a step-by-step approach that leverages the factored form of a quadratic function. This method allows us to directly incorporate the given zeros into the function's expression. Let's break down the process into clear, manageable steps.

Step 1: Write the Factored Form

The factored form of a quadratic function with zeros at x = a and x = b is given by f(x) = k(x - a) (x - b), where k is a non-zero constant. In our case, the zeros are x = -2 and x = 5. So, we can substitute these values into the factored form:

f(x) = k(x - (-2)) (x - 5)

Simplifying the expression inside the parentheses, we get:

f(x) = k(x + 2) (x - 5)

At this point, we have a general form of the quadratic function with the desired zeros. The constant k can be any non-zero number, which means there are infinitely many quadratic functions with these zeros. For simplicity, we often choose k = 1, but it's important to remember that other values of k would also produce valid quadratic functions with the same zeros.

Step 2: Expand the Factored Form

To obtain the standard form of the quadratic function, which is f(x) = ax² + bx + c, we need to expand the factored form. Expanding the expression involves multiplying the binomials (x + 2) and (x - 5):

f(x) = k [(x + 2) (x - 5)]

Using the distributive property (also known as the FOIL method), we multiply the terms:

f(x) = k [x(x - 5) + 2(x - 5)]

f(x) = k [x² - 5x + 2x - 10]

Combining like terms, we get:

f(x) = k [x² - 3x - 10]

Step 3: Choose a Value for k

As mentioned earlier, the constant k can be any non-zero number. For simplicity, let's choose k = 1. This choice will give us a basic quadratic function with the specified zeros. Substituting k = 1 into the expression, we have:

f(x) = 1 * (x² - 3x - 10)

f(x) = x² - 3x - 10

So, the quadratic function with zeros at x = -2 and x = 5 is f(x) = x² - 3x - 10. This is a specific quadratic function that satisfies the given conditions. If we were to choose a different value for k, we would obtain a different quadratic function, but it would still have the same zeros.

Step 4: Verify the Zeros

To ensure that our function is correct, we can verify that f(-2) = 0 and f(5) = 0. Let's substitute these values into the function:

For x = -2:

f(-2) = (-2)² - 3(-2) - 10

f(-2) = 4 + 6 - 10

f(-2) = 0

For x = 5:

f(5) = (5)² - 3(5) - 10

f(5) = 25 - 15 - 10

f(5) = 0

Since f(-2) = 0 and f(5) = 0, we have verified that our function f(x) = x² - 3x - 10 indeed has zeros at x = -2 and x = 5. This step is crucial to ensure the accuracy of the constructed function and to reinforce the understanding of the relationship between zeros and the function's expression.

Practical Applications

Understanding how to construct a function with specific zeros has numerous practical applications in mathematics and related fields. This skill is particularly valuable in solving equations, modeling real-world phenomena, and analyzing graphs. Let's explore some specific examples to illustrate the breadth of these applications.

Solving Quadratic Equations

One of the most direct applications of constructing functions with known zeros is in solving quadratic equations. A quadratic equation is an equation of the form ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0. The solutions to this equation are the zeros of the corresponding quadratic function f(x) = ax² + bx + c. If we can identify the zeros of the function, we have found the solutions to the equation. For example, consider the equation x² - 3x - 10 = 0. We constructed the function f(x) = x² - 3x - 10 and found that it has zeros at x = -2 and x = 5. Therefore, the solutions to the equation x² - 3x - 10 = 0 are x = -2 and x = 5.

Modeling Real-World Phenomena

Quadratic functions are often used to model real-world phenomena that exhibit parabolic behavior. Examples include the trajectory of a projectile, the shape of a suspension bridge cable, and the profit function in economics. In these applications, the zeros of the quadratic function can represent significant points, such as the time when a projectile hits the ground or the break-even points in a profit-loss scenario. By constructing a quadratic function with specific zeros, we can create a mathematical model that accurately reflects the real-world situation. For instance, if we know the points where a projectile is launched and where it lands, we can construct a quadratic function to describe its path.

Analyzing Graphs

The zeros of a function provide crucial information about its graph. For a quadratic function, the zeros are the x-intercepts of the parabola. Knowing the zeros allows us to sketch the graph of the function and to identify key features, such as the vertex (the highest or lowest point on the parabola) and the axis of symmetry. The zeros, along with the coefficient of the x² term, determine the shape and position of the parabola. If we have a quadratic function in factored form, we can immediately identify the zeros and use them to sketch the graph. Conversely, if we have the graph of a quadratic function, we can read off the zeros from the x-intercepts and use this information to construct the function's equation.

Optimization Problems

In optimization problems, we often seek to find the maximum or minimum value of a function. Quadratic functions are frequently encountered in these problems, and the zeros can play a role in finding the optimal value. The vertex of a parabola, which represents the maximum or minimum value of the quadratic function, lies on the axis of symmetry, which is located midway between the zeros. Therefore, knowing the zeros can help us find the vertex and solve the optimization problem. For example, if we want to maximize the area of a rectangular garden with a fixed perimeter, we can set up a quadratic function for the area in terms of the dimensions of the garden. The zeros of this function can help us determine the dimensions that maximize the area.

Conclusion

In conclusion, finding a quadratic function with specified zeros is a fundamental concept in mathematics with a wide range of applications. By understanding the relationship between the zeros and the factored form of a quadratic function, we can construct functions that meet specific criteria. The step-by-step methodology outlined in this article provides a clear and effective approach to this process. Whether we are solving equations, modeling real-world phenomena, or analyzing graphs, the ability to construct functions with predetermined zeros is an invaluable skill. The practical applications discussed highlight the versatility and importance of this concept in various mathematical contexts. Mastering this skill enhances our ability to tackle a diverse set of problems and deepen our understanding of quadratic functions and their behavior.

Answer

The correct answer is D. f(x) = x² - 3x - 10.