Which Function Has Zeros Of And 2

Finding a function with specific zeros is a fundamental concept in algebra and calculus. When we say a function has zeros at certain values, it means that the function's output is zero when those values are plugged in as input. In this case, we want to find a function that equals zero when x = -3 and x = 2. This involves understanding the relationship between roots, factors, and polynomial functions. We will explore various methods to construct such a function, delve into the underlying principles, and examine some examples to solidify our understanding.

Constructing a Basic Function

The simplest approach to finding a function with zeros at -3 and 2 is to construct a polynomial function. If a function f(x) has a zero at x = a, then (x - a) is a factor of f(x). Therefore, if f(x) has zeros at x = -3 and x = 2, then (x + 3) and (x - 2) are factors of f(x).

To create the simplest polynomial function with these zeros, we multiply these factors together:

f(x) = (x + 3)(x - 2)

Expanding this expression gives us:

f(x) = x^2 + x - 6

This is a quadratic function that crosses the x-axis at x = -3 and x = 2. To verify this, we can substitute these values into the function:

For x = -3:

f(-3) = (-3)^2 + (-3) - 6 = 9 - 3 - 6 = 0

For x = 2:

f(2) = (2)^2 + (2) - 6 = 4 + 2 - 6 = 0

Both values result in f(x) = 0, confirming that -3 and 2 are indeed the zeros of this function.

Generalizing the Function

While f(x) = x^2 + x - 6 is a valid function with zeros at -3 and 2, it is not the only one. We can multiply the function by any non-zero constant k without changing its zeros. This is because if f(x) = 0, then k f(x) = 0 as well, provided that k ≠ 0.

Thus, a more general form of the function is:

f(x) = k(x + 3)(x - 2)

Where k is any non-zero constant. Expanding this gives:

f(x) = k(x^2 + x - 6)

For example, if k = 2:

f(x) = 2(x^2 + x - 6) = 2x^2 + 2x - 12

The zeros remain at x = -3 and x = 2. The constant k simply scales the function vertically, affecting its amplitude but not its roots.

Introducing Higher-Order Polynomials

We can also construct higher-order polynomial functions with the same zeros by including additional factors. For instance, we can add a factor (x - a), where a is any value other than -3 and 2, to create a cubic function.

Let's add a factor (x - 1):

f(x) = (x + 3)(x - 2)(x - 1)

Expanding this expression gives us:

f(x) = (x^2 + x - 6)(x - 1) = x^3 - x^2 + x^2 - x - 6x + 6 = x^3 - 7x + 6

This is a cubic function with zeros at x = -3, x = 2, and x = 1. Notice that adding the factor (x - 1) introduced a new zero at x = 1 but did not change the existing zeros at x = -3 and x = 2.

In general, any function of the form:

f(x) = (x + 3)(x - 2) * g(x)

where g(x) is any polynomial function, will have zeros at x = -3 and x = 2. The function g(x) may introduce additional zeros, but it will not remove the zeros at x = -3 and x = 2.

Repeated Zeros

Another way to modify the function is by introducing repeated zeros. A repeated zero occurs when a factor is raised to a power greater than 1. For example, consider the function:

f(x) = (x + 3)^2 (x - 2)

This function has a zero at x = -3 with multiplicity 2 and a zero at x = 2 with multiplicity 1. The multiplicity of a zero affects the behavior of the graph of the function at that zero.

Expanding this expression gives us:

f(x) = (x^2 + 6x + 9)(x - 2) = x^3 - 2x^2 + 6x^2 - 12x + 9x - 18 = x^3 + 4x^2 - 3x - 18

At x = -3, the graph of the function touches the x-axis but does not cross it, due to the even multiplicity. At x = 2, the graph crosses the x-axis.

In general, a function with zeros at x = -3 and x = 2 can be written as:

f(x) = (x + 3)^m (x - 2)^n * h(x)

Where m and n are positive integers representing the multiplicities of the zeros at x = -3 and x = 2, respectively, and h(x) is any polynomial function that does not have zeros at x = -3 or x = 2.

Non-Polynomial Functions

While polynomial functions are the most straightforward way to construct functions with specific zeros, it is also possible to create non-polynomial functions with zeros at x = -3 and x = 2.

Consider the function:

f(x) = sin((x + 3)π) * (x - 2)

The sine function has zeros at integer multiples of π. Therefore, sin((x + 3)π) has zeros when (x + 3) is an integer. In particular, it has a zero at x = -3. The factor (x - 2) ensures that the function also has a zero at x = 2.

Another example is:

f(x) = e^(x + 3 - 2) - 1

Which simplifies to:

f(x) = e^(x + 1) - 1

This function will not have a zero at x = -3 or x = 2. To correct it:

f(x) = (e^(x+3) - 1) * (x - 2)

This function has a zero at x = 2, and when x = -3:

f(-3) = (e^0 - 1) * (-3 - 2) = (1 - 1) * -5 = 0

However, these types of functions are more complex to construct and analyze than polynomial functions.

Practical Applications

The ability to construct functions with specific zeros has many practical applications in mathematics, science, and engineering. Here are a few examples:

• Curve Fitting: In data analysis, we often want to find a function that approximates a set of data points. If we know that the function should have specific zeros, we can use this information to constrain the curve-fitting process and obtain a more accurate model.

• Control Systems: In control systems engineering, we design systems to regulate the behavior of physical processes. The zeros of the system's transfer function determine the system's response to different inputs. By carefully choosing the zeros, we can achieve desired performance characteristics such as stability and speed of response.

• Signal Processing: In signal processing, we analyze and manipulate signals such as audio and video. The zeros of a signal's z-transform determine the signal's frequency content. By placing zeros at specific frequencies, we can filter out unwanted noise or enhance desired features.

• Root Finding: Many numerical algorithms for finding the roots of equations rely on constructing a function with the same roots as the original equation. By manipulating the function to make it easier to find its roots, we can solve complex equations that would otherwise be intractable.

Examples and Exercises

To solidify our understanding, let's work through some examples and exercises.

Example 1: Find a quadratic function with zeros at x = -1 and x = 4 that passes through the point (0, 8).

Solution: First, we construct the general form of the quadratic function with the given zeros:

f(x) = k(x + 1)(x - 4)

Next, we use the fact that the function passes through the point (0, 8) to find the value of k:

f(0) = k(0 + 1)(0 - 4) = 8

-4k = 8

k = -2

Therefore, the quadratic function is:

f(x) = -2(x + 1)(x - 4) = -2(x^2 - 3x - 4) = -2x^2 + 6x + 8

Example 2: Find a cubic function with zeros at x = -3, x = 2, and x = 0.

Solution: The function has factors (x + 3), (x - 2), and x. Thus, the cubic function is:

f(x) = k(x + 3)(x - 2)(x)

Assuming k = 1 for simplicity:

f(x) = x(x + 3)(x - 2) = x(x^2 + x - 6) = x^3 + x^2 - 6x

Exercise 1: Find a quadratic function with zeros at x = 1 and x = 3.

Exercise 2: Find a cubic function with zeros at x = -2, x = 1, and x = 4.

Exercise 3: Find a function with zeros at x = -5 and x = 2 and a vertical asymptote at x = 0.

Conclusion

Constructing functions with specific zeros is a fundamental skill in mathematics with numerous applications. By understanding the relationship between roots, factors, and polynomial functions, we can create functions that meet specific requirements. Whether it's constructing a simple quadratic function or a more complex non-polynomial function, the principles remain the same. Through practice and experimentation, we can master this skill and apply it to solve a wide range of problems.