Problem 15 - A Graphical Approach to College Algebra

Question

Use the intermediate value theorem to show that each function has a real zero between the two numbers given. Then, use your calculator to approximate the zero to the nearest hundredth. $$P(x)=-x^{4}+2 x^{3}+x+12 ; \quad 2.7 \text { and } 2.8$$

Step-by-Step Solution

Answer

A zero exists between 2.7 and 2.8, approximately at 2.75.

Step 1: Understand the Intermediate Value Theorem

The Intermediate Value Theorem states that if a continuous function \( f(x) \) takes values \( f(a) \) and \( f(b) \) at each end of an interval \([a, b]\), and \( f(a) \) and \( f(b) \) have opposite signs, then there is at least one real number \( c \) in the interval \((a, b)\) such that \( f(c) = 0 \). This indicates a real zero exists between \( a \) and \( b \).

Step 2: Evaluate the Function at Given Points

Calculate \( P(2.7) \) and \( P(2.8) \) using the polynomial \( P(x) = -x^4 + 2x^3 + x + 12 \). Substitute 2.7 and 2.8 into the equation: \[ P(2.7) = -(2.7)^4 + 2(2.7)^3 + 2.7 + 12 \]Calculate this to find \( P(2.7) \).Similarly, \[ P(2.8) = -(2.8)^4 + 2(2.8)^3 + 2.8 + 12 \]Calculate this to find \( P(2.8) \).

Step 3: Check Signs and Apply Theorem

Find the sign of \( P(2.7) \) and \( P(2.8) \) calculated previously. If \( P(2.7) \) and \( P(2.8) \) have opposite signs, the Intermediate Value Theorem guarantees a zero in the interval (2.7, 2.8).

Step 4: Approximate Zero Using Calculator

Use a graphing calculator or numerical method to find the point between 2.7 and 2.8 where \( P(x) = 0 \). Adjust the x-value according to calculated values until you approximate the zero to the nearest hundredth.

Key Concepts

Real Zero

A real zero of a function is the x-value where the function itself crosses the x-axis, meaning the function equals zero at this point. In mathematical terms, it is where the output or the y-value of the function is zero.

• This concept is significant in functions, especially because it tells us where the function changes its sign, from positive to negative or vice versa.
• In the context of polynomial functions, real zeros can be found by evaluating the function within given intervals. This information is integral to modeling real-world phenomena, as many processes are represented by polynomials.

Understanding real zeros is crucial when dealing with algebraic equations, as they provide solutions to the equation. Many mathematical tools such as the Intermediate Value Theorem help in confirming the existence of real zeros, especially when calculating them analytically might be complex.

Continuous Function

A continuous function is one where small changes in the input lead to small changes in the output. This means the function's graph can be drawn without lifting your pencil from the paper. A key property of continuous functions is that they do not have breaks, jumps, or holes over their domain.

• This concept becomes especially important when using the Intermediate Value Theorem, as it applies only to continuous functions.
• For example, the polynomial function in our problem, \( P(x) = -x^4 + 2x^3 + x + 12 \), is continuous across all real numbers, which allows us to apply the theorem.

This continuity allows us to make certain conclusions about the behavior of the function within a given interval, such as the guarantee of a zero if the function takes on opposite signs at the endpoints of this interval. Without continuity, predictions made using the Intermediate Value Theorem wouldn't be reliable.

Polynomial Function

Polynomial functions are mathematical expressions that use the sum of powers of a variable, each multiplied by a coefficient, to describe a curve on a graph. For example, in the function \( P(x) = -x^4 + 2x^3 + x + 12 \), each term is a power of \( x \) with a corresponding coefficient.

• Polynomials are crucial in mathematics because they form the basis for many types of functions and equations.
• The simplicity of their algebraic structure makes them useful in calculus and numerical approximation methods.

These functions are a vital class of functions used in various fields, such as physics, engineering, and economics, due to their flexible and easily understandable nature. They are also perfect illustrations of concepts such as continuity and the Intermediate Value Theorem, as they are inherently continuous across their entire domain.

Graphing Calculator

A graphing calculator is a handheld device capable of plotting graphs, solving simultaneous equations, and performing other tasks with variables. It's a powerful tool used to visually confirm the analytical work done in finding function solutions.

• In exploring the concept of real zeros, a graphing calculator allows for a visual representation of the polynomial function, making it easier to identify where the curve crosses the x-axis.
• By zooming in on specific intervals, such as between 2.7 and 2.8 in our exercise, you can approximate the exact point of zero with greater accuracy.

Using this tool complements mathematical theory and provides a practical method to solve complex equations. It is especially helpful for visual learners, allowing them to see the immediate consequences of changing variables or coefficients in a function.