Problem 39 - Beginning and Intermediate Algebra

Question

Fill in the blank with the correct term. Some of the given choices will not be used. Descartes' rule of signs a vertical asymptote the leading-term test \(\quad\) an oblique the intermediate \(\quad\) asymptote value theorem direct variation the fundamental \(\quad\) inverse variation theorem of algebra a horizontal line a polynomial function a vertical line a rational function \(\quad\) parallel a one-to-one function \(\quad\) perpendicular a constant function a horizontal asymptote Two lines with slopes \(m_{1}\) and \(m_{2}\) are if and only if the product of their slopes is \(-1\)

Step-by-Step Solution

Answer

Two lines with slopes \(m_1\) and \(m_2\) are \(\textbf{perpendicular}\) if and only if the product of their slopes is \(-1\).

Step 1: Identify the relationship between the slopes of two lines

We are given a statement about the relationship between the slopes of two lines. We need to find the correct term to complete the statement. The statement is: "Two lines with slopes \(m_1\) and \(m_2\) are __________ if and only if the product of their slopes is \(-1\)."

Step 2: Review the choices provided

The given choices are: 1. Descartes' rule of signs 2. A vertical asymptote 3. The leading-term test 4. An oblique asymptote 5. The intermediate value theorem 6. Direct variation 7. The fundamental theorem of algebra 8. Inverse variation 9. A horizontal line 10. A polynomial function 11. A vertical line 12. A rational function 13. Parallel 14. A one-to-one function 15. Perpendicular 16. A constant function 17. A horizontal asymptote

Step 3: Select the most appropriate term

Most of the provided terms do not relate to the given statement about the relationship between the slopes of two lines. In geometrical terms, two lines are said to be perpendicular if their slopes have a product of -1.

Step 4: Fill in the blank with the correct term

Now that we have identified the term that fits the statement, we can fill in the blank: "Two lines with slopes \(m_1\) and \(m_2\) are **perpendicular** if and only if the product of their slopes is \(-1\)."

Key Concepts

Slope

In geometry, the slope is a crucial concept used to measure the steepness or inclination of a line. You can think of it as rises over runs or simply how much a line goes up for how much it goes across.
Mathematically, slope is represented by the letter "m" and is calculated using the formula:

• \(m = \frac{y_2 - y_1}{x_2 - x_1}\)

Here,

• \((x_1, y_1)\) and \((x_2, y_2)\) are coordinates of any two points on the line.

A positive slope indicates the line rises as it moves from left to right, while a negative slope means it falls.
A slope of zero suggests a horizontal line, and an undefined slope corresponds to a vertical line. Recognizing and calculating the slope is vital for describing and analyzing lines in the coordinate plane.

Perpendicular Lines

Perpendicular lines are two lines that intersect at a right angle, meaning they form a 90-degree angle with each other. A defining feature of perpendicular lines is their slopes.
Specifically, if line one has a slope of \(m_1\), and line two is perpendicular to it, then the slope of line two \(m_2\) will satisfy the equation:

• \(m_1 \times m_2 = -1\)

This relationship tells us that the slopes are negative reciprocals of each other.
Understanding perpendicularity is essential in geometric constructions and proving theorems, as it plays a role in forming shapes like rectangles and squares. Furthermore, knowing how to find the slope of perpendicular lines assists in parallel and perpendicular line equations, which are fundamental in geometry.

Polynomial Functions

Polynomial functions are expressions consisting of variables and coefficients, combined using addition, subtraction, multiplication, and non-negative integer exponents. They take the general form:

• \(a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0\)

Where:

• \(a_n, a_{n-1}, \ldots, a_1, a_0\) are coefficients.
• \(n\) is the highest power of the variable and called the degree of the polynomial.

Polynomial functions are significant in mathematics due to their broad applicability across numerous problems, including optimization and calculus.
They come in various degrees, such as linear (first degree), quadratic (second degree), and cubic (third degree).
Understanding polynomial functions helps in modeling real-world scenarios and performing operations like differentiation and integration in calculus.

Descartes' Rule of Signs

Descartes' Rule of Signs is a handy technique in algebra, allowing us to predict the number of positive and negative real roots a polynomial function can have.
To use this rule, examine the polynomial's terms:

• Count how many times the signs (plus or minus) change in the polynomial's coefficient list. This number gives the maximum possible number of positive real roots.
• Then, replace \(x\) with \(-x\) and recount the sign changes to find the maximum number of negative real roots.

This rule doesn't guarantee the exact number of roots, as it only provides possibilities. Yet, it's a helpful guide when analyzing polynomial equations, especially higher-degree ones.
Understanding Descartes' Rule of Signs contributes to efficient problem-solving, as it reduces the complexity of root finding by narrowing down the potential solutions to test.