Problem 80 - Brief Applied Calculus

Question

Fill in the missing words: If a line slants downward as you go to the right, then its_______ is________.

Step-by-Step Solution

Answer

slope is negative.

Step 1: Understand the Problem

We need to determine the sign (positive or negative) of the slope of a line that slants downward as you move from left to right on a coordinate plane.

Step 2: Recall What Slope Represents

Slope is a measure of the steepness and direction of a line. Mathematically, it is represented as the ratio of the change in the vertical direction (rise) to the change in the horizontal direction (run), often written as \( m = \frac{\Delta y}{\Delta x} \).

Step 3: Determine the Sign of the Slope for a Downward-Slanting Line

For a line that slants downward as you move from left to right, the change in \( y \) (rise) is negative while the change in \( x \) (run) is positive. This results in a negative slope, because \( \frac{\text{negative}}{\text{positive}} \) equals negative.

Step 4: Fill in the Blanks

Given the information from the previous steps, we can say: If a line slants downward as you go to the right, then its **slope** is **negative**.

Key Concepts

Negative Slope

In the world of mathematics, understanding the slope of a line helps us grasp how the line behaves on a graph. Imagine a line that slopes downward from left to right. This line has what we call a "negative slope." Simply put, this means the line declines as it moves towards the right.

Why is this slope negative? It's because of how the change in the vertical direction, or "rise," compares to the change in the horizontal direction, or "run." When the line descends, the rise is actually a drop, making it a negative number. Pairing this with a positive run gives us a negative ratio. This concept of negative slope is crucial in understanding how lines interact and behave on a graph.

Coordinate Plane

The coordinate plane is a fundamental part of understanding algebra and geometry. It's like a flat map where every point is defined by a pair of numbers. These numbers, called coordinates, help us find specific locations on the graph.

This plane is divided into four sections, known as quadrants. The point where both axes meet is the origin, marked as (0, 0). The horizontal line is the x-axis, and the vertical line is the y-axis. Lines and curves that we study, including those with a negative slope, are drawn across this plane.

• The coordinate plane helps visualize mathematical relationships.
• It allows us to plot solutions to equations.
• Understanding it is key to navigating the world of graphs and slopes.

Steepness and Direction

When we talk about steepness and direction of a slope, we're referring to how a line rises or falls as it moves horizontally. The steepness tells us how quickly the line rises or drops, while the direction indicates whether the slope goes up or down.

A steep slope means the line rises or drops sharply. A flat line has a gentle slope, or possibly zero slope. If a line rises as it moves from left to right, it has a positive slope. If it falls, like that negative slope we discussed earlier, the slope is negative.

• Steepness affects how quickly the dependent variable changes.
• The direction tells if the line is increasing or decreasing.
• Recognizing these helps interpret data represented by lines.

Mathematical Representation of Slope

Slope is mathematically represented by the formula:\[ m = \frac{\Delta y}{\Delta x} \]This formula is essential to calculate the slope of any line on a graph.

• "m" is the symbol for slope.
• "\(\Delta y\)" stands for the change in the y-values (vertical change).
• "\(\Delta x\)" represents the change in the x-values (horizontal change).

When applying this formula, notice that the ratio tells us whether the slope is positive or negative. For instance, a negative slope arises when there's a downward shift in elevation, paired with a move to the right.

Each time we interpret graphs or equations, this formula helps us understand the relationship between variables.