Problem 59 - Introductory Algebra for College Students

Question

Make Sense? In Exercises \(57-60\), determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. I visualize slope as walking along a line from left to right. If I'm walking uphill, the slope is positive, and if I'm walking downhill, the slope is negative.

Step-by-Step Solution

Answer

'Makes Sense. The reasoning is correct, when you travel from left to right, uphill indicates a positive slope while downhill indicates a negative slope.'

Step 1 Claim Review

Review the claim carefully. If moving from left to right on a line graph, 'walking uphill' would indeed mean a positive slope, and 'walking downhill' would indicate a negative slope. This is because slope represents the change in y over the change in x (often described as 'rise over run'). Thus, a 'rise' (or 'walk uphill') represents a positive change, and a 'run' (or 'walk downhill') indicates a negative change.

Step 2 Verification

Verify this claim with the common understanding of slopes. If the line goes up from left to right, the slope is positive. Conversely, if it goes down from left to right, the slope is negative.

Step 3 Conclusion

Conclude based on the review and the verification. The claim 'makes sense' because walking uphill (rising y-values) when moving from left to right represents a positive slope, while walking downhill (decreasing y-values) stands for a negative slope.

Key Concepts

Positive Slope

A positive slope on a line graph is like walking uphill. Imagine you're strolling from the left side to the right side on a hill. With every step you take, you're moving higher. That's what a positive slope is all about: as the x-values increase, the y-values rise too.

This concept is shown mathematically as the ratio of the change in y (vertical change) over the change in x (horizontal change). If both changes are in the same direction, either both increasing or both decreasing, the ratio results in a positive number.

Here's an easy way to remember:

• Walking uphill = positive change.
• Graph line slopes upwards from left to right.

This means the line is climbing towards the sky, representing growth or increase as you look at the graph from left to right.

Negative Slope

A negative slope, on the other hand, is like strolling downhill. Picture walking down a gentle slope; from left to right, every step takes you lower. This is how a negative slope acts on a line graph: as x-values increase, the y-values decrease.

The slope formula, again the change in y over the change in x, yields a negative value here because you're losing height with each step forward (increasing x), making the rise negative compared to the run, which is positive.

To summarize:

• Walking downhill = negative change.
• Graph line points downwards from left to right.

It depicts scenarios where there's a decline or reduction when moving along the graph.

Line Graph

A line graph is a powerful tool in visualizing relationships between two variables. Each point on the graph corresponds to a particular combination of x and y coordinates. When plotting a line graph, you effectively create a visual map showing how values change in relation to each other. This is why understanding slopes is crucial.

In practical terms, the slope tells us how steep the line is and the direction it heads.

• A steep line indicates a larger slope, whether positive or negative, while a flatter line suggests a smaller slope.
• The slope also shows whether the relationship is increasing (positive) or decreasing (negative) as you move from left to right.

Using line graphs to observe such changes can aid in making predictions, finding trends, or understanding dependencies between variables in various real-world contexts.