Question

1. How is the ratio of outputs to inputs, y x, different from the slope ratio?
2. When a linear graph passes through the origin, why is the ratio of outputs to inputs, y x the same as the slope?
3. How can you tell from a graph whether a relationship is inverse variation or direct variation? Does this change when we look at direct square variation versus inverse square variation? Why or why not?
4. What is a horizontal asymptote? Do direct or inverse variation graphs have asymptotes? Why?
5. Explain how to tell whether a table varies directly, inversely, or neither. Include what to watch for to determine if a function varies directly or inversely with the square.

Answer 1

1. The slope is ratio of the change in y and the change in x, that is, the slope is , but the ratio of outputs is . Now, depending on the function, this ratio might not always be equal to the slope. For example, take the function
Then the slope of the function is 15, while the ratio of the outputs to the inputs is something more than 15 as:

Therefore, the ratio of the outputs to inputs is different from the slope ratio.

2. When a linear graph passes through the origin, it has no constant term, that is, the equation of the graph is simply , and in this case, the ratio of the outputs to inputs is , which is the same as the slope.

3. When a graph is direct variation, then the graph is a straight line.

When a graph is inverse variation, the output decreases as the input increases so we have a hyperbolic curve.

Direct square variation has a parabolic curve

while inverse square variation looks similar to the inverse variation, except that the output is positive even for negative inputs, as:

A horizontal asymptote of a graph is a line such that the graph becomes very close to the line as x approaches infinity, but does not meet the graph.

Direct variation graphs don't have asymptotes, but inverse variation graphs have asymptotes. Direct variation graphs don't have asymptotes, as they are straight lines, but inverse variation graphs do have asymptotes, as x goes to infinity, the ratio decreases constantly and becomes 0.