Question

Solve problems comparing functions and write about the conclusions.
What information can be used to compare linear relationships? Explain why.

Answer 1

To compare two linear relationships you need to know how quickly each one changes (the rate of change) and where each one starts (the initial value). For linear functions those two pieces of information are exactly the slope and the y‑intercept. More generally you can use any representation that gives the same information: an equation, a graph, a table of values, or two (or more) points.

What to use and why
- Slope (m in y = mx + b): tells the rate of change — how much y changes for a one‑unit change in x. If one slope is larger (in absolute value) that line is steeper; positive slope = increasing, negative = decreasing.
- y‑intercept (b): gives the value when x = 0 — the starting or initial value.
- Equation: y = mx + b immediately shows slope and intercept.
- Two points (x1,y1) and (x2,y2): let you compute slope = (y2 − y1)/(x2 − x1); with slope and one point you can get the intercept.
- Table of values: compute the constant difference in y for equal increments in x to get the slope; the row for x = 0 (if present) gives the intercept.
- Graph: visually compare slopes (steepness), intercepts (where they cross the y‑axis), intersections (where lines meet), and whether lines are parallel (same slope) or identical (same slope and intercept).
- Special cases: vertical lines have undefined slope (not functions of form y = mx + b); perpendicular lines in the plane have slopes whose product is −1 (if neither is vertical).

Short examples and conclusions
1) Equations: y = 3x + 2 and y = 3x − 4.
- Slopes: both 3 → same rate of change.
- Intercepts: 2 and −4 → different starting values.
- Conclusion: parallel lines (same slope, different intercept); they never meet.

2) Equations: y = 0.5x + 1 and y = 1x + 0.
- Slopes: 0.5 and 1 → the second increases faster (steeper).
- Intercepts: 1 and 0 → the first starts higher at x = 0.
- Conclusion: the second line grows faster; for small x the first may be larger, but for large x the second will be larger because of the larger slope.

3) Solve intersection: y = 2x + 3 and y = −x + 6.
- Set equal: 2x + 3 = −x + 6 → 3x = 3 → x = 1.
- y = 2(1) + 3 = 5.
- Conclusion: the lines intersect at (1, 5). They are not parallel because slopes 2 and −1 are different.

4) Table example: A: (1,4), (3,8) → slope = (8−4)/(3−1) = 2. B: (1,5), (3,9) → slope = (9−5)/(3−1) = 2.
- Both have slope 2; compare intercepts by extending to x = 0: A intercept = 2, B intercept = 3.
- Conclusion: parallel lines with same slope but different intercepts; B is always 1 unit above A.

Summary
To compare linear relationships use slope (rate of change) and y‑intercept (initial value). These fully characterize a line in slope–intercept form, and they explain whether lines are steeper, increase/decrease, are parallel, coincide, or intersect and where.