Question

when we describe a line, the first thing we notice is its steepness. mathematically we measure steepness with slope. slope is a type of rate of change. it compares how much the line moves up or down (the
ise\) with how much it moves left or right (the
un\). a line that goes upward as you move to the right has a positive slope. a line that goes downward as you move to the right has a negative slope. a line that is flat has a slope of zero, and a vertical lines slope is undefined.
there are different ways to write the equation of a line, and each form is useful in a different situation. in slope - intercept form, y = mx + b, the slope is visible as m, and the number b shows where the line crosses the y - axis. in point - slope form, y - y1 = m(x - x1), the slope and a single known point are used to describe the line. in standard form, ax + by = c, both intercepts can be found easily, and fractions are often avoided.
even though they look different, slope - intercept, point - slope, and standard form all describe the same line. they are just different ways of organizing the same information. lines can also be compared to each other: parallel lines never meet because they have the same slope, while perpendicular lines meet at a right angle because their slopes are negative reciprocals.
questions

1. what does slope compare when describing a line?
2. how can we tell from a graph whether the slope is positive, negative, zero, or undefined?
3. in slope - intercept form, what does the number b represent?
4. when is point - slope form most useful?
5. why might someone prefer standard form instead of slope - intercept form?
6. how can three different forms of an equation still describe the same line?
7. what must be true about the slopes of parallel and perpendicular lines?

Explanation:

1. Compares vertical change to horizontal change.
2. Positive: rises right; negative: falls right; zero: flat; undefined: vertical.
3. y - intercept.
4. When a point and slope are known.
5. Useful for some algebraic operations.
6. They represent same linear relationship.
7. Parallel: equal slopes; perpendicular: negative - reciprocals.

Answer:

1. Vertical change to horizontal change.
2. Positive if rises to the right, negative if falls to the right, zero if flat, undefined if vertical.
3. y - intercept.
4. When a point on the line and its slope are known.
5. For algebraic manipulations and finding intercepts easily.
6. They are different ways to represent the same linear relationship.
7. Parallel lines have equal slopes, perpendicular lines have negative - reciprocal slopes.