Lines
A line is the one graph completely described by two numbers: where it is and how steeply it climbs. Everything in this section is those two numbers written down in a different order. The skill worth building is not memorising three equations but knowing, from what a question gives you, which one to start from — that choice is most of the work.
Calculus — graphing and functions
Graphing calculator- From two pointsm = y2 − y1x2 − x1
Rise over run. Either point can be first, as long as the same one is first on the top and the bottom.
- What the sign saysm > 0 rises, m < 0 falls
Read left to right, as you read everything else.
- Horizontaly = b, m = 0
No rise for any run. y is stuck at one value, whatever x does.
- Verticalx = a, m is undefined
Run of zero, so the slope formula divides by zero. Undefined, which is not the same as zero.
- Parallelm1 = m2
Same steepness, so they never meet.
- Perpendicularm1m2 = −1, m2 = −1m1
The negative reciprocal: flip it over and change the sign. Both, not one. A horizontal line and a vertical line are the exception the rule cannot state, since one slope does not exist.
- Slope-intercepty = mx + b
Best for graphing and for comparing two lines: the slope m and the y-intercept b are both sitting there in plain sight.
- Point-slopey − y1 = m(x − x1)
Best for building a line, because it takes exactly what most questions give you: one point and a slope. Note both minus signs.
- StandardAx + By = C
Best for finding both intercepts quickly, and the only one of the three that can write a vertical line.
- Interceptsx-intercept: set y = 0. y-intercept: set x = 0.
An intercept is where the line crosses an axis, and on an axis the other coordinate is zero. That is the whole rule.
- 1
Two points. Slope first, from the formula. Then point-slope with either of the two points — it does not matter which, and the answers will agree once tidied.
- 2
A point and a slope. Point-slope, straight in. This is what the form exists for.
- 3
A slope and the y-intercept. Slope-intercept, straight in: the two numbers are m and b.
- 4
Parallel or perpendicular to a given line. Get the given line's slope first — put it in slope-intercept form if it is not already. Take it as is for parallel, or the negative reciprocal for perpendicular. Then you have a point and a slope.
- 5
Finish in the form you were asked for. Point-slope is a legitimate answer unless the question says otherwise. If it asks for slope-intercept, solve for y; if standard, clear the fractions and move the x term across.
- m = y2 − y1x1 − x2m = y2 − y1x2 − x1
Both subtractions have to run in the same direction. Reversing one of them flips the sign of the slope, which turns a line that rises into one that falls.
- A vertical line has slope 0A vertical line has no slope. A horizontal line has slope 0.
A horizontal line has a rise of zero, so m = 0run = 0. A vertical line has a run of zero, so the formula divides by zero and there is no such number. Undefined and zero are different answers.
- Perpendicular to m = 25 is m = −25Perpendicular to m = 25 is m = −52
Negating alone is not enough: 25 · −25 = −425, not −1. The fraction has to be turned over as well.
- 5y = −2x + 20 ⇒ y = −2x + 45y = −2x + 20 ⇒ y = −25x + 4
Dividing by 5 has to reach every term on the right, not only the constant. Halving the work here halves the slope.
Type an answer and it gets marked. What you get right is kept on this device.
- 1.Slope through (1, −3) and (5, 9)
- 2.Through (−4, 2) with slope −12
- 3.Through (0, −5) and (2, 1)
- 4.Perpendicular to y = 4x + 1 through (8, −2)
- 5.Intercepts of 5x + 2y = −10
- 6.Vertical line through (6, −1)