Depth · one concept from first principles · step 2 of 4

Rotate a line about its fixed point

Every line y=ax-1 passes through (0, -1), whatever a is. Find all slopes a for which this line touches the segment from A(-2, 4) to B(-5, 1).

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How to solve

Strategy Draw a Diagram — Setting x=0 shows every such line goes through P(0,-1), so changing a just rotates the line about P. Draw it: the line sweeps across the segment between the position through B and the position through A. Find those two boundary slopes by substituting each endpoint, then take everything in between.

1STEP 1

Find the fixed point

Put x=0: the a term vanishes, so every line in the family passes through the same point.

x=0 → y=a · 0-1=-1

2STEP 2

Slope when the line hits A

Substitute A(-2, 4) into y=ax-1.

4=-2a-1 → a=-5/2

3STEP 3

Slope when the line hits B

Substitute B(-5, 1) into y=ax-1.

1=-5a-1 → a=-2/5

4STEP 4

Sweep between the two positions

The segment is left of the y-axis, so turning the line from A to B sweeps every point, with slope running from -5/2 to -2/5.

-5/2 ≤ a ≤ -2/5

The two endpoints mark the extreme positions of the turning line.

8.F.A.3Draw A Diagram

The line $y=ax-1$ meets segment $AB$ exactly when $-\frac{5}{2}\le a\le-\frac{2}{5}$ .

Why?

Every line in the family passes through $(0,-1)$ , so changing $a$ only turns the line around that point.

Why?

At $x=0$ the term $a\times0$ is $0$ groups of $a$ , which is nothing, so $y=-1$ whatever $a$ is.

🧱Multiplication as equal groupsa times b means a equal groups of b — it is just repeated adding of the same amount.

Why?

A point lies on the line exactly when its coordinates make $y=ax-1$ true, so passing through $A$ or $B$ fixes $a$ at $-\frac{5}{2}$ or $-\frac{2}{5}$ .

🧱A solution makes the equation trueA solution is a number that, put in for the unknown, makes both sides come out equal; that is all being a solution means.

Why?

The segment sits to the left of the $y$ -axis, so the line turning about $(0,-1)$ from $A$ to $B$ never passes vertical, and the lines that hit $AB$ are exactly those with slopes between the slopes to $A$ and $B$ .

Why?

For a point to the left of $(0,-1)$ the run is negative, so multiplying by it turns the order over, and the point lies above the line of slope $a$ exactly when its own slope from $(0,-1)$ is smaller than $a$ .

🧱Scaling an inequalityMultiplying both sides by the same positive number stretches or shrinks the gaps on the number line but keeps the order; multiplying by the same negative number also flips the line around zero, so the inequality turns over.

Why?

So for $-\frac{5}{2}\le a\le-\frac{2}{5}$ the end $A$ is on or above the line and $B$ is on or below it, while for $a$ outside that range both ends sit strictly on the same side.

🧱Transitivity of orderIf the first is smaller than the second and the second smaller than the third, the first is smaller than the third, so pairwise comparisons chain into one full order.

Why?

Along the segment the gap between its height and the line's height changes by the same amount at every step, so it moves steadily between its two end values and reaches zero exactly when those values are not both of the same strict sign.

🧱A constant step reaches any termWhen a list climbs by the same amount every time, any term is the starting value plus that step repeated, so you never have to write out the ones between.

Answer

-5/2 ≤ a ≤ -2/5

Try a=-1 (inside the range): y=-x-1 meets segment AB, which lies on y=x+6, where -x-1=x+6, so x=-7/2, between -5 and -2. Try a=-3 (outside): -3x-1=x+6 gives x=-7/4, which is not between -5 and -2, so it misses.

Takeaway

When a line is pinned at one point, changing its slope just swings it like a door -- the ends of the target tell you how far it can swing.

Find the fixed point
Slope when the line hits A
Slope when the line hits B
Sweep between the two positions

Where next?

Level upsuggested

Bisect a pentagon by a line from one vertex Pivot a line past a triangle's extreme vertices

Another one like thissuggested

12 practice problems

Step back and solidifysuggested

A line through the origin sweeps between endpoints

A parent dashboard for the family lives at sensimlab.com.

▶ Practice — 12 problems