AMC 10 · 2004 · #5
Grade 8 geometry-2d
Pick an answer.
The five choices cut the number line for mb into sign regions and size regions, so the work splits into two easy questions instead of one hard one (Tool #3): what sign does mb have, and is its size above or below 1? Both can be answered from bounds alone. Reading the picture (Tool #1) gives the direction of the line and the two gridlines its intercepts sit between — the sturdy facts a graph can actually support. Then pushing those bounds to their extremes (Tool #14) shows the conclusion holds for every slope and intercept consistent with the picture, so nothing depends on eyeballing an exact value.
Read the y-intercept as a range
The picture certifies only a range: the intercept lies between the gridlines, so 0 ＜ b ＜ 1.
The y-intercept is just the height of the line where it crosses the vertical axis, so the gridlines above and below it bracket its value.
8.F.B.4Draw A DiagramRead the slope as a range
The line falls but stays above the axis at one, giving -1 ＜ m ＜ 0.
Slope is the drop per one step right, so a line that needs more than one step to fall its own intercept's height has a slope shallower than -1.
8.EE.B.6Draw A DiagramSettle the sign of the product
A negative times a positive is negative, which already kills three choices.
One negative factor flips the sign of a product, so a falling line with a positive intercept always has mb negative.
7.NS.A.2Eliminate PossibilitiesSettle the size of the product
Both factors are under one in size, so the product stays above negative one: -1 ＜ mb ＜ 0, choice (B).
Multiplying by a number less than 1 makes things smaller, so two such factors can never reach 1.
Both factors are smaller than one in size, so their product can never reach one.
▸ Why?
Multiplying by a number below one shrinks a quantity, so doing it twice shrinks it further still.
▸ Why?
A value that starts below one and only shrinks can never climb back past one, so the bound holds all the way.
A graph gives you reliable inequalities, not exact numbers — a falling line with its y-intercept between 0 and 1 has m negative and both |m| and b below 1, so mb is negative but never as far down as -1.
- Read the y-intercept as a range
- Read the slope as a range
- Settle the sign of the product
- Settle the size of the product