LEARN · EXPLAIN · REVISE

Lines through O meet the segment from A=(t,1) to B=(t,3). Find the slope range.

Read the idea, work independently, then explain what changed.

高二選擇性必修第一册(A版).pdf · 2.1 · PDF 56 / printed page 51

Revisit first: Exponents

TOPIC 01

Inclination and slope of a line

Connect slope, inclination and parallel/perpendicular directions, including vertical lines.

Connect slope, inclination and parallel/perpendicular directions, including vertical lines.
Justify the method and check the conditions in a new situation.

Use one hint at a time. A correction explains what changed, not just the final answer.

01 / Standard#Your turn

t=12

A vertical line has no finite slope; m1m2=−1 applies only when both slopes exist.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Translate the geometric condition into a coordinate equation.

Hint 2

Use this intermediate relation.

y=mt,\quad 1\le y\le3

Worked solution

Translate the geometric condition into a coordinate equation.
Apply the stated relation and retain its conditions.
$1\le 12m\le3$
Apply the stated relation and retain its conditions.
$1/12\le m\le3/12$
A segment gives a closed range; replacing it by the full supporting line would change the answer.

The requested relation or conclusion is shown below.

\frac1{12}\le m\le\frac3{12}

Checks and common pitfalls: A segment gives a closed range; replacing it by the full supporting line would change the answer.

Reasoning checklist · self / teacher assessment

Think first. Reveal a hint when the class is ready.

Teacher preparation and assessment

Connect slope, inclination and parallel/perpendicular directions, including vertical lines.
Which condition is essential in inclination and slope of a line?
Does a steeper line always have a larger slope?

Slope: For a nonvertical line, slope is rise divided by run and equals tan of the inclination.
$m=(y_2-y_1)/(x_2-x_1)$
Exceptional directions: A vertical line has no finite slope; m1m2=−1 applies only when both slopes exist.

A vertical line is perpendicular to a horizontal line even though its slope is undefined.