MathLab with Mr. Ahmed

MathLab with Mr. Ahmed is a math learning website by Ahmed Amasha, a mathematics teacher teaching the American curriculum. Explore Algebra 2, Precalculus, Calculus, and SAT and PSAT Math Prep through lessons, focused practice and printable resources.

MathLabwith Mr. Ahmed
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1.10 Lines

Focus on slope and parallel slopes.

Before you solve

Ideas in this lesson

Review the ideas below. Select one to see worked steps and five practice questions.

01Compute and read slopesWorked steps · five questions02Write lines from points and slopesWorked steps · five questions03Use slope-intercept form and line familiesWorked steps · five questions04Handle horizontal and vertical linesWorked steps · five questions05Reason about parallel and perpendicular linesWorked steps · five questions06Connect standard form and interceptsWorked steps · five questions07Interpret and build linear rate modelsWorked steps · five questions08Use slopes to justify coordinate geometryWorked steps · five questions09Graph and analyze parameterized line familiesWorked steps · five questions10Construct perpendicular bisectors and circle tangentsWorked steps · five questions11Derive intercept form and justify coordinate argumentsWorked steps · five questions12Build and interpret physically restricted linear modelsWorked steps · five questions
Lesson brief and rules

1.10 Lines

Slope is change in yy divided by change in xx; in y=mx+b,my=mx+b, m is slope.

Worked example 11
01 / Find the slope through (−2,4)(-2,4) and (2,−2)(2,-2), and interpret its sign.
02 / Compute change in yy using the same point order as change in xx.
03 / Here m=−2−42−(−2)=−64=−32m=\dfrac{-2-4}{2-(-2)}=\dfrac{-6}{4}=-\dfrac32.
04 / The negative slope means moving right along the line moves downward.
Worked example 2 · Compute and read slopes

Read the two marked points on the graph. What is the slope, and what rise/run describes it?

Coordinate graphA line passes through marked points P=(-3,0) and Q=(1,2).-8-8-6-6-4-4-2-222446688xyPQ
A line passes through marked points P=(-3,0) and Q=(1,2).

Moving from PP to QQ increases xx by 44 and yy by 22.
Slope is vertical change divided by horizontal change: m=24=12m=\dfrac24=\dfrac12.
The scale on both axes must be read from the labels rather than from the apparent angle.

Read the graphs

Compute and read slopes

Read the marked points, calculate the slope, and explain the signed rise for a move 33 units right.
Read slope from the marked pointsRead slope from the marked points; window (-8, 8, -8, 8). Line L: slope -2/3, vertical intercept 1. P=(-3,3). Q=(3,-1). -8-6-4-202468-8-6-4-22468PQxy
Read slope from the marked points

Read P=(−3,3)P=(-3,3) and Q=(3,−1)Q=(3,-1).
The rise is −1−3=−4-1-3=-4 and run is 3−(−3)=63-(-3)=6.
Thus m=−46=−23m=-\dfrac46=-\dfrac23. A run of 33 corresponds to rise −2-2.

Answer: −23-\dfrac23; the line falls 22 units.

Write lines from points and slopes

Read the two marked points and derive an equation of their line.
Read two points to determine the lineRead two points to determine the line; window (-8, 8, -4, 12). Line L: slope 3/2, vertical intercept 5. P=(-2,2). Q=(2,8). -8-6-4-202468-4-224681012PQxy
Read two points to determine the line

Calculate m=64=32m=\dfrac{6}{4}=\dfrac32.
Using the first point, point-slope form is y−2=32(x−(−2))y-2=\dfrac32(x-(-2)).
Replace subtracting a negative by addition.
Substitute both original points to verify the resulting line.

Answer: y−2=32(x+2)y-2=\dfrac32(x+2)

Use slope-intercept form and line families

The graph shows three members of the family y=mx+2y=mx+2, where mm varies. What do all lines in this family have in common?
Coordinate graphThree lines have slopes −1, 0, and 1, with common y-intercept (0,2).-6-6-4-4-2-2224466xyP
Three lines have slopes −1, 0, and 1, with common y-intercept (0,2).

Set x=0x=0 in the family equation; the term mxmx becomes zero for every mm.
Every line therefore has the same yy-intercept (0,2)(0,2).
Changing mm changes the slope, so the lines generally rotate around that common point rather than remain parallel.

Answer: They pass through (0,2)(0,2).

15 questions covering every idea, with a worked answer for each.Start practice ↗
Textbook reference: Stewart, Redlin & Watson, Precalculus: Mathematics for Calculus, 7th ed. (Cengage, 2016). The explanations and practice here are original, organized by textbook section.

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