Unit 1: Algebra 1 Key Concepts Revisited | Algebra 2 | Welch Teaching & Coaching Website

Lesson Map

Each lesson matches one subsection of Chapter 1 in the W.A. Algebra 2 text. The goal is not to relearn Algebra 1 from scratch; it is to make the linear skills dependable enough to use throughout Algebra 2.

Lesson	Textbook subsection	Focus	You should be able to…
Lesson 1	1.1	Solving linear equations & inequalities	Solve multi-step equations, compound inequalities, and absolute-value equations and inequalities; graph solution sets and check possible extraneous solutions.
Lesson 2	1.2	Properties of linear functions & their graphs	Identify slope and intercepts; graph from slope-intercept, point-slope, and standard form; state domain and range.
Lesson 3	1.3	Creating linear equations from two points	Find slope from two points and write an equation in point-slope and slope-intercept form.
Lesson 4	1.4	Parallel, perpendicular, vertical & horizontal lines	Compare slopes, write related lines through a point, and recognize equations of vertical and horizontal lines.
Lesson 5	1.5	Function notation	Evaluate a function from an input, solve for an input from an output, read function tables, and interpret values in context.
Lesson 6	1.6	Applications of linear functions	Create a linear model from a constant rate or two data points, use it to predict values, and explain slope and intercept in context.
Performance Task	Unit synthesis	Constant-velocity motion	Collect position-time data, build and test a linear motion model, and interpret its slope as velocity.
Assessment	Chapter 1	Unit 1 review	Move comfortably among equations, tables, graphs, and contexts, and explain what each number means.

Content Summary

The idea first, then the method, then a worked example in the format students should be able to reproduce.

Lesson 1 • Section 1.1

Solving Linear Equations and Inequalities

Solving an equation means isolating the variable by undoing operations. Work in reverse order of operations, keep both sides balanced, and check the result in the original equation. If variables appear on both sides, collect the variable terms on one side first. If parentheses appear, distribute before combining like terms.

3(x − 4) + 7 = 22 → 3x − 12 + 7 = 22 → 3x = 27 → x = 9

Inequalities use the same algebra, with one critical rule: multiplying or dividing by a negative reverses the inequality sign. A solution to an inequality is usually a set of values, not one number.

−4x + 6 > 18 → −4x > 12 → x < −3

A compound AND inequality asks for the overlap of two conditions. An OR inequality accepts values that satisfy either condition. Absolute value represents distance from zero, so an equation such as |expression| = positive number usually creates two cases.

|2x − 3| = 9 → 2x − 3 = 9 or 2x − 3 = −9 → x = 6 or x = −3

Check absolute-value answers. Algebra can create a candidate that does not satisfy the original equation. A failed check is an extraneous solution.

Lesson 2 • Section 1.2

Properties of Linear Functions and Their Graphs

The slope of a line is its constant rate of change: how much the dependent variable changes for each one-unit change in the independent variable.

m = change in y ÷ change in x = Δy ÷ Δx

Slope-intercept form

y = mx + b

m is slope and b is the y-intercept. Plot b first, then use rise over run.

Point-slope form

y − y₁ = m(x − x₁)

Start at the known point (x₁, y₁), then move using the slope.

Standard form

Ax + By = C

Set y = 0 for the x-intercept and x = 0 for the y-intercept.

In slope-intercept form, the intercept gives the starting point and the slope gives the repeated movement.

For any nonvertical line extending forever, both domain and range are all real numbers. A horizontal line still has all real x-values, but its range is only one y-value. A vertical line has only one x-value and is not a function of x.

Lesson 3 • Section 1.3

Creating Linear Equations from Two Points

If you know two points, the first job is to find their slope. Keep the subtraction order consistent in the numerator and denominator.

m = (y₂ − y₁) ÷ (x₂ − x₁)

Then use either point in point-slope form. Once the equation is correct, distribute and isolate y if slope-intercept form is requested.

Worked example. Write the equation through (2, 7) and (6, 15).

m = (15 − 7) ÷ (6 − 2) = 8/4 = 2

y − 7 = 2(x − 2) → y − 7 = 2x − 4 → y = 2x + 3

Fast check: substitute both original points into the final equation. A line built from two points must contain both of them.

Lesson 4 • Section 1.4

Parallel, Perpendicular, Vertical and Horizontal Lines

Line relationships are slope relationships. Parallel lines have the same slope but different intercepts. Perpendicular nonvertical lines have slopes that are negative reciprocals, so their product is −1.

Parallel

m₁ = m₂

Perpendicular

m₁ × m₂ = −1

To write a line parallel or perpendicular to a given line, first rewrite the given line so you can identify its slope. Choose the required slope, then use the provided point in point-slope form.

Worked example. Write a line through (4, −1) perpendicular to y = (2/3)x + 5.

m⊥ = −3/2

y + 1 = −(3/2)(x − 4) → y = −(3/2)x + 5

Horizontal

y = c, m = 0

Vertical

x = c, slope undefined

Lesson 5 • Section 1.5

Function Notation

Function notation gives an equation a name. If f(x) = 3x − 2, then f(5) means “use 5 as the input in function f.”

f(5) = 3(5) − 2 = 13

Sometimes the output is given instead. If f(x) = 19, replace the whole output with 19 and solve for the input.

19 = 3x − 2 → 21 = 3x → x = 7

Function notation also works with tables and nested expressions. In f(g(2)), evaluate the inside function first; its output becomes the input of the outside function.

f(x) does not mean f × x. It names the output produced by function f when the input is x.

Lesson 6 • Section 1.6

Applications of Linear Functions

A situation is linear when the rate of change is constant. In a useful model, every piece of the equation should have a meaning.

output = (rate of change)(input) + starting value

Worked example. A temperature sensor reads 18°C at t = 0 and warms at a constant 2.5°C per minute.

T(t) = 2.5t + 18

The slope 2.5 means the measured temperature rises 2.5°C each minute. The intercept 18 is the initial temperature. To predict the temperature after 8 minutes, evaluate T(8) = 38°C. To find when the sensor reaches 43°C, solve 43 = 2.5t + 18 to get t = 10 minutes.

If the model comes from two measured data points instead of a stated rate, first find the slope from the points, then create the equation exactly as in Lesson 3.

Context matters. A mathematically correct linear equation can make physically unreasonable predictions outside the measured interval. Always choose a reasonable domain for an applied model.

Vocabulary

These terms connect the symbolic work to graphs and applications.

Solution

A value or set of values that makes an equation or inequality true.

Extraneous solution

A candidate produced during algebra that fails when checked in the original equation.

Compound inequality

Two inequalities connected by AND or OR.

Absolute value

The distance a number is from zero; distance is never negative.

Slope / rate of change

The ratio Δy ÷ Δx. In an application, include units.

x-intercept

The point where y = 0 and the graph crosses the x-axis.

y-intercept

The point where x = 0 and the graph crosses the y-axis; often a starting value in an application.

Domain

The allowable input values of a relation or function.

Range

The resulting output values.

Point-slope form

A line written as y − y₁ = m(x − x₁), useful when a point and slope are known.

Parallel lines

Coplanar lines with the same slope and different intercepts, so they never meet.

Perpendicular lines

Lines meeting at 90°; nonvertical slopes are negative reciprocals.

Function notation

Notation such as f(x) that names the output of a particular function for input x.

Linear model

A function with constant rate of change used to approximate or describe a relationship.

Sample Problems

Eighteen problems, three per lesson, in homework and assessment format. Work each one on paper first — the solutions are one click away, which makes them very easy to read too early.

Lesson 1 — Equations & Inequalities

1

Solve a multi-step equation

Solve 5(2x − 3) + 4 = 3x + 24.

Show solution

10x − 15 + 4 = 3x + 24 → 10x − 11 = 3x + 24 → 7x = 35 → x = 5

Check: both sides equal 39.

2

Solve and graph a compound inequality

Solve −5 ≤ 2x + 3 < 11.

Show solution

−8 ≤ 2x < 8 → −4 ≤ x < 4

Closed circle at −4, open circle at 4, shade between them.

3

Solve an absolute-value equation

Solve |3x + 1| = 10.

Show solution

3x + 1 = 10 or 3x + 1 = −10 → x = 3 or x = −11/3

Both solutions check in the original equation.

Lesson 2 — Linear Graphs

4

Graph from slope-intercept form

For y = −(3/4)x + 6, identify the slope and y-intercept.

Show solution

m = −3/4, b = 6

Plot (0, 6), then go down 3 and right 4 to get another point.

5

Graph from standard form using intercepts

Find the intercepts of 3x + 2y = 12.

Show solution

Set y = 0: 3x = 12, so x = 4. Set x = 0: 2y = 12, so y = 6.

x-intercept (4, 0), y-intercept (0, 6)

6

Read point-slope form

For y + 3 = 2(x − 5), name a point and the slope.

Show solution

point (5, −3), slope m = 2

Remember y + 3 means y − (−3).

Lesson 3 — Equations from Points

7

Find slope from two points

Find the slope between (−3, 4) and (5, −8).

Show solution

m = (−8 − 4) ÷ (5 − (−3)) = −12/8 = −3/2

8

Write the line through two points

Write the equation through (1, 5) and (4, 14).

Show solution

m = (14 − 5) ÷ (4 − 1) = 3

y − 5 = 3(x − 1) → y = 3x + 2

9

Use a point and a slope

Write a line with slope −2/5 through (10, 3).

Show solution

y − 3 = −(2/5)(x − 10) → y = −(2/5)x + 7

Lesson 4 — Related & Special Lines

10

Classify two lines

Classify y = 4x − 7 and 8x − 2y = 10 as parallel, perpendicular, or neither.

Show solution

Rewrite the second equation: −2y = −8x + 10 → y = 4x − 5.

same slope 4, different intercepts → parallel

11

Write a perpendicular line

Write the line through (2, 6) perpendicular to y = (1/3)x − 4.

Show solution

m⊥ = −3

y − 6 = −3(x − 2) → y = −3x + 12

12

Special line

Write the equation of the vertical line through (−5, 8).

Show solution

x = −5

Every point on the line has x-coordinate −5.

Lesson 5 — Function Notation

13

Evaluate an input

If f(x) = −2x + 9, find f(7).

Show solution

f(7) = −2(7) + 9 = −5

14

Find the input from an output

If g(x) = 5x − 8, find x when g(x) = 27.

Show solution

27 = 5x − 8 → 35 = 5x → x = 7

15

Compose two functions

If f(x) = 2x + 1 and g(x) = x² − 3, find f(g(2)).

Show solution

g(2) = 4 − 3 = 1, then f(1) = 2(1) + 1 = 3

Lesson 6 — Linear Models

16

Build a model from a rate and starting value

A 120-liter tank drains at 6 liters per minute. Write V(t), then find V(12).

Show solution

V(t) = 120 − 6t

V(12) = 120 − 72 = 48 liters

The slope −6 L/min is the drain rate and the intercept 120 L is the initial volume.

17

Build a model from two measurements

A cart is 0.40 m from the origin at 1.0 s and 1.60 m from the origin at 4.0 s. Model position x(t).

Show solution

m = (1.60 − 0.40) ÷ (4 − 1) = 1.20/3 = 0.40 m/s

x − 0.40 = 0.40(t − 1) → x(t) = 0.40t

The slope is the cart’s constant velocity.

18

Use and limit a model

A candle is 24 cm tall and burns at 1.5 cm per hour. When will it reach 6 cm? Give a physically reasonable domain for h(t) = 24 − 1.5t.

Show solution

6 = 24 − 1.5t → −18 = −1.5t → t = 12 h

The candle reaches height zero at 16 h, so a reasonable model domain is 0 ≤ t ≤ 16.

Equations & Rules

Every formula and rule for the unit on one printable page.

Slope

m = (y₂ − y₁) ÷ (x₂ − x₁)

Slope-intercept form

y = mx + b

Point-slope form

y − y₁ = m(x − x₁)

Standard form

Ax + By = C

Parallel lines

m₁ = m₂

Perpendicular lines

m₁ × m₂ = −1

Horizontal line

y = c; m = 0

Vertical line

x = c; slope undefined

Absolute-value equation

|u| = k → u = k or u = −k

Inequality sign rule

Multiplying or dividing both sides by a negative number reverses the inequality.

Function notation

f(a) = output when the input is a

Linear model

output = (rate)(input) + starting value

Unit 1 Review Guide

Before moving on, you should be able to move comfortably among equations, tables, graphs, and contexts. The important question is not only “Can I solve this?” but “What does this number mean?”

Algebra skills

Solve multi-step equations with distribution and variables on both sides.
Solve and graph simple, compound, and absolute-value inequalities.
Check absolute-value solutions in the original equation.
Rearrange equations into a useful form.

Linear-function skills

Find slope and intercepts from equations, graphs, or points.
Graph all three common forms of a line.
Create equations from a point and slope, or from two points.
Use parallel and perpendicular slope relationships.
Evaluate and interpret function notation.
Build and explain linear models in context.

Before you turn in the assessment

Every inequality checked for a negative multiply or divide — did the sign flip?
Both cases written for each absolute-value equation, then each candidate checked in the original.
Slope subtractions taken in the same order top and bottom.
Point-slope signs handled correctly when a coordinate is negative.
Perpendicular slopes flipped and negated, not just one of the two.
f(x) read as an output name, never as multiplication.
Every applied answer given with units.
Every model given a domain that makes physical sense.

Performance Task: Constant Velocity

A hands-on physics task that turns a set of measurements into a linear model — and then makes the model defend itself.

Build a Motion Model

Driving question: Can you use a short set of physical measurements to build a linear equation that predicts where a moving object will be later?

Students release a battery-powered toy car, dynamics cart, or other approximately constant-speed vehicle along a straight track. They measure position as a function of time, build a linear model, test the model against new measurements, and explain the physical meaning of every part of the equation.

Materials

Constant-speed toy car or dynamics cart
3–5 m straight floor or track space
Meter sticks or measuring tape
Masking tape for position marks
Stopwatch or phone video timer
Graph paper or spreadsheet

Physics connection

For constant velocity, position changes linearly with time:

x(t) = vt + x₀

That is the same structure as y = mx + b. The slope is velocity v, and the y-intercept is initial position x₀.

Part A — Collect the data

Create a straight motion lane and choose an origin, x = 0.
Place the vehicle at a measured starting position that does not have to be zero.
Record the vehicle’s position at a minimum of six equally spaced times. Video analysis is encouraged if hand timing is inconsistent.
Repeat at least one trial. Keep units consistent: seconds and meters are recommended.

Part B — Build the model

Plot position x versus time t.
Select two well-separated data points and calculate slope using Δx ÷ Δt.
Write the model first in point-slope form, then convert it to x(t) = vt + x₀.
State a reasonable domain and range based on the actual track and measurement interval.
Explain, with units, what v and x₀ mean physically.

Part C — Make and test predictions

Use function notation to predict the position at a time you did not use to create the model.
Run the cart again and measure that actual position.
Calculate the absolute prediction error and discuss why the model is not perfect.
Solve the model backward: choose a position on the track and predict the time at which the cart should reach it. Test that prediction.

Part D — Compare motion lines

Run a second trial with the same vehicle speed but a different starting position. If the speeds are genuinely the same, the position-time lines should be approximately parallel: same slope, different intercepts. Explain whether the data support that claim.

Then create, as a purely mathematical extension, the equation of a line through one measured point that is perpendicular to your motion-model line.

Safety: Use a floor or low tabletop with a stop block. Do not create a steep ramp or launch a vehicle toward people, glass, electronics, or the edge of a table.

Required submission

Data table with units
Position-time graph
Slope calculation from two points
Point-slope and slope-intercept/function forms
Domain and range

Interpretation of slope and intercept
One forward prediction and test
One backward prediction and test
Parallel-line comparison
Error analysis and conclusion

Suggested Rubric — 20 points

Category	4 — Strong	3 — Proficient	2 — Developing	1 — Incomplete
Data & graph	Six or more usable measurements, correct units, clear scaled graph.	Complete data and readable graph with minor issues.	Some missing or inconsistent data or graph features.	Insufficient data or graph.
Linear model	Correct slope, point-slope form, final function, domain and range.	Model mostly correct with one minor error.	Partial model or repeated algebra errors.	No defensible model.
Physics meaning	Correctly explains slope as velocity and intercept as starting position, with units.	Meanings correct but explanation or units incomplete.	Partial interpretation.	Interpretation missing or incorrect.
Predictions & testing	Forward and backward predictions are both tested and error is quantified.	Both predictions tested; limited error analysis.	Only one prediction is meaningfully tested.	Predictions not tested.
Comparison & conclusion	Uses slopes and intercepts to analyze parallel trials and gives a thoughtful limitation discussion.	Comparison and conclusion are basically correct.	Comparison is mostly descriptive rather than mathematical.	Conclusion unsupported.

Teacher note. This task intentionally uses a simple constant-velocity model. Real carts accelerate briefly at startup and measurements contain timing and position error, so students should not expect every point to land exactly on one line. That mismatch is useful: the algebraic model is an approximation of a physical system, not the system itself.

Textbook Basis

Where this unit comes from, and what was rewritten for the web.

This page is an instructional adaptation of Chapter 1, Algebra 1 Key Concepts Revisited, from the W.A. Algebra 2 (Summer 2016) CK-12 FlexBook. Section sequence and learning targets follow the source text; worked and practice examples on this page have been rewritten or newly created for web instruction.

CK-12 Foundation content is provided under the Creative Commons Attribution-NonCommercial 3.0 license unless otherwise noted. Visit CK-12 for platform and licensing information.

Where to Go Next

Unit 1 makes the linear toolkit automatic so the rest of Algebra 2 can lean on it.

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All Algebra 2 Units

The full Algebra 2 course map, with every content summary, sample problem set, equation sheet, and performance task in one place.

Back to Algebra 2 →

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Physics: Kinematics

The performance task is constant-velocity motion. The physics side of the same idea, including the graphs and the vocabulary.

Open Kinematics →