Linear Radar Slides
ALGEBRA FOUNDATIONS
MODULE 1 • LESSON 1
Linear Functions
Spotting straight-line relationships across tables, graphs, equations, and real-world stories.
Tables
Constant Δ
Graphs
Straight Line
Equations
\(y = mx + b\)
Contexts
Steady Rate
The Fundamental Rule
What Makes a Function Linear?
One golden requirement
Core Definition
A constant rate of change between variables.
Every equal step forward in \(x\) produces the exact same change in \(y\).
Linear Behavior STEADY
- Rate never speeds up or slows down
- Graph is an unbroken, single straight line
- Formula has input variable to 1st power: \(x^1\)
Non-Linear Behavior CHANGING
- Rate accelerates, decelerates, or curves
- Graph bends, arcs, jumps, or waves
- Formula has powers, roots, or products: \(x^2\), \(2^x\)
Representation 1: Tables
Look for Constant Differences
Rate = Δy / Δx
Linear Table Δy / Δx = +3 / +1 = 3
\(x\)
\(y\)
Δ Diff
0
4
—
1 (+1)
7
+3
2 (+1)
10
+3
3 (+1)
13
+3
Verdict: Equal steps of \(+1\) in \(x\) give constant \(+3\) in \(y\). Linear!
Non-Linear Table Δy changes each step
\(x\)
\(y\)
Δ Diff
0
1
—
1 (+1)
2
+1
2 (+1)
4
+2
3 (+1)
8
+4
Verdict: Differences double (+1, +2, +4). Accelerating, not linear!
Pro Tip: Always verify that \(x\) values advance by equal increments before checking differences in \(y\)!
Representation 2: Graphs
Straight Lines with Constant Slope
Slope \(m = \frac{\text{Rise}}{\text{Run}}\)
Linear Graph Constant Steepness
Run = +2 Rise = +3
The slope triangle is identical anywhere along the line.
Non-Linear Graph Changing Curvature
Flat slope Steep slope
The steepness keeps changing as you move along the curve.
Visual rule of thumb: If a ruler cannot touch every single point simultaneously, it is not linear.
Representation 3: Equations
Identify Degree 1 Equations
Slope-Intercept: \(y = mx + b\)
y
Output Variable
=
m
Slope / Rate
x1
Exponent is 1
b
Initial Value (\(y\)-int)
Linear Examples (Passes)
\(y = 4x - 7\) \(m = 4, b = -7\)
\(3x + 2y = 12\) Standard form
\(y = -0.5x\) Direct variation (\(b=0\))
Red Flags (Non-Linear)
\(y = x^2 + 5\) Exponent \(> 1\) (Quadratic)
\(y = 3^x\) Variable in exponent
\(y = \frac{8}{x}\) Division by variable
Representation 4: Situations
Spotting Rates in Real Life
Keywords: "per", "each", "every", "flat fee"
Rideshare Taxi LINEAR
“A taxi costs a $3.00 flat pickup fee plus $2.50 per mile traveled.”
Constant Rate: Increases by exactly $2.50 for each mile.
Equation: \(C = 2.50m + 3.00\)
Bacteria Colony NON-LINEAR
“A petri dish starts with 100 bacteria and doubles in population every single hour.”
Accelerating Rate: Hour 1 gains 100, Hour 2 gains 200, Hour 3 gains 400!
Equation: \(B = 100(2^t)\)
Linear Clues: “constant speed” “fixed hourly wage” “draining at steady pace” “equal additions”
Worked Example
The 4-Way Verification
Connecting the Dots
Context: A 50-gallon water barrel is draining water through a spout at a steady rate of 5 gallons per minute.
1. Table
| Min (\(t\)) | Gal (\(V\)) | Δ |
|---|
| 0 | 50 | — |
| 1 | 45 | -5 |
| 2 | 40 | -5 |
| 3 | 35 | -5 |
2. Equation
\(V = -5t + 50\)
• Slope \(m = -5\) (drain rate)
• Intercept \(b = 50\) (starts full)
• Degree = 1 (linear!)
3. Graph
50 gal 10 min
Straight line sloping down
Conclusion: Constant decrease of 5 gal/min confirms this is a Linear Function across every format.
Key Takeaways
Linear Function Identification Radar
Summary Checklist
In Tables
Equal steps in \(x\) yield constant differences in \(y\). The ratio \(\frac{\Delta y}{\Delta x}\) is unchanging.
In Graphs
Always a straight line. No curves or waves. Slope (\(\frac{\text{rise}}{\text{run}}\)) is identical everywhere.
In Equations
Highest power of input is exactly 1 (\(y = mx + b\)). No squared terms, no exponents with \(x\), no \(x\) in denominators.
In Situations
Describes an activity with a constant unit rate (e.g., speed, wage per hour, steady drainage).
Turn & Talk: Is \(y = 7\) a linear function? Why or why not? What is its slope?
Think: \(m = 0\)
