Grade 8 Functions – Anchor Charts 8.F.1 & 8.F.3

8.F.1

Understanding Functions

A function assigns to each input exactly one output. The graph is the set of ordered pairs (input, output).

📌 A function is a rule that assigns to each input exactly one output. Functions describe real-world situations where one quantity determines another — like how the number of items you buy determines the total cost. The graph of a function is made up of all ordered pairs (input, output) plotted on the coordinate plane.

🔍 Function or NOT a Function?

✔ IS a Function

Each input → exactly ONE output

Input (Domain)

→

Output (Range)

No input maps to more than one output ✅
Outputs CAN repeat — that's okay!

✗ NOT a Function

One input → TWO different outputs!

Input (Domain)

→

Output (Range)

x = 2 maps to BOTH 8 and 11 ❌

📋 4 Ways to Represent a Function

📊 Table of Values

x (input)	y (output)
1	3
2	6
3	9

No repeated x-values = function ✅
If ANY x repeats → NOT a function

📐 Ordered Pairs

✅ { (1, 3), (2, 6), (3, 9) }

No x repeats → IS a function

❌ { (1, 3), (2, 6), (2, 9) }

x = 2 repeats → NOT a function

🗺️ Mapping Diagram

Domain

⟶

Range

Each domain value has exactly one arrow

📈 Graph + Vertical Line Test (VLT)

Vertical line crosses ONE point = Function!
Crosses 2+ points = NOT a function

⭐ Key Concepts to Know

📦 Input & Output

The input (domain) is the value you put into the function. The output (range) is the result. A function machine takes inputs and produces outputs based on a rule.

📍 Coordinate Plane

Ordered pairs are graphed as (x, y) points. The x-axis holds the inputs; the y-axis holds the outputs. Each point represents one input–output pair.

f(x) Notation

f(x) = y is read "f of x." It tells you the output when x is the input. Example: if f(x) = 3x, then f(4) = 12.

🧠 Remember

Every INPUT has exactly ONE output.
Outputs CAN repeat — that's fine!

🚫 Watch Out

Any x-value that appears twice in a table → NOT a function. Any graph that fails the VLT → NOT a function.

🌍 Real World

Functions describe patterns: distance from speed, cost from quantity, temperature over time.

🎯 Check Your Understanding — 8.F.1

Score: 0 / 6

What is the definition of a function?

A rule where every output has exactly one input

A rule where every input has exactly one output

A set of ordered pairs where y-values never repeat

A graph that always forms a straight line

Which set of ordered pairs represents a function?

{ (1,2), (1,5), (3,6) }

{ (2,3), (4,7), (2,9) }

{ (1,4), (2,4), (3,4) }

{ (5,1), (5,2), (5,3) }

How do you use the Vertical Line Test to check if a graph represents a function?

If a horizontal line crosses the graph more than once, it is a function

If any vertical line crosses the graph at more than one point, it is NOT a function

If the graph is a straight line, it passes the vertical line test

If the graph has no negative values, it is a function

If f(x) = 4x − 1, what is f(3)?

A table shows x = 2 paired with y = 5, and x = 2 paired with y = 9. What can you conclude?

The relation is a function because both outputs are positive

The relation is NOT a function because x = 2 has two different outputs

The relation is a function because the outputs (5 and 9) are different

Not enough information to decide

Which of the following is TRUE about the domain and range of a function?

The domain contains output values and the range contains input values

Both domain and range must contain the same values

Range values (outputs) can never repeat in a function

Domain values (inputs) each map to exactly one range value (output)

✏️ Open Response: Describe a real-world situation that represents a function. Explain why it qualifies as a function.

8.F.3

Linear vs. Non-Linear Functions

Interpret y = mx + b as a linear function; identify and give examples of functions that are NOT linear

📌 A linear function has a graph that is a straight line and can be written as y = mx + b. The rate of change (slope) is constant. A non-linear function does NOT graph a straight line — its rate of change varies. Example: A = s² (area of a square) is non-linear because the points (1,1), (2,4), (3,9) are not on a straight line.

📌 The Linear Equation: y = mx + b

y = mx + b

SLOPE
rate of change
rise ÷ run

INPUT
independent
variable

Y-INTERCEPT
crosses y-axis
when x = 0

OUTPUT
dependent
variable

⚡ Linear vs. Non-Linear Side-by-Side

✔ LINEAR — y = 2x + 1

Straight line · Constant rate of change · Fits y = mx + b

x	0	1	2	3
y	1	3	5	7

Δy = +2 every step ✅ Constant rate!

✗ NON-LINEAR — y = x²

Curved graph · Rate of change is NOT constant · Does NOT fit y = mx + b

x	0	1	2	3
y	0	1	4	9

Δy: +1, +3, +5 — changes every step ❌

📝 Examples & Special Cases

Linear Examples

y = 3x − 2
y = −x + 5
y = ½x
y = 4 (slope = 0)

Exponent of x is 1 · No x in denominator or root · Rate of change is constant

Non-Linear Examples

y = x²
y = x³ + 1
y = 1/x
A = s² (from standard)

Exponent ≠ 1 · x in denominator or root · Rate of change varies

Special Cases

y = 4 → Linear ✅
(horizontal, slope = 0)
x = 3 → Not a Function ❌
(vertical, fails VLT)

Linear functions may NOT always look like y = mx + b — learn to find rate of change in other forms too!

📐 Finding Rate of Change

From a Table

Divide the change in y by the change in x:

m = Δy / Δx

If Δy/Δx is the same for all pairs → LINEAR
If it changes → NON-LINEAR

From Two Points

m = (y₂ − y₁) / (x₂ − x₁)

Example: Points (2, 5) and (4, 11)
m = (11 − 5) / (4 − 2) = 6/2 = 3

Slope = 3 means: for every 1 unit right, go up 3 units

✅ Linear vs. Non-Linear Checklist

📏

LINEAR

Straight line · Constant rate of change · Fits y = mx + b

0 linear checks

〰️

NON-LINEAR

Curved graph · Changing rate of change · Does NOT fit y = mx + b

0 non-linear checks

📏 Slope (m)

m = rise/run = (y₂−y₁)/(x₂−x₁)
Positive ↗ · Negative ↘ · Zero → · Undefined ↕

📍 Y-Intercept (b)

Where line crosses y-axis when x = 0
Written as the point (0, b)

🔑 How to Tell

Table → constant Δy? Linear!
Graph → straight line? Linear!
Equation → no exponents on x? Linear!

🎯 Check Your Understanding — 8.F.3

Score: 0 / 6

In the equation y = mx + b, what does b represent?

The slope of the line

The y-intercept — where the line crosses the y-axis

The input value

The rate of change between two points

A table shows Δy values of +4, +4, +4 for equal Δx steps. What does this tell you?

The function is linear because the rate of change is constant

The function is non-linear because the y-values are all the same

The function is non-linear because Δy is always 4

Not enough information to determine linearity

Which equation represents a NON-LINEAR function?

y = 5x + 2

y = −3x

y = x² + 1

y = 7

What is the slope of the line passing through (1, 3) and (4, 12)?

Why is A = s² (area of a square) considered a non-linear function?

Because it uses the variable s instead of x

Because the area is always a positive number

Because the points (1,1), (2,4), (3,9) do not lie on a straight line

Because it does not have a y-intercept

A line has a slope of −2 and a y-intercept of 5. Which equation represents this line?

y = 5x − 2

y = 2x + 5

y = −2x + 5

y = −2x − 5

✏️ Open Response: Create your own table of values for a non-linear function. Explain how you know it is non-linear by analyzing the rate of change.

Grade 8 Functions – Anchor Chart 8.F.2

8.F.2

Comparing Functions

Compare properties of two functions each represented in a different way — algebraically, graphically, numerically in tables, or by verbal descriptions

📌 The same function can be shown in 4 different ways. To compare two functions, you must be able to identify key properties — like slope (rate of change), y-intercept, and overall behavior — no matter how each function is represented. For example: given a linear function as a table and another as an equation, you can determine which has the greater rate of change.

📋 The 4 Ways Functions Are Represented

📐 Algebraically

y = 2x + 3

Read the equation directly:
• Slope (m) = 2
• Y-intercept (b) = 3
• Substitute x to find y

📊 In a Table

x	0	1	2	3
y	1	4	7	10

Find slope: Δy/Δx = 3/1 = 3
Y-intercept: when x=0, y=1

📈 Graphically

Find slope: count rise over run
Y-intercept: where line crosses y-axis

💬 Verbal Description

"A plumber charges $50 to show up, then $25 for each hour of work."

Translate to equation:
y = 25x + 50
Slope = $25/hr · Y-int = $50

🔍 How to Find Key Properties From Each Representation

Property	📐 Equation	📊 Table	📈 Graph	💬 Verbal
Slope / Rate of Change	The value of m in y = mx + b	Calculate Δy ÷ Δx between rows	Count rise over run between two points	Look for "per" or "each" — that's the rate
Y-Intercept	The value of b in y = mx + b	Find y when x = 0	Where line crosses the y-axis	The "starting" or "flat fee" amount
Is It a Function?	Yes, if written y = f(x)	No repeated x-values	Passes the VLT	Each input gives one output

⚔️ Side-by-Side Comparison Example

Function A — Table

x	0	2	4	6
y	−1	5	11	17

📐 Slope: Δy/Δx = 6/2 = 3
📍 Y-intercept: x=0 → y = −1
✏️ Equation: y = 3x − 1

Function B — Equation

y = 5x + 2

📐 Slope: m = 5
📍 Y-intercept: b = 2
✏️ Already in slope-intercept form

Property	Function A (Table)	Function B (Equation)	Winner
Slope / Rate of Change	3	5	B is steeper ↗
Y-Intercept	−1	2	B starts higher
Value when x = 4	11	22	B is greater
Linear?	✅ Yes (constant Δy)	✅ Yes (y=mx+b form)	Both linear

🪜 Steps to Compare Two Functions

Identify the Form

Is each function given as an equation, table, graph, or description?

Find the Slope

Extract rate of change from each representation using the right method

Find the Y-Intercept

Find where each function starts (when x = 0)

Compare & Conclude

Use the properties to answer questions about which is greater, faster, or higher

💬 Translating Verbal Descriptions

Key Phrases → Slope

"per hour" → m = hourly rate
"each item" → m = price per item
"for every" → m = rate
"increases by" → m = amount

Key Phrases → Y-Intercept

"starts at" → b = starting value
"flat fee" → b = one-time cost
"initial amount" → b
"before any work" → b

Quick Check

"A taxi charges $3 per mile plus a $5 base fee."

Slope = $3/mile
Y-intercept = $5
Equation: y = 3x + 5

🎯 The Goal

Compare the rate of change AND y-intercept of two functions — even when they are shown in different forms.

📏 Rate of Change

Equation → m value
Table → Δy ÷ Δx
Graph → rise ÷ run
Words → "per" or "each" phrase

📍 Y-Intercept

Equation → b value
Table → y when x = 0
Graph → crossing point on y-axis
Words → starting/flat fee amount

⚠️ Watch Out

A higher y-intercept doesn't mean faster growth! Always compare the SLOPE separately from the starting point.

🎯 Check Your Understanding — 8.F.2

Score: 0 / 7

Function A is given by the equation y = 4x + 1. Function B is shown in a table where x = 0, 1, 2 gives y = 3, 8, 13. Which function has a greater rate of change?

Function A, because its slope is 4

Function B, because its slope is 5 (Δy/Δx = 5/1)

They are equal because both start at positive y-values

Function A, because the equation is easier to read

How do you find the y-intercept of a function shown in a table?

Find the largest y-value in the table

Find the y-value when x = 0

Calculate Δy ÷ Δx for the first two rows

Find the x-value when y = 0

A graph shows a line crossing the y-axis at (0, 6) with a slope of 2. A verbal description says: "A candle starts 10 inches tall and burns down 1 inch per hour." Which function has the greater y-intercept?

The graph, because its slope is positive

The verbal description, because it starts at 10 inches (y-intercept = 10)

The graph, because 6 > −1

They have the same y-intercept

Function A: y = 3x − 2. Function B: passes through (0, 4) and (2, 10) on a graph. Which has the greater rate of change?

Function A, because its slope is 3

Function B, because its slope is 3… wait — they're EQUAL! Both have slope = 3

Function B, because it has a higher y-intercept

Function A, because equations always have higher slopes than graphs

A plumber charges $40 per hour with no call-out fee. An electrician charges $25 per hour but has a $60 call-out fee. After 4 hours, who charges more?

The plumber — 4 hrs: $160 vs electrician: $25(4) + 60 = $160… they're equal!

The electrician, because the call-out fee makes it always higher

The plumber, because $40/hr is always more than $25/hr

Cannot be determined without a graph

Which of the following correctly describes how to find the slope from a graph?

Divide the x-coordinate by the y-coordinate of any point

Find where the line crosses the x-axis

Count how many units the line goes up (rise) for every unit it goes right (run)

Multiply the y-intercept by the x-intercept

Function A has a slope of 2 and y-intercept of 10. Function B has a slope of 6 and y-intercept of −4. At what x-value will Function B's output EQUAL Function A's output?

x = 2

x = 3

x = 3.5 (2x+10 = 6x−4 → 14 = 4x → x = 3.5)

x = 4

✏️ Open Response: You are comparing two savings plans. Plan A is shown in a table; Plan B is described verbally. Explain step-by-step how you would determine which plan saves more money after 10 weeks. What properties would you look for?