Unit 1 · 9th Grade Mathematics

Functions, Graphs
& Features.

Study guide for all 11 lessons — with practice questions after every sub-unit.

Start studying Topic B

Unit 1 — Lesson Video

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Topic A: Features of Functions

Lessons 1–7.

01 — Lesson 1

Modeling Situations Graphically

Represent real-world situations as graphs by choosing the right variables and scale.

  • Independent variable (x-axis): what you control — usually time.
  • Dependent variable (y-axis): what changes because of x.
  • Steep: fast change. Flat: no change. Downward: decreasing.
  • Identify key events: when does the situation start, change, and end?
Example: Sylvia bikes 8 miles home — starts fast, stops at a store (flat section), then finishes slowly. Graph: steep rise → flat → slow rise to 8.
Practice — Lesson 1
Q1 · Multiple Choice

Sylvia's bike ride graph has a flat section in the middle. What does the flat section mean?

Q2 · Multiple Choice

On a distance-vs-time graph, which is the independent variable?

02 — Lesson 2

Functions & Function Notation

A function assigns exactly one output to every input.

  • Vertical line test: if a vertical line hits the graph twice → not a function.
  • f(x) is read "f of x" — it replaces y in an equation.
  • f(3) means: substitute x = 3, then calculate.
f(x) = 2x + 1  →  f(3) = 2(3)+1 = 7  ·  f(0) = 1
Practice — Lesson 2
Q3 · Short Answer

If f(x) = 3x − 4, what is f(5)?

Q4 · Multiple Choice

Which of the following is NOT a function?

03 — Lesson 3

Intercepts in Context

  • Y-intercept = f(0): the output when input is zero — usually the starting value.
  • X-intercept: when f(x) = 0 — when the output hits zero (runs out, ends, lands).
Candle: f(t) = 8 − 2t
f(0) = 8 → starts 8 in. tall  ·  8−2t=0 → t=4 → burns out at 4 hrs
Practice — Lesson 3
Q5 · Short Answer

Using f(t) = 10 − 2t, at what hour does the candle burn out? (Find the x-intercept.)

Q6 · Multiple Choice

For f(t) = 10 − 2t, what does f(0) = 10 tell us in context?

04 — Lesson 4

Domain & Range

Domain = all valid inputs (x). Range = all possible outputs (y).

  • On a graph: domain = left-to-right; range = bottom-to-top.
  • In context: ask if negatives make sense (can't have −3 hours).
  • Written as inequalities, e.g. 0 ≤ x ≤ 40.
Student works 0–40 hrs/week at $10/hr:
Domain: 0 ≤ x ≤ 40  ·  Range: 0 ≤ y ≤ 400
Practice — Lesson 4
Q7 · Multiple Choice

A fundraiser earns money only from 8am to 3pm. If x = hours since midnight, what is the domain?

Q8 · Multiple Choice

Which best describes the range of a function?

05 — Lesson 5

Rate of Change

How fast the output changes per unit of input — the slope between two points.

  • Formula: (y₂ − y₁) ÷ (x₂ − x₁)
  • With notation: [f(b) − f(a)] ÷ (b − a)
  • Positive: increasing  ·  Negative: decreasing  ·  Zero: flat
Car at mile 100 (t=2) and mile 160 (t=4):
(160−100)÷(4−2) = 60÷2 = 30 mph
Practice — Lesson 5
Q9 · Multiple Choice

A runner is at mile 2 at t = 20 min and mile 6 at t = 44 min. What is her average rate of change?

Q10 · Short Answer

If f(x) = 2x + 6, what is the x-intercept? (Set f(x) = 0 and solve.)

06 — Lesson 6

Describing & Sketching Functions

Use all features together to fully describe what a function is doing.

  • Domain & Range
  • Y-intercept & X-intercept(s)
  • Increasing / Decreasing intervals
  • Maximum / Minimum values
  • Where f(x) > 0 or f(x) < 0 (above/below x-axis)
"Increases on 0 ≤ x ≤ 3, max at f(3)=9, decreases on 3 ≤ x ≤ 6. X-intercepts at x=0 and x=6."
Practice — Lesson 6
Q11 · Multiple Choice

A function increases on 1 ≤ x ≤ 5, then decreases on 5 ≤ x ≤ 9. Where is the maximum?

Q12 · Multiple Choice

When is a function f(x) positive?

07 — Lesson 7

Analyzing & Modeling Situations

Apply all Topic A skills to build or interpret a graph from a real-world description.

Step-by-step process
  • 1. Identify the quantities (what varies, what causes it).
  • 2. Label axes with variables and units.
  • 3. Choose a sensible scale for the data range.
  • 4. Identify key events: start, changes, end.
  • 5. Sketch the shape.  6. Label key points.
Practice — Lesson 7
Q13 · Multiple Choice

You're graphing the temperature of coffee as it cools over 30 minutes. What should go on the x-axis?

Q14 · Multiple Choice

When sketching the coffee cooling graph, what shape would you expect?

Topic B: Nonlinear Functions

Lessons 8–11.

08 — Lesson 8

Quadratic Functions

Creates a U-shaped curve called a parabola. Models situations that rise then fall (or vice versa).

  • Opens up (∪): has a minimum. Opens down (∩): has a maximum.
  • Vertex: the highest or lowest point of the parabola.
  • Axis of symmetry: vertical line through the vertex.
  • X-intercepts (zeros/roots): 0, 1, or 2 crossings.
  • Grows faster than linear as x gets large.
Example: A kicked ball rises to a peak (vertex = max height), then falls to the ground (x-intercept).
Practice — Lesson 8
Q15 · Multiple Choice

A ball is kicked into the air. Which function type best models its height over time?

Q16 · Multiple Choice

A parabola opens downward (∩). What does this tell us?

09 — Lesson 9

Exponential Functions

Grows or shrinks by a constant multiplying factor — not by adding. Faster than linear or quadratic.

  • Form: f(x) = a · bˣ  (a = start, b = growth/decay factor)
  • b > 1: growth — curves steeply upward
  • 0 < b < 1: decay — falls toward zero, never reaches it
  • Y-intercept: always f(0) = a (the starting value)
  • No x-intercept for standard exponential functions
100 bacteria doubling each hour: f(t) = 100·2ᵗ
After 3 hrs: f(3) = 100·8 = 800
Practice — Lesson 9
Q17 · Multiple Choice

A bacteria colony starts with 50 and doubles every hour. Which function models this?

Q18 · Short Answer

Using f(t) = 50 · 2ᵗ, how many bacteria are there after 3 hours?

10 — Lesson 10

Systems of Functions

Two or more functions on the same graph. Where they meet = the solution.

  • Solution: (x, y) where f(x) = g(x) — same output for the same input.
  • Graphically: the intersection point(s).
  • In context: the moment two quantities are equal.
  • Can mix any function types: linear + quadratic, linear + exponential, etc.
f(x) = 3x and g(x) = x + 4:
3x = x + 4 → 2x = 4 → x = 2, point (2, 6)
Practice — Lesson 10
Q19 · Short Answer

Find the x-value where f(x) = 2x and g(x) = x + 5 intersect. (Set them equal and solve.)

Q20 · Multiple Choice

What does the intersection point of two function graphs represent in context?

11 — Lesson 11

Parent Functions & Representing Situations

The simplest version of each function family — the "base model" before any changes.

  • Linear f(x) = x: straight line, constant rate of change.
  • Quadratic f(x) = x²: parabola, rate of change increases.
  • Exponential f(x) = 2ˣ: rapid growth curve, multiplies by factor.
How to identify
  • Constant rate of change → Linear
  • Goes up then down / has a vertex → Quadratic
  • Output multiplies repeatedly → Exponential
Practice — Lesson 11
Q21 · Multiple Choice

Which parent function has a constant rate of change?

Q22 · Multiple Choice

A town's population triples every 10 years. Which parent function best models this?