MCF3M Functions & Applications · Unit 3 — Quadratic Functions: Models and Forms
Lesson 3.2
Relating the Standard and Factored Forms
Move between f(x) = ax² + bx + c and f(x) = a(x − r)(x − s), and use each form to find zeros, the axis of symmetry, the vertex and the y-intercept.
Estimated time: 45 minutes
Learning goals
- recognize standard form and factored form
- convert between the forms when factoring is possible
- identify zeros from factored form and the y-intercept from standard form
- find the axis of symmetry and vertex
- explain why equivalent forms have the same graph
- solve contextual problems using the most useful form
Step 1 of 4
Warm Up
Success criteria
Step 2 of 4
Learn
Look first
See it — the axis of symmetry
Look first. The formula comes last.
f(x) = (x − 2)(x − 6)
Meet the parabola
See this U-shaped curve? It's called a parabola.
See it — the vertex
The vertex is the turning point. It's where the parabola changes direction.
f(x) = (x − 2)(x − 6)
Tap the lowest point of the curve.
The vertex is always on the axis of symmetry.
See it — zeros and x-intercepts
Zeros are the x-values where the graph touches or crosses the x-axis. At these points, y = 0.
Tap a factor
f(x) =
Each bracket gives one zero.
Why? If one bracket equals 0, the whole product is 0 — so y = 0.
Your turn: g(x) = (x + 3)(x − 1)
What are the zeros?
Standard form vs factored form
Different equations. Same parabola. Each form makes different information easier to see.
The curve didn't move. Expand (x − 2)(x − 6) and you get x² − 8x + 12 — the same function.
Two ways to write one parabola
Standard form: f(x) = ax² + bx + c. The constant c is the y-intercept, because f(0) = c.
Factored form: f(x) = a(x − r)(x − s). The zeros are x = r and x = s, because a bracket equal to 0 makes the whole product 0.
In both forms, a tells the direction: a > 0 opens up, a < 0 opens down.
The axis of symmetry is halfway between the zeros. Substitute that x-value to find the vertex.
Axis: x = (r + s) ÷ 2
Worked example
Example A: f(x) = (x − 2)(x − 6). Write it in standard form and find its key features.
Expand
x² − 6x − 2x + 12 = x² − 8x + 12
Zeros
x − 2 = 0 and x − 6 = 0
x = 2, x = 6
Axis
x = (2 + 6) ÷ 2 = 4
Vertex
f(4) = (2)(−2) = −4 → (4, −4)
y-intercept
c = 12 in standard form
(0, 12)
Answer: x² − 8x + 12; zeros 2, 6; axis x = 4; vertex (4, −4); y-intercept 12
Worked example
Example B: g(x) = −x² + 6x + 7. Write it in factored form and find its key features.
Factor out −1
g(x) = −(x² − 6x − 7)
Factor the trinomial
Product −7, sum −6: −7 and 1
g(x) = −(x − 7)(x + 1)
Zeros
x = 7, x = −1
Axis
x = (7 + (−1)) ÷ 2 = 3
Vertex
g(3) = −9 + 18 + 7 = 16 → (3, 16)
y-intercept
(0, 7)
Answer: −(x − 7)(x + 1); zeros 7, −1; axis x = 3; vertex (3, 16) — a maximum
Two Forms, One Parabola
Pick a function or move the zeros. The graph and both equations always describe the same parabola.
Factored form
f(x) = (x − 2)(x − 6)
Standard form
f(x) = x² − 8x + 12
- Zeros: x = 2 and x = 6 (read from factored form — flip the signs)
- Axis: x = (2 + 6) ÷ 2 = 4
- Vertex: f(4) = −4 → (4, −4)
- y-intercept: (0, 12) (the constant in standard form)
- Opens: up (a > 0) — vertex is a minimum
Challenges
1. Which standard form matches (x − 4)(x + 3)?
2. You need the zeros fast. Which form is most useful?
3. You need the y-intercept fast. Which form is most useful?
4. Graph-to-equation: which function is graphed below?
Choose the equation.
Not every quadratic factors
Real-world idea
Example C — Water rocket
A water rocket is launched from the ground. Its height is h(t) = −5t² + 20t metres after t seconds.
When does it land, and how high does it go?
Worked example
h(t) = −5t² + 20t. Find when the rocket lands, its maximum height, and the practical domain and range.
Factor
Common factor −5t
h(t) = −5t(t − 4)
Zeros
t = 0 (launch) and t = 4 (landing)
Axis
t = (0 + 4) ÷ 2 = 2
Maximum
h(2) = −5(2)(−2) = 20
Opens down
a = −5 < 0: the rocket rises, then falls — the vertex is a maximum.
Context
Time can't be negative and the rocket stops at the ground, so 0 ≤ t ≤ 4 and 0 ≤ h ≤ 20.
Answer: Lands at 4 s; max 20 m at 2 s; domain 0 ≤ t ≤ 4; range 0 ≤ h ≤ 20
Graph features recap
One parabola, f(x) = (x + 1)(x − 3). Tap each feature to see it before you practise.
Step 3 of 4
Practice
Build it
Apply it
Challenge
Explain it
Step 4 of 4
Mastery Check
Lesson 3.2 mastery check
5 questions · no hints during the check. Answer them all, then check your score — you can retry as many times as you like.