🎯 Learning Objectives
- The three forms of a quadratic: and what each one tells you.
- How to find the vertex (maximum or minimum point).
- How to find the axis of symmetry, x-intercepts, and y-intercept.
- When the parabola opens up vs. down.
- Strategies for matching parabolas to equations.
- Practice questions with worked solutions.
1. The 3 Forms of a Quadratic
STANDARD form
y = a·x² + b·x + c
VERTEX form → vertex at (h, k)
y = a·(x − h)² + k
FACTORED form → x-intercepts at p and q
y = a·(x − p)·(x − q)
2. What Each Form Tells You at a Glance
Form | Shows Directly |
Standard (ax² + bx + c) | y-intercept = c |
Vertex (a(x − h)² + k) | Vertex = (h, k) |
Factored (a(x − p)(x − q)) | x-intercepts at x = p and x = q |
3. Up or Down?: The Sign of a
❓ Sign of the leading coefficient a? | ||
a > 0 ▼ | a < 0 ▼ | |
Opens UP (smile): has a MINIMUM | Opens DOWN (frown): has a MAXIMUM | |
4. Finding the Vertex from Standard Form
Then plug into the equation to get y
x-coordinate of vertex = −b / (2·a)
This same x-value is the AXIS OF SYMMETRY: the vertical line that splits the parabola in half.
5. Anatomy of a Parabola
Feature | How to Find It |
Vertex | Use x = −b/(2a), then compute y. |
Axis of symmetry | x = −b/(2a) (a vertical line through the vertex). |
y-intercept | Plug x = 0 → y = c. |
x-intercepts (zeros) | Set y = 0; factor or use quadratic formula. |
Maximum / Minimum | The y-coordinate of the vertex. |
📝 Example: Sketch y = x² − 6x + 5
Opens: a = 1 > 0 → UP.
Vertex x: x = −b/(2a) = 6/2 = 3.
Vertex y: y = 9 − 18 + 5 = −4. So vertex = (3, −4).
y-intercept: (0, 5).
x-intercepts: (x − 1)(x − 5) = 0 → x = 1 and x = 5.
6. Strategies
⚡ STRATEGY 1: Identify the form first
Before doing anything, ask: is this in standard, vertex, or factored form? You may already have the answer (vertex or x-intercepts) in plain sight.
⚡ STRATEGY 2: Symmetry shortcut for vertex
If you know the two x-intercepts (p and q), the vertex's x-coordinate is exactly halfway: x = (p + q)/2.
⚡ STRATEGY 3: Plug 0 in for y-intercept
Always set x = 0 and compute y. Skip every other step when only the y-intercept is asked.
⚠️ Common Mistakes
- Forgetting that vertex form (x − h)² has a MINUS sign: so y = (x − 4)² has vertex at x = 4, not x = −4.
- Confusing axis of symmetry (a vertical line) with the vertex (a point).
- Reading 'maximum' as the x-value when it's actually the y-value.
- Forgetting the sign of a controls direction (up/down).
7. Summary
📌 Key Takeaways
- Standard: y = ax² + bx + c. y-int = c.
- Vertex form: y = a(x − h)² + k. Vertex = (h, k).
- Factored form: y = a(x − p)(x − q). x-ints at p, q.
- a > 0: Opens up (minimum).
- a < 0: Opens down (maximum).
- Vertex x: x = −b/(2a).
- Symmetry strategy: Vertex x = (p + q)/2 if you know x-intercepts.
8. Practice: 10 Questions
Q1. The vertex of y = (x − 3)² + 4 is…
A) (3, 4) B) (−3, 4) C) (3, −4) D) (−3, −4)
Hint: Vertex form: (h, k) = (3, 4).
Q2. The y-intercept of y = 2x² − 7x + 11 is…
A) 2 B) −7 C) 11 D) 0
Hint: y-int = c = 11.
Q3. The parabola y = −3(x − 2)(x + 4) opens…
A) Up B) Down C) Sideways D) Cannot tell
Hint: a = −3 < 0 → DOWN.
Q4. The x-coordinate of the vertex of y = x² − 8x + 12 is…
A) 4 B) −4 C) 8 D) −8
Hint: x = −b/(2a) = 8/2 = 4.
Q5. The x-intercepts of y = (x − 2)(x + 5) are…
A) 2 and 5 B) −2 and 5 C) 2 and −5 D) −2 and −5
Hint: Set each factor to 0: x = 2 or −5.
Q6. The minimum value of y = x² + 4x + 7 is…
A) 4 B) 3 C) 2 D) 7
Hint: Vertex x = −2, y = 4 − 8 + 7 = 3.
Q7. The axis of symmetry of y = x² − 6x + 5 is…
A) x = 3 B) x = −3 C) x = 6 D) y = 3
Hint: x = −b/(2a) = 6/2 = 3.
Q8. A parabola has x-intercepts at 1 and 7. The x-coordinate of its vertex is…
A) 3 B) 4 C) 5 D) 6
Hint: Halfway: (1 + 7)/2 = 4.
Q9. Which equation has a maximum value?
A) y = x² + 1 B) y = (x−2)² + 3 C) y = −2x² + 5 D) y = 3x² − 7
Hint: Maximum requires a < 0. Only C has negative a.
Q10. The vertex of y = 2(x + 1)² − 8 is…
A) (1, 8) B) (−1, −8) C) (1, −8) D) (−1, 8)
Hint: (x − h) means h = −1. Vertex (−1, −8).
Answer Key & Worked Solutions
# | Answer | Reasoning |
Q1. | A) (3, 4) | Read directly from vertex form. |
Q2. | C) 11 | y-int = c. |
Q3. | B) Down | a < 0. |
Q4. | A) 4 | −b/(2a) = 4. |
Q5. | C) 2 and −5 | Factored form gives roots. |
Q6. | B) 3 | Min y at vertex. |
Q7. | A) x = 3 | Same as vertex x. |
Q8. | B) 4 | Midpoint of x-ints. |
Q9. | C) y = −2x² + 5 | Negative a → opens down → max. |
Q10. | B) (−1, −8) | Watch the sign inside (x + 1) = (x − (−1)). |