Grade 11 · Lesson 12
Quadratic graphs
Three useful forms of the same parabola: y = ax² + bx + c, y = a(x + b)² + c, and y = (x + a)(x + b).
General form
The expanded form y = ax² + bx + c. Here a sets width and direction, c is the y-intercept, and the axis of symmetry is x = −b/(2a).
Live graph
y = x² − 2x − 3
Axis: x = 1 · Vertex: (1, -4). Y-intercept: -3.
Key idea
- c is where the curve meets the y-axis.
- Axis x = −b/(2a); vertex lies on that axis.
- a > 0 opens up; a < 0 opens down.
Table of values
Compute y for each x, then plot. The axis of symmetry sits halfway between matching y-values.
| x | -3 | -2 | -1 | 0 | 1 | 2 | 3 |
|---|---|---|---|---|---|---|---|
| y | 12 | 5 | 0 | -3 | -4 | -3 | 0 |
(1, -4) lies on the curve
Which form when?
Same parabola, different “readouts.” Pick the form that makes the feature you need obvious.
ax² + bx + c
Best for y-intercept and expanding/simplifying. Find the axis with −b/(2a), then the vertex.
a(x + b)² + c
Best for sketching from the turning point. Vertex (−b, c) is written into the equation.
(x + a)(x + b)
Best when roots matter. Cross the x-axis at −a and −b, then find the midpoint for the axis.
Check yourself · y = ax² + bx + c
Type your answer, then check. You can write x^2 or x²; spaces are flexible.
Question 1 of 3
What is the y-intercept of y = 2x² − 3x + 5?
Remember
- ax² + bx + c → c is y-intercept; axis x = −b/(2a).
- a(x + b)² + c → vertex (−b, c); a sets width/direction.
- (x + a)(x + b) → roots −a and −b; y-intercept = a·b.