Grade 10 · Lesson 21
Quadratic graphs
First master parabolas through the origin — y = ax². Then shift them with the constant — y = ax² + b.
Parabolas through the origin
Every equation y = ax² is a parabola that passes through (0, 0). The number a controls how wide it is and whether it opens up or down.
Live graph
y = x²
1
This parabola opens upwards (a > 0). Larger |a| makes it same width as y = ±x².
Key idea
- Always through the origin — when x = 0, y = 0.
- a > 0 opens up; a < 0 opens down.
- Larger |a| → narrower; smaller |a| → wider.
Table of values
Pick x-values, compute y = ax², then plot. Notice the symmetry about the y-axis.
| x | -3 | -2 | -1 | 0 | 1 | 2 | 3 |
|---|---|---|---|---|---|---|---|
| y | 9 | 4 | 1 | 0 | 1 | 4 | 9 |
x = 2
(2, 4) lies on the curve
By symmetry, so does (-2, 4).
Check yourself · y = ax²
Type your answer, then check. Spelling and spaces are flexible — you can write x^2 or x².
Question 1 of 3
A parabola through the origin opens upwards with a = 2. Write its equation.
Remember
- y = ax² → through (0, 0); a sets width and up/down.
- y = ax² + b → same shape; b is the vertex y-value.
- Symmetric about the y-axis. Plot vertex, then use a table.