Graphs & equations

Japanese school year: Junior high 3

What you learn

You will explore quadratic functions of the form y=ax², examining how the coefficient affects the opening and width of parabolas. These curves are widely used to model accelerating motion, such as freely falling objects. Prior knowledge of squaring numbers and plotting coordinates is helpful for understanding this topic.

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Key points

This is the standard equation where $y$ is proportional to the square of $x$ . The constant $a$ is the proportionality constant, and the graph forms a parabola with its vertex at the origin $(0, 0)$ .

y = ax^2 \quad (a \neq 0)

This identity shows that substituting $-x$ yields the exact same $y$ -value as $x$ . Consequently, the parabola is symmetric with respect to the $y$ -axis.

a(-x)^2 = ax^2

When $a > 0$ , the parabola opens upward, and the origin is the lowest point (minimum). A larger value of $a$ makes the opening narrower.

a > 0 \implies y \ge 0

When $a < 0$ , the parabola opens downward, and the origin is the highest point (maximum). A larger absolute value $|a|$ makes the opening narrower.

a < 0 \implies y \le 0

Choose a set to practice.

Practice 1Coming soon