Parabolas & lines

Japanese school year: Junior high 3

What you learn

Learn to find the coordinates of intersection points between a parabola and a line by solving a system of equations. This technique is frequently used to analyze geometric relationships between graphs. Before starting, it is helpful to have a solid understanding of linear functions and solving quadratic equations.

Go to practice

Key points

This equation determines the $x$ -coordinates of the intersection points between the parabola $y = ax^2$ and the line $y = mx + n$ . Setting the two expressions equal produces a quadratic equation to solve.

ax^2 = mx + n

This formula gives the slope $m$ of a line intersecting the parabola $y = ax^2$ at two points with $x$ -coordinates $\alpha$ and $\beta$ . The slope equals $a(\alpha + \beta)$ .

m = a(\alpha + \beta)

This formula determines the $y$ -intercept $n$ of the line intersecting the parabola $y = ax^2$ at points with $x$ -coordinates $\alpha$ and $\beta$ . It is calculated directly as $-a\alpha\beta$ .

n = -a\alpha\beta

Choose a set to practice.

Practice 1Coming soon