Graphs and equations

Japanese school year: Math I

What you learn

You will learn how to determine quadratic equations from geometric conditions and understand how parabolas shift on the coordinate plane. This knowledge is applied in trajectory analysis and modeling data. Prior familiarity with finding vertices and basic coordinate geometry is recommended.

Go to practice

Key points

Use this vertex form to set up the quadratic function when the vertex $(p, q)$ or axis $x = p$ is known. Plug in another point to determine coefficient $a$ .

y = a(x - p)^2 + q

When three distinct points on the parabola are given, substitute them into this general form to create and solve a system of equations for $a, b$ , and $c$ .

y = ax^2 + bx + c

This factored form is convenient when the two $x$ -intercepts $(\alpha, 0)$ and $(\beta, 0)$ where the graph crosses the $x$ -axis are known.

y = a(x - \alpha)(x - \beta)

This equation represents the parabola $y = ax^2$ shifted horizontally by $p$ and vertically by $q$ .

y - q = a(x - p)^2

Choose a set to practice.

Practice 1Coming soon