Lesson 1: Parabolas

Property

The graph of a quadratic equation $y = ax^2 + bx + c$ is called a parabola.
For the graph of $y = ax^2$: The parabola opens upward if $a > 0$ and downward if $a < 0$. The magnitude of $a$ determines how wide or narrow the parabola is.
For the graph of $y = x^2 + c$: The graph is shifted upward by $c$ units if $c > 0$ and downward if $c < 0$.
For the graph of $y = ax^2 + bx + c$, the $x$-coordinate of the vertex is $x_v = \frac{-b}{2a}$.

Examples

• For the parabola $y = x^2 - 6x + 5$, the vertex's x-coordinate is $x_v = \frac{-(-6)}{2(1)} = 3$. To find the y-coordinate, plug $x=3$ back in: $y = (3)^2 - 6(3) + 5 = 9 - 18 + 5 = -4$. The vertex is at $(3, -4)$.

• The graph of $y = -3x^2$ opens downward because $a=-3$ is less than 0. It is also narrower than the basic parabola $y=x^2$ because the magnitude of $a$ is greater than 1.

Property

A quadratic equation $y = ax^2 + bx + c$, $a \neq 0$, can be written in the vertex form

where the vertex of the graph is $(x_v, y_v)$. To convert from standard form, complete the square.

Examples

• The equation $y = 3(x - 5)^2 + 1$ is in vertex form. By comparing it to $y = a(x - x_v)^2 + y_v$, we can see the vertex is at $(5, 1)$.

• For the equation $y = -4(x + 2)^2 - 7$, we can rewrite it as $y = -4(x - (-2))^2 - 7$. The vertex is at $(-2, -7)$.