Lesson 2: Standard Form of a Quadratic Function

Property

For the graph of $y = ax^2 + bx + c$, the $x$-coordinate of the vertex is

To find the $y$-coordinate of the vertex, substitute the value of $x_v$ into the equation for $y$.

Examples

• To find the vertex of $y = x^2 + 8x + 10$, we identify $a=1$ and $b=8$. The x-coordinate is $x_v = \frac{-8}{2(1)} = -4$. The y-coordinate is $y_v = (-4)^2 + 8(-4) + 10 = -6$. The vertex is $(-4, -6)$.

• For the graph of $y = -2x^2 - 12x + 5$, we have $a=-2$ and $b=-12$. The x-coordinate of the vertex is $x_v = \frac{-(-12)}{2(-2)} = -3$. The y-coordinate is $y_v = -2(-3)^2 - 12(-3) + 5 = 23$. The vertex is $(-3, 23)$.

Property

The graph of the function $f(x) = ax^2 + bx + c$ is a parabola where:

• the axis of symmetry is the vertical line $x = -\frac{b}{2a}$.
• the vertex is a point on the axis of symmetry, so its $x$-coordinate is $-\frac{b}{2a}$.
• the $y$-coordinate of the vertex is found by substituting $x = -\frac{b}{2a}$ into the quadratic equation.

Examples

• For $f(x) = x^2 - 6x + 11$, the axis of symmetry is $x = -\frac{-6}{2(1)} = 3$. The vertex is $(3, f(3))$, which is $(3, 2)$.
• For $f(x) = -2x^2 - 8x - 5$, the axis of symmetry is $x = -\frac{-8}{2(-2)} = -2$. The vertex is $(-2, f(-2))$, which is $(-2, 3)$.
• For $f(x) = 4x^2 - 8$, the axis of symmetry is $x = -\frac{0}{2(4)} = 0$. The vertex is $(0, f(0))$, which is $(0, -8)$.

Explanation

The axis of symmetry is an invisible vertical line that splits the parabola into two perfect mirror images. The vertex is the parabola's turning point (either the very bottom or very top), and it always sits right on this line.