Graphing Quadratic Functions

Property The equation of the axis of symmetry for a quadratic function in the form $f(x) = ax^2 + bx + c$ is $x = \frac{b}{2a}$. To find the vertex, substitute this $x$ value back into the function.

Explanation Think of the axis of symmetry as the parabola's spine! This magic formula pinpoints the center line, leading you directly to the vertex—the function's peak or valley. It's the ultimate shortcut to sketching a perfect parabola without plotting a million points. Find the center, find the vertex, and you're halfway to a flawless graph!

Examples Find the vertex of $y = x^2 + 6x + 8$. Axis of symmetry: $x = \frac{6}{2(1)} = 3$. Vertex: $y = ( 3)^2 + 6( 3) + 8 = 1$. The vertex is $( 3, 1)$. Find the vertex of $y = 2x^2 + 12x + 10$. Axis of symmetry: $x = \frac{12}{2(2)} = 3$. Vertex: $y = 2( 3)^2 + 12( 3) + 10 = 8$. The vertex is $( 3, 8)$. Find the vertex of $y = 4x^2 + 8$. Since $b=0$, the axis of symmetry is $x = \frac{0}{2(4)} = 0$. The vertex is $(0, 8)$.