Finding the Vertex of a Quadratic Function

Quadratic Functions and Their Graphs

Vertex of a Parabola

The vertex of a quadratic function is the highest or lowest point on its graph, depending on whether the parabola opens upward or downward. For a function in the form f(x) = ax^2 + bx + c, the vertex can be found using a specific formula.

• Standard Form: $f(x) = ax^2 + bx + c$

• Vertex Formula: The x-coordinate of the vertex is $x = -\frac{b}{2a}$

• To find the y-coordinate, substitute this x-value back into the function: $y = f\left(-\frac{b}{2a}\right)$

Example: Find the Vertex of $f(x) = 2x^2 - 16x + 37$

• Identify coefficients: $a = 2$, $b = -16$, $c = 37$

• Find x-coordinate: $x = -\frac{-16}{2 \times 2} = \frac{16}{4} = 4$

• Find y-coordinate: Substitute $x = 4$ into the function: $f(4) = 2(4)^2 - 16(4) + 37$ $= 2 \times 16 - 64 + 37$ $= 32 - 64 + 37$ $= (32 + 37) - 64$ $= 69 - 64$ $= 5$

• Vertex: $(4, 5)$

Summary Table: Vertex of a Quadratic Function

Form	Vertex Formula	Example	Vertex
$f(x) = ax^2 + bx + c$	$\left(-\frac{b}{2a},\ f\left(-\frac{b}{2a}\right)\right)$	$f(x) = 2x^2 - 16x + 37$	$(4, 5)$

• Application: The vertex represents the minimum point of the parabola since $a > 0$ (the parabola opens upward).