Solution: The vertex form of a parabola $ y = ax^2 + bx + c $ has its vertex at $ x = -\frac{b}{2a} $. Here, $ a = -2 $, $ b = 8 $, so $ x = -\frac{8}{2(-2)} = 2 $. Substituting $ x = 2 $ into the equation: $ y = -2(2)^2 + 8(2) - 5 = -8 + 16 - 5 = 3 $. Thus, the vertex is at $ (2, 3) $. \boxed{(2, 3)}
Mastering Parabolas: Finding the Vertex Using Vertex Form
Understanding the shape and position of a parabola is essential in algebra and calculus. One key feature is the vertex — the highest or lowest point depending on the parabola’s direction. For quadratic equations in standard form $ y = ax^2 + bx + c $, calculating the vertex coordinates has a simple, efficient solution known as the vertex formula.
What Is the Vertex of a Parabola?
The vertex of a parabola described by $ y = ax^2 + bx + c $ occurs at $$ x = - rac{b}{2a} $$ This formula is derived from completing the square and reveals the axis of symmetry. Once the $ x $-coordinate is found, substituting it back into the equation gives the $ y $-coordinate, pinpointing the exact vertex.
Applying the Vertex Formula
Let’s apply this method step-by-step using a concrete example.
Given the quadratic function: $$ y = -2x^2 + 8x - 5 $$ Here, the coefficients are $ a = -2 $, $ b = 8 $, and $ c = -5 $.
- Find the $ x $-coordinate of the vertex:
$$ x = - rac{b}{2a} = - rac{8}{2(-2)} = - rac{8}{-4} = 2 $$
- Determine the $ y $-coordinate by substituting $ x = 2 $:
$$ y = -2(2)^2 + 8(2) - 5 = -2(4) + 16 - 5 = -8 + 16 - 5 = 3 $$
So, the vertex of this parabola is at $ (2, 3) $.
Why This Method Matters
Using $ x = - rac{b}{2a} $ avoids the labor of completing the square manually and provides a direct route to the vertex. This efficiency is especially useful when analyzing symmetry, gradients, or optimization problems involving parabolas.
Final Vertex Summary
Vertex: $ (2, 3) $ Axis of Symmetry: $ x = 2 $
Mastering this vertex form and formula empowers students and professionals alike in algebra, physics, and engineering — where parabolic motion and curve modeling are key.
Try it yourself: Try any quadratic in standard form — apply $ x = - rac{b}{2a} $ to find the vertex quickly using the vertex formula!
oxed{(2, 3)}
