Vertex form of a quadratic function
Convert quadratic functions to vertex form y=a(x-h)²+k in Grade 10 algebra by completing the square, identify vertex (h,k) and axis of symmetry, and graph with transformations.
Key Concepts
The vertex form of a quadratic function is $f(x) = a(x h)^2 + k$ where $a \neq 0$, and $(h, k)$ represents the vertex of the parabola. The value of $h$ indicates a horizontal shift and $k$ indicates a vertical shift.
In $f(x) = (x 5)^2 + 2$, the vertex is at $(5, 2)$, representing a shift 5 units right and 2 units up. For the function $g(x) = 2(x + 1)^2 4$, the vertex is at $( 1, 4)$. For $y = x^2 + 6$, which is $y = (x 0)^2 + 6$, the vertex is at $(0, 6)$, a vertical shift of 6 units.
This handy formula is a cheat code for graphing parabolas! Instead of plotting tons of points, you can instantly find the vertex at $(h, k)$. The 'h' value slides the graph left or right, and 'k' moves it up or down. It's the ultimate shortcut to finding the parabola's home base on the coordinate plane.
Common Questions
What is the vertex form of a quadratic and what does each part mean?
f(x) = a(x-h)²+k. The vertex is (h,k). The sign of a determines direction (up if a>0, down if a<0). |a| controls width (larger |a| is narrower).
How do you convert x²-6x+5 to vertex form by completing the square?
x²-6x+5 = (x²-6x+9)-9+5 = (x-3)²-4. Vertex form: f(x)=(x-3)²-4, vertex at (3,-4).
How do you write the vertex form of the quadratic with vertex (-1,3) that passes through (1,7)?
f(x)=a(x+1)²+3. Substitute (1,7): 7=a(4)+3, so 4a=4, a=1. Equation: f(x)=(x+1)²+3.