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A quadratic function is said to be written in vertex form if it has the following format.
y=a(x-h)^2+k
Here, a, h, and k are real numbers with a≠ 0. The value of a gives the direction of the parabola. When a > 0, the parabola faces upward, and when a < 0, it faces downward. The vertex of the parabola lies at (h,k), and the axis of symmetry is the vertical line x=h. Consider the graph of y=- 14(x-4)^2+8.
Comparing the generic vertex form with the example function, the values of a, h, and k can be identified. Vertex Form:& y= a(x- h)^2+ k Example Function:& y= - 1/4(x- 4)^2+ 8 These values determine the characteristics of the parabola shown in the graph.
| Direction | Vertex | Axis of Symmetry |
|---|---|---|
| a= - 1/4 | h= 4 and k= 8 | h= 4 |
| Since - 14 is less than 0, the parabola opens downward. | The vertex is located at ( 4, 8). | The axis of symmetry is the vertical line x= 4. |
| Function 1 | Function 2 | Function 3 |
|---|---|---|
| y=2(x+1)^2+7 ⇕ y= 2(x-( - 1))^2+ 7 |
y=(x-3)^2+1 ⇕ y= 1(x- 3)^2+ 1 |
y=5(x-2)^2-3 ⇕ y= 5(x- 2)^2+( - 3) |