Convert between expanded quadratic coefficients and a(x − h)² + k parameters. Beside vertex form, the output shows the operation used to obtain it.
Vertex form isolates the turning point (h,k) and vertical scale a of a parabola.
For a Vertex Form audit, retain Quadratic coefficient a and Linear coefficient b. Decide the likely direction of Vertex form before rerunning Vertex Form. Change only Constant c; the response in Vertex form can then be traced within the Vertex Form setup.
x²+6x+5=(x+3)²−4, so h=−3 and k=−4.
It makes graph shifts, extrema, symmetry, and range easier to read than expanded form.
The leading coefficient cannot be zero. The sign shown inside parentheses is opposite the sign of h.
Link Quadratic coefficient a to its Vertex Form role. Link Linear coefficient b to its Vertex Form role. The retained Vertex Form formula identifies the Vertex Form model.
The Vertex Form meaning depends on Quadratic coefficient a. The Vertex Form meaning also depends on Linear coefficient b. Carry those Vertex Form roles into any later Vertex Form work.
Calculate h=−b/(2a), evaluate k at h, and write a(x−h)²+k. To continue from Vertex Form, try transformation method.
The vertex should lie on the original quadratic and x=h should be its symmetry line. Values equally spaced from h must produce equal outputs.
For Vertex Form, after checking the sample, vary one meaningful input and predict the direction of change before recalculating. A result that moves oppositely deserves review. Label the final value with its geometric or algebraic role so it is not confused with a neighboring length, area, root, or coefficient.
Substitute the Vertex Form solution into Quadratic coefficient a. This Vertex Form check rejects false Vertex Form branches and forbidden denominators.
Choose an easy Quadratic coefficient a value before running Vertex Form. Predict Linear coefficient b, then compare it with the Vertex Form output.
Because the form uses x−h.
x=h.
It is k when a<0.
Yes, distribute the square and combine terms.