Math · College algebra · Worked example
Write a quadratic from its vertex
A parabola has vertex (−1, 4) and passes through (1, −4). Find its equation in vertex form and in standard form.
Start from vertex form
The vertex (−1, 4) gives h = −1 and k = 4, so f(x) = a(x + 1)² + 4. Only a is unknown.
Use the other point
Substitute (1, −4) and solve for a.
Write both forms
f(x) = −2(x + 1)² + 4. Expanding gives the standard form.
Check
In standard form, −b/(2a) = −(−4)/(2(−2)) = −1, matching the vertex, and f(1) = −2 − 4 + 2 = −4.
Result
f(x) = −2(x + 1)² + 4, or −2x² − 4x + 2 in standard form.
Your turn
Find the quadratic function with vertex (2, −3) whose graph passes through (0, 5).
Show the answer and explanation
f(x) = 2(x − 2)² − 3, or 2x² − 8x + 5.
f(x) = a(x − 2)² − 3, and f(0) = 4a − 3 = 5 gives a = 2.
Keep exploring
Graph plots the parabola through its vertex (−1, 4) and the point (1, −4).
Return to the concept →Sources and scope
Authored study material. Tool results depend on the stated inputs and model assumptions.
Try in the workspace
Open the example inputs, change a value and keep a useful result on your board.
Graph the parabola Check the expansion in Math Open worked example on a board Vertex formula in Math ReferenceYour existing work stays on this device. Examples open as editable copies.