Math · College algebra · Worked example
Transform the graph of y = x²
Describe how to get the graph of g(x) = 2(x − 3)² + 1 from the graph of f(x) = x², and find where the vertex and the points (±1, 1) end up.
Read a, h and k
Compare g with a·f(x − h) + k: a = 2, h = 3 and k = 1. There is no horizontal scaling, so b = 1.
Shift right 3
The x − 3 inside moves every point right 3. The vertex goes from (0, 0) to (3, 0), and (1, 1) goes to (4, 1).
Stretch vertically by 2
Doubling the output doubles each height: (4, 1) becomes (4, 2). The vertex, at height 0, stays at (3, 0).
Shift up 1
Adding 1 lifts every point: the vertex lands at (3, 1), and (1, 1) and (−1, 1) land at (4, 3) and (2, 3).
Check with the formula
Evaluate g at the three new x-values: each gives the predicted height.
Result
Shift right 3, stretch vertically by 2, then shift up 1. The vertex moves to (3, 1), and (±1, 1) move to (4, 3) and (2, 3).
Your turn
Describe the graph of h(x) = −(x + 2)² + 4 compared with y = x², and give its vertex.
Show the answer and explanation
Reflect across the x-axis, shift left 2 and up 4. The vertex is (−2, 4).
h(x) = −f(x − (−2)) + 4, so a = −1 flips the parabola to open downward, h = −2 shifts it left 2 and k = 4 lifts it 4. The vertex (0, 0) moves to (−2, 4).
Keep exploring
In Function Transformations, set the output scale a to −2. The parabola flips to open downward, and its vertex stays at (3, 1).
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.
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