Transformations of Nonlinear Graphs
Writing a function as a times f(x minus h) plus k shows exactly how its graph moves: h slides it sideways, k slides it up or down, and a flips or stretches it vertically.
Every nonlinear function you'll transform follows the same template: a times f(x - h) + k. Each letter controls one move. The k outside shifts the whole graph up or down, since you're adding to every output after it's computed. The h inside does something that trips people up: it shifts left or right, but backwards from what the sign suggests. That's because h changes what input gets plugged into f. To get the same output you got at x = 0 before, you now need x - h = 0, meaning x = h. So replacing x with (x - 3) means you need x = 3 to reproduce the old behavior at 0 — the graph moved right. Adding 3 instead, as in f(x+3), means x = -3 reproduces the old zero behavior, so the graph moves left.
The a out front scales outputs vertically and flips the graph over the x-axis if negative. It stretches the graph away from the x-axis if |a| > 1, and squashes it toward the axis if |a| < 1. Because a only multiplies the output, it never touches an x-intercept — any height of 0 stays 0 no matter what you multiply by. That's different from h and k, which move the whole graph and can create or destroy intercepts.
This form connects directly to vertex form for parabolas and to asymptote behavior for exponentials. For y = -2(x+3)2 - 5, the vertex sits at (-3, -5): h = -3 so shift left 3, k = -5 so shift down 5, and a = -2 flips the parabola and stretches it, making -5 the maximum value. For an exponential like y = 2x, the horizontal asymptote y = 0 shifts straight down with k, so dropping the graph 3 units moves the asymptote to y = -3.
Key terms
- Vertex form
- The form a(x-h)2 + k for a parabola, where (h, k) is the vertex.
- Horizontal asymptote
- A horizontal line that a graph approaches but never crosses as x goes to infinity or negative infinity.
- x-intercept
- A point where a graph crosses the x-axis, where the output equals 0.
- Vertical stretch
- Scaling a graph's outputs by a factor with magnitude greater than 1, pulling it away from the x-axis.
Watch it explained
- How it worksHow do the slots in a times f of (x minus h), plus k, each move, flip, or stretch the original curve?
- Worked exampleWalk me through every transformation taking y = x2 to y = -2(x+3)2 - 5, and find the new vertex.
- Worked exampleFind the equation after y = 2x is shifted 4 to the right and 1 down.
- Common mistakeWhy doesn't multiplying a function by 2 move any of its x-intercepts?
- Going furtherWhat happens to the asymptote of an exponential curve when the whole graph drops 3 units?
Check yourself
End of topic.