See parent and transformed functions
A function transformation changes a familiar parent graph without making you redraw it. This calculator keeps the parent curve visible in gray and draws the transformed curve in your selected color. Choose from linear, quadratic, absolute value, cubic, square root, cube root, reciprocal, exponential, sine, or cosine, then adjust one input at a time. The live summary translates the equation into words so a shift, stretch, compression, or reflection is easy to recognize.
Read a, b, h, and k
The workspace uses g(x) = a · f(b(x − h)) + k. The value h moves the graph right when positive and left when negative. The value k moves it up when positive and down when negative. The magnitude of a controls vertical scale, while a negative value reflects the graph across the x-axis. The magnitude of b controls horizontal scale, while a negative b reflects the graph across the y-axis. Values between zero and one stretch a graph in the corresponding direction.
Use presets, then explore
Transformation presets provide a quick starting point for common classroom examples, including right and left shifts, vertical movement, reflections, and scale changes. After selecting a preset, type any finite value directly into the number fields for a more precise problem. Use the zoom controls when a large shift or scale moves the graph out of view, hover for coordinates, and compare the gray parent with the active curve to check your reasoning.
Transformation rules in plain language
The form g(x) = a · f(b(x − h)) + k separates vertical and horizontal changes. A positive h moves the parent graph right, k moves it up, a negative a reflects it across the x-axis, and a negative b reflects it across the y-axis. Magnitudes greater than one create a vertical stretch through a or a horizontal compression through b; values between zero and one create the opposite effect. The parent curve stays visible so you can identify one change at a time.
Use transformations to predict coordinates
Choose a recognizable point on the parent graph before changing the parameters. For example, the origin on y = x² moves to (h, k) in a(x − h)² + k, while the point (1, 1) also changes according to the horizontal and vertical scales. Make a prediction, select a preset, then adjust a, b, h, or k one at a time. Comparing predicted and plotted coordinates is a reliable way to distinguish a horizontal shift from a horizontal scale change.
Transformation example with a quadratic
Start with f(x) = x² and choose g(x) = 2(x − 3)² − 1. The point (0, 0) moves to (3, −1), the graph becomes twice as narrow vertically, and the vertex becomes the minimum (3, −1). If you instead use g(x) = −2(x − 3)² − 1, the vertex stays in the same place but the parabola opens downward. Use the parent curve to separate movement from shape change.
Transformation example with a sine wave
For g(x) = 3sin(2(x − π/4)) + 1, the amplitude is 3, the period is π, the phase shift is π/4 right, and the midline is y = 1. Change only one parameter at a time and observe which feature moves. This approach is useful when a transformed function looks unfamiliar but still belongs to a familiar parent family.
