Graph Transformations

Understanding Graph Transformations

Transformations visually alter the original shape of a graph. The primary ones are translations, reflections, dilations, and stretches.
A translation moves a graph to a new location without changing its size or orientation. This transformation is described by the vector (h, v), where ‘h’ denotes horizontal shift and ‘v’ denotes vertical shift.
A reflection mirrors the graph about a specific line. Reflecting a graph over the x-axis changes the sign of all y-values, while reflecting over the y-axis alters the sign of all x-values.
A dilation resizes a graph. When the scale factor is greater than 1, the graph expands; when it’s between 0 and 1, the graph reduces in size.
A stretch is a transformation that expands or compresses the graph in one direction. Vertical stretching elongates the graph in the y-direction, while horizontal stretching does the same along the x-axis.

Recognising Graph Transformations

One can identify a graph translation by examining changes in the graph’s coordinates. If all points on the graph have moved ‘h’ units to the right/left and ‘v’ units up/down, that’s a translation.
Reflections can be spotted by investigating the symmetry of a graph about an axis. If the shape of the graph repeats itself on either side of a line, there’s a reflection.
Recognising a dilation can be tricky. If every point on the graph is farther from the origin than on the original graph, it’s a dilation with a scale factor greater than 1. If the points are closer to the origin, the scale factor is between 0 and 1.
To spot a stretch, look for elongation or compression of the graph’s shape in a specific direction. If it appears vertically elongated, it’s a vertical stretch. Similarly, horizontal compression indicates a horizontal stretch.

Effects of Graph Transformations on Equations

A graph of y = f(x) translated h units horizontally and v units vertically has the equation y = f(x - h) + v.
The equation of a graph reflected over the x-axis is y = -f(x), while it’s y = f(-x) when reflected over the y-axis.
The equation for a dilation of y = f(x) by a factor of k is y = kf(x) for a vertical dilation, and y = f(kx) for a horizontal dilation.
Vertical and horizontal stretches are represented by y = f(x/k) and y = kf(x) respectively, where ‘k’ is the stretch factor.

Remember that the above transformations can be combined to produce more complex changes to a graph’s appearance. Mastering their identification and use is key to manipulating and understanding diverse mathematical functions.