Linear transformations of graphs and points, including translations by vectors, vertical and horizontal stretches, vertical and horizontal translations, reflections in the axes, and combined transformations such as y=f(ax+b).
The graph of y=f(x)+b can be obtained from the graph of y=f(x) by a vertical translation b units upwards (if b<0, the transformation may also be called a translation ∣b∣ units down).
The graph of y=f(x−a) can be obtained from the graph of y=f(x) by a horizontal translation a units to the right (if a<0, the transormation may also be called a translation ∣a∣ units to the left).
If a point P is translated by a vector (ab), apply a translation a units to the right and b units up:
The graph of y=pf(x) can be obtained from the graph of y=f(x) by a vertical stretch with scale factor p.
The graph of y=f(qx) can be obtained from the graph of y=f(x) by a horizontal stretch with scale factor q1.
The graph of y=−f(x) can be obtained from the graph of y=f(x) by a reflection in the x-axis.
The graph of y=f(−x) can be obtained from the graph of y=f(x) by a reflection in the y-axis.
The graph of y=f(ax+b)=f[a(x+ab)] can be obtained from the graph of y=f(x) by a horizontal translation b units to the left, followed by a horizontal stretch with scale factor a1.
OR a horizontal stretch with scale factor a1 followed by a horizontal translation ab units to the left.
Nice work completing Linear Transformations, here's a quick recap of what we covered:
Exercises checked off
Linear transformations of graphs and points, including translations by vectors, vertical and horizontal stretches, vertical and horizontal translations, reflections in the axes, and combined transformations such as y=f(ax+b).
The graph of y=f(x)+b can be obtained from the graph of y=f(x) by a vertical translation b units upwards (if b<0, the transformation may also be called a translation ∣b∣ units down).
The graph of y=f(x−a) can be obtained from the graph of y=f(x) by a horizontal translation a units to the right (if a<0, the transormation may also be called a translation ∣a∣ units to the left).
If a point P is translated by a vector (ab), apply a translation a units to the right and b units up:
The graph of y=pf(x) can be obtained from the graph of y=f(x) by a vertical stretch with scale factor p.
The graph of y=f(qx) can be obtained from the graph of y=f(x) by a horizontal stretch with scale factor q1.
The graph of y=−f(x) can be obtained from the graph of y=f(x) by a reflection in the x-axis.
The graph of y=f(−x) can be obtained from the graph of y=f(x) by a reflection in the y-axis.
The graph of y=f(ax+b)=f[a(x+ab)] can be obtained from the graph of y=f(x) by a horizontal translation b units to the left, followed by a horizontal stretch with scale factor a1.
OR a horizontal stretch with scale factor a1 followed by a horizontal translation ab units to the left.
Nice work completing Linear Transformations, here's a quick recap of what we covered:
Exercises checked off