Answer Key for Lesson 1 Three Transformations of Function Graphs

Shift the curve horizontally by adding or subtracting a constant to the variable inside the equation. For instance, f(x – 3) shifts the graph 3 units to the right, while f(x + 3) moves it 3 units to the left. Pay close attention to the sign of the constant–positive shifts left, negative shifts right.

Vertical shifts occur when a constant is added or subtracted outside the function. For example, f(x) + 2 will move the graph 2 units upward, while f(x) – 2 moves it 2 units downward. This shift doesn’t affect the shape, only the position on the y-axis.

Reflections are caused by negating the variable or the entire function. A negative inside the function, -f(x), reflects the graph over the x-axis, while f(-x) flips it over the y-axis. Carefully analyze the direction of the flip to understand the effect on the curve’s orientation.

Mastering these operations will help manipulate and understand the behavior of mathematical curves. Adjusting each part of the equation directly impacts the graph’s position and direction, providing a systematic approach to altering and analyzing curves with precision.