# 1.7 | Transformations of Functions

Let $$c$$ be a positive real number. Vertical and horizontal shifts in the graph of $$y = f(x)$$ are represented as follows:

1. Vertical shift $$c$$ units upward:
$$h(x) = f(x) + c$$
2. Vertical shift $$c$$ units downward:
$$h(x) = f(x) – c$$
3. Horizontal shift $$c$$ units to the right:
$$h(x) = f(x-c)$$
4. Horizontal shift $$c$$ units to the left:
$$h(x) = f(x+c)$$
Reflections in the coordinate axes of the graph of $$y = f(x)$$ are represented as follows.

1. Reflection in the $$x$$-axis:
$$h(x) = -f(x)$$
2. Reflection in the $$y$$-axis:
$$h(x) = f(-x)$$
A vertical stretch of the graph $$y = f(x)$$ is represented by $$g(x) = c f(x),$$ where $$c>1$$. A vertical shrink is represented by $$g(x) = c f(x),$$ where $$0 < c < 1$$. A horizontal stretch is represented by $$g(x) = f(c x),$$ where $$0 < c < 1$$. A horizontal shrink is represented by $$g(x)= f(c x),$$ where $$c > 1.$$