Understanding Shifts and Reflections in Linear Graphs

When we look at linear graphs, we can notice some changes called shifts and reflections. Let’s break these down!

Shifts

1. Vertical Shift:

   When we add a number (let's call it $k$) to the function, the whole graph moves up or down.

   • If $k$ is positive (greater than zero), the graph goes up.
   • If $k$ is negative (less than zero), the graph goes down.

   Example:

   • For the equation $y = 2x + 3$, if we change it to $y = 2x + 5$, the graph moves up by 2 units.

2. Horizontal Shift:

   Here, we subtract a number (let's call it $h$) from $x$. This will move the graph left or right.

   • If $h$ is positive, the graph goes left.
   • If $h$ is negative, the graph goes right.

   Example:

   • For the equation $y = 2(x - 1) + 3$, if we change it to $y = 2(x - 2) + 3$, the graph moves to the right by 1 unit.

Reflections

1. Reflection across the x-axis:

   If we change the sign of $y$, the whole graph flips upside down.

   Example:

   • The equation $y = 2x + 3$ becomes $y = -2x + 3$.

2. Reflection across the y-axis:

   If we change the sign of $x$ in the function, the graph flips over the y-axis.

   Example:

   • The equation $y = 2x + 3$ changes to $y = -2x + 3$.

These shifts and reflections are like tools that help us understand how to change and work with linear equations!