When you're trying to understand how to graph different functions, it's important to know about shifts, stretches, and reflections. These changes can really change how a graph looks, and knowing how each one works can help you picture them better.

Shifts

Shifts move the whole graph up, down, left, or right.

• Vertical Shifts: If you add or subtract a number from the function, it shifts the graph up or down. For example, if you have $f(x) + k$, the graph moves up $k$ units. If you use $f(x) - k$, it moves down instead.

• Horizontal Shifts: When you add or subtract a number inside the function, the graph shifts left or right. For example, with $f(x - h)$, the graph moves to the right by $h$ units. But with $f(x + h)$, the graph shifts to the left.

Stretches

Stretches change how big or small the graph is. You can stretch it either vertically or horizontally:

• Vertical Stretch: If you multiply the function by a number greater than 1, like $2f(x)$, it stretches the graph away from the x-axis.

• Horizontal Stretch: This happens when you multiply the x variable by a fraction. For instance, with $f(\frac{1}{2}x)$, your graph gets squished toward the y-axis.

Reflections

Reflections flip the graph over a line:

• Over the x-axis: You can flip the whole graph downwards by multiplying the function by -1, like $-f(x)$.

• Over the y-axis: For this flip, you use $f(-x)$ to turn the graph sideways.

Getting a grip on these transformations makes graphing a lot easier and more understandable!