Easy Guide to Changing Quadratic Graphs

When we look at quadratic graphs, there are a few ways we can change or move them around. Here are some simple techniques you can use:

1. Vertical Shifts:

   • If we add a number, called $k$, to our equation, it will push the whole graph up or down.
   • For example, if we write:
     [f(x) = ax^2 + bx + c + k]
     The graph moves up by $k$ units if $k$ is positive. If $k$ is negative, it moves down.

2. Horizontal Shifts:

   • Changing the input of the function can shift the graph to the right or left.
   • This is shown like this:
     [f(x) = a(x - h)^2 + k]
     If $h$ is positive, the graph slides to the right. If $h$ is negative, it goes to the left.

3. Reflections:

   • To flip the graph upside down, we can reflect it across the x-axis.
   • We can do this by using the equation:
     [f(x) = -ax^2]
     This makes the graph turn upside down.

4. Stretching and Compressing:

   • Vertical Stretch/Compression:
     • If we have a value $a$ that is greater than 1, it stretches the graph.
     • If $a$ is between 0 and 1, it squishes the graph.
     • This is shown in the equation:
       [f(x) = a x^2]
   • Horizontal Stretch/Compression:
     • Here, if $|a|$ is greater than 1, the graph gets narrower.
     • If $|a|$ is between 0 and 1, the graph spreads out wider.
     • This can be seen in the equation:
       [f(x) = a(x - h)^2 + k]

These techniques help us understand how to move and change quadratic graphs in a simple way!