When you learn about quadratic equations, it's really important to know how these equations look when we draw them on a graph.

Quadratic equations look like this:

[ y = ax^2 + bx + c ]

Here, ( a ), ( b ), and ( c ) are just numbers we use in the equation. When we graph these equations, we get a shape called a parabola. This parabola can either point up or down, depending on the number ( a ).

But did you know that we can move the graph around without changing its basic shape? Let’s explore how we can do this!

Understanding Basic Shifts

1. Vertical Shifts: A vertical shift happens when we add or subtract a number from the whole equation.

   For example, if we start with the equation ( y = x^2 ) and change it to ( y = x^2 + 3 ), the graph moves upwards by 3 units. On the other hand, if we change it to ( y = x^2 - 2 ), the graph moves downwards by 2 units.

   Example:

   • Original Equation: ( y = x^2 )
   • New Equation: ( y = x^2 + 3 )
   • This makes the entire graph go up by 3 units.

2. Horizontal Shifts: Horizontal shifts happen when we change the ( x ) value in the equation.

   For instance, if we write ( y = (x - 2)^2 ), the graph moves to the right by 2 units. If we write ( y = (x + 1)^2 ), the graph moves to the left by 1 unit.

   Example:

   • Original Equation: ( y = x^2 )
   • New Equation: ( y = (x - 2)^2 )
   • Here, every point on the graph moves 2 units to the right.

Combining Shifts

You can even mix both vertical and horizontal shifts together!

Let’s say we start with ( y = x^2 ). If we want to move it 2 units to the right and 3 units up, our new equation will be:

[ y = (x - 2)^2 + 3 ]

This new equation shifts every point on the graph 2 units to the right and 3 units up at the same time.

Summary and Visualizing

To recap:

• Vertical shifts: ( y = x^2 + k ) (moves up if ( k > 0 ), down if ( k < 0 ))
• Horizontal shifts: ( y = (x - h)^2 ) (moves to the right if ( h > 0 ), to the left if ( h < 0 ))
• Combined shifts: ( y = (x - h)^2 + k )

A Simple Example

Think about the graph of ( y = x^2 ). When you draw it, it looks like a U shape.

Now, if we apply the changes in the equation ( y = (x - 1)^2 + 2 ):

1. First, shift the whole curve to the right by 1 unit.
2. Then, move it up by 2 units.

When you graph these steps, you will notice that the U shape stays the same, but it just moves to a new spot.

By exploring these shifts, you not only learn about quadratic equations but also get better at predicting how changes will affect their graphs!