Understanding Quadratic Graph Shifts

When we move a quadratic graph sideways, it changes the equation a bit. A quadratic equation usually looks like this:

$$y = ax^2 + bx + c$$

Moving Left or Right

1. Moving Left:

   • If we shift the graph to the left by $h$ units, the equation changes to: $$y = a(x + h)^2 + b(x + h) + c$$
   • This means all the points on the graph move to the left, but the shape of the graph stays the same.

2. Moving Right:

   • If we move the graph to the right by $h$ units, the equation changes to: $$y = a(x - h)^2 + b(x - h) + c$$
   • Just like the left shift, this changes the x-coordinate for every point on the graph by $h$, while keeping its shape.

Effects on the Vertex

The vertex is the highest or lowest point on the quadratic graph, shown as $(h, k)$ in the simpler form of the quadratic equation $y = a(x - h)^2 + k$. Here's how moving changes the vertex:

• Move Left: The x-coordinate of the vertex goes down, going from $(h, k)$ to $(h + h, k)$.
• Move Right: The x-coordinate goes up, changing from $(h, k)$ to $(h - h, k)$.

Wrapping It Up

In short, when you shift the graph left or right, it changes where it sits on the x-axis. But, the overall shape and direction (it opens up if $a > 0$ or down if $a < 0$) stay the same. Knowing about these transformations helps us see how changes in the equation show up on the graph. This idea is really important for understanding quadratic equations in 8th-grade math.