When a shape on a graph is u shape opening upward and another shape that is a transformation that is wider and shifts to the left or right How do you count the units

Answer 1

To analyze the transformation of U-shaped graphs (commonly represented by quadratic functions like $y=a{x}^2$), you need to consider a few aspects: the width of the graph and the horizontal shifts. Here's how to count the units for these transformations:

1. Normal U-shape: The standard form of a U-shaped graph is $y=a{x}^2$ where $a>0$. If $a=1$, the graph is "normal" in width.

2. Wider Shape: A wider shape can be represented by $y=a{\left(\frac{x}{k}\right)}^2$ where $k>1$. Here, $a$ still controls the direction of the U (upward or downward) and $k$ affects the width:

   • If $k>1$, the shape is wider compared to $y=a{x}^2$.
   • The value of $k$ indicates how many times wider the graph is. For example, if $k=2$, the graph is twice as wide.

3. Horizontal Shift: To shift the graph left or right, translate the variable $x$ appropriately. The transformation has the form $y=a(x-h{)}^2$ where:

   • $h>0$ shifts the graph $h$ units to the right.
   • $h<0$ (or simply a $y=a(x+c{)}^2$ style) shifts the graph $c$ units to the left.

Summary Steps to Count Units:

• Identify the Vertex: In the standard form $y=a(x-h{)}^2+k$, the vertex is at point $(h,k)$.
• Counting Units:
  • For width, consider the constant $k$ used in $y=a{\left(\frac{x}{k}\right)}^2$ (the greater the $k$ value, the wider the graph).
  • For horizontal shifts, look at $h$:
    • Positive $h$ = shift right by $h$ units.
    • Negative $h$ or the addition of a positive in standard form = shift left by $c$ units.

For practical examples:

• If the original parabola is $y=x^2$ and the transformation is $y=\frac{1}{4}(x-3{)}^2$:
  • The graph opens upward, shifts $3$ units to the right, and the $\frac{1}{4}$ makes it wider than the original.

Remember, carefully analyze each term in the equation to count shifts and width correctly!