To analyze the transformation of U-shaped graphs (commonly represented by quadratic functions like \(y = ax^2\)), 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:
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Normal U-shape: The standard form of a U-shaped graph is \(y = ax^2\) where \(a > 0\). If \(a = 1\), the graph is "normal" in width.
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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 \(y = ax^2\).
- The value of \(k\) indicates how many times wider the graph is. For example, if \(k = 2\), the graph is twice as wide.
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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!