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Vertical stretching is a transformation that scales a function along the y-axis by multiplying the function by a constant. In our example, the base function is a semicircle equation, and we apply vertical stretching through the parameter \( c \). When \( c = 1 \), there is no vertical stretching since \( f(x) = \sqrt{4 - x^2} \) remains unchanged. However, for \( c = 3 \), we have \( f(x) = 3 \sqrt{4 - x^2} \), meaning that the semicircle is stretched vertically. The maximum height of the semicircle changes from \( 2 \) to \( 6 \), effectively tripling the y-values of all points on the curve.

• Vertical stretching changes the height and shape of a graph.
• The greater the value of \( c \), the more the graph stretches vertically.
• It's important to note that endpoints along the x-axis remain unchanged.

This transformation might make a graph appear taller or flatter, depending on the magnitude of the factor.

Reflection across the x-axis involves flipping a graph over the x-axis. This occurs when the parameter \( c \) in our function is negative. For our semicircle function, this is represented mathematically as \( f(x) = c \cdot \sqrt{4 - x^2} \) where \( c < 0 \). Consider \( c = -2 \) from our exercise. The function becomes \( f(x) = -2 \sqrt{4 - x^2} \). First, we reflect the positive semicircle across the x-axis, turning all positive y-values negative. Then, due to the magnitude (\( 2 \)), the semicircle stretches twice as much downward.

• Reflection flips all positive y-values to negative.
• It results in a mirror image being formed below the x-axis.
• Reflection influences both shape and position on the coordinate plane.

This transformation helps to see graphs from an inverse perspective, which can be useful in analysis.

The semicircle graph is a simple yet important geometric shape often used in transformations. Represented by \( \, \sqrt{4 - x^2} \) in its basic form, it depicts the top half of a circle.This graph is centered at the origin with a radius of \( 2 \), reaching from \( -2 \) to \( 2 \) on the x-axis. By identifying the original semicircle graph, one can understand transformations better.

• Endpoints lie at (-2, 0) and (2, 0).
• The peak (or highest point) is at (0,2) for the untransformed graph.
• Transforms such as stretching or reflection change the graph's shape but not its x-values.

Recognizing the semicircle layout simplifies sketching transformations quickly and accurately.

Graph sketching involves plotting a visual representation of functions on a coordinate plane. It is both an art and a science which facilitates understanding of how different values transform a base graph.To sketch graphs with varying \( c \) values, combine knowledge of vertical stretching, reflection, and the semicircle graph.

• First, sketch the base function, here it's: \( \sqrt{4 - x^2} \).
• Apply the transformation for each \( c \): \(-2\), \(1\), and \(3\).
• Ensure distinct sketches for each transformation on the same plane.

Through practice, this method becomes intuitive and aids in forecasting the impact of transformations. Graph sketching is essential for visualizing changes in functions, helping clarify the implications of mathematical manipulations.