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Chapter 1: Problem 62

Graph the function using transformations. $$y=-\frac{1}{5} \sqrt{x}$$

Short Answer

Expert verified
The graph is a reflected and compressed version of \(y=\sqrt{x}\).

Step by step solution

01

Identify the Parent Function

The parent function for this transformation is the square root function, which is \( y = \sqrt{x} \). This is the simplest form of the function we are dealing with.
02

Vertical Stretch/Compression and Reflection

The function \( y = -\frac{1}{5} \sqrt{x} \) includes a coefficient of \(-\frac{1}{5}\) in front of the square root. This coefficient affects the graph in two ways: the negative sign indicates a reflection across the x-axis, and the \(\frac{1}{5}\) indicates a vertical compression by a factor of 5. This means that each point on the graph is moved to \(\frac{1}{5}\) of its original distance from the x-axis, and flipped to the opposite side.
03

Graph the Transformation

Begin graphing the original parent function \( y = \sqrt{x} \), which starts at the origin (0,0) and passes through points like (1,1), (4,2), and (9,3). Now, apply the transformations: reflect these points over the x-axis and compress each y-value by a factor of \(\frac{1}{5}\). This results in new points like: (1, -0.2), (4, -0.4), and (9, -0.6). Plot these transformed points to get the new graph. The transformed graph should be a downward sloping curve starting at the origin and going through the transformed points, showing the reflection and compression.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Square Root Function
The square root function is a fundamental mathematical concept used in many fields. It is represented by the equation \( y = \sqrt{x} \). This function forms a curve that plots from the origin and moves to the right, extending indefinitely, forming a gentle upward slope.

Key characteristics:
  • The starting point is at the origin, (0,0).
  • The function passes through points such as (1,1), (4,2), and (9,3) because the square roots of these x-values are integers.
  • It always remains in the first quadrant since both x and y are positive.
Understanding this baseline鈥攈ow the square root function graphically appears鈥攊s pivotal for grasping more intricate transformations like vertical compression and reflection. Once you establish this curve, any transformation builds on this shape.
Vertical Compression
Vertical compression occurs when the y-values of a function are multiplied by a factor between 0 and 1. For the given function, \( y = -\frac{1}{5} \sqrt{x} \), the compression factor is \( \frac{1}{5} \). This compresses the graph vertically by shrinking the output of the square root function.

Features of vertical compression:
  • This transformation decreases the height of the graph, making it "flatter." For example, a point like (4,2) on the parent function becomes (4,0.4) on the compressed graph.
  • The graph retains its original shape but all y-coordinates are scaled down by the compression factor.
  • The x-values remain unchanged, so the horizontal positioning on the graph is unaffected.
Through this transformation, one can observe that the graph approaches the x-axis more rapidly as x increases, visually shortening the vertical reach of the graph. This is why the curve looks closer to the x-axis than its parent form.
Reflection Across the X-Axis
Reflection across the x-axis in a function lets you invert the graph as if you were "flipping" it upside down. This is signified by a negative sign before the function, as seen in \( y = -\frac{1}{5} \sqrt{x} \).

Reflection aspects:
  • Every positive y-value from the square root function becomes negative.
  • Points that were above the x-axis are now mirrored below it. For instance, the point (1,0.2) from the vertically compressed graph is turned to (1, -0.2) after reflection.
  • This does not affect the x-coordinates or the horizontal spread of the graph.
Reflection drastically changes the direction in which the graph travels on the plane, turning what once was an upward curving graph into a downward-sloping one.

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