Problem 41 - College Algebra with Modeling and Visualization

Question

Use transformations to explain how the graph of \(f\) can be found by using the graph of \(y=x^{2}\) \(y=\sqrt{x},\) or \(y=|x| .\) You do not need to graph \(y=f(x)\). \(f(x)=2 \sqrt{-x}\)

Step-by-Step Solution

Answer

Reflect \(y=\sqrt{x}\) over the y-axis and then stretch it vertically by a factor of 2.

Step 1: Identify the Base Function

The function provided is \(f(x) = 2 \sqrt{-x}\). The base function that we will use for transformations is \(y = \sqrt{x}\), which represents the graph of a square root function.

Step 2: Apply Reflection on the x-axis

In the function \(f(x) = 2 \sqrt{-x}\), the negative inside the square root function, \(-x\), indicates a reflection over the y-axis. This changes the input \(x\) to \(-x\), flipping the graph horizontally.

Step 3: Apply Vertical Stretch

The coefficient '2' outside the square root, in \(2\sqrt{-x}\), indicates a vertical stretch by a factor of 2. This means that each y-coordinate of the points on the base graph \(y = \sqrt{x}\) gets multiplied by 2, stretching the graph vertically.

Key Concepts

Reflection Over Y-Axis

In mathematics, a reflection over the y-axis involves flipping a graph horizontally. For the function \( f(x) = 2\sqrt{-x} \), the negative inside the square root, i.e., \( -x \), is the indicator of this transformation. This can be visualized as taking the graph of \( y = \sqrt{x} \) and reflecting it over the y-axis. Essentially, every point \( (x, y) \) on the original graph is transformed to \( (-x, y) \).

• This transformation affects the domain of the function. Instead of considering \( x \geq 0 \) as with \( y = \sqrt{x} \), we consider \( x \leq 0 \) in this context.
• Reflections along the y-axis, like mirrors, preserve the y-coordinate while changing the sign of the x-coordinate.

Understanding reflections helps in predicting how graphs behave when negatives are introduced inside functions.

Vertical Stretch

A vertical stretch changes the appearance of a graph by multiplying the output values by a scale factor. In \( f(x) = 2\sqrt{-x} \), the coefficient "2" outside the square root is the scale factor. To apply this transformation, each y-coordinate of the square root function \( y = \sqrt{x} \) is multiplied by 2.

• This stretch affects the steepness or sharpness of the graph's rise. It pulls the points on the curve further away from the x-axis.
• For example, if the base graph at some point \( (x, y) \) had a y-value of 1, after the vertical stretch, its new y-value becomes 2.
• Keep in mind, the stretch factor must be positive to ensure it remains a stretch and not a reflection over the x-axis.

Recognizing this transformation helps in understanding how graphs can be stretched or compressed along the vertical axis.

Square Root Function

The square root function is a fundamental component in graph transformations. It serves as the base function in transformations like in \( f(x) = 2\sqrt{-x} \). The standard form, \( y = \sqrt{x} \), is characterized by its gradual increase and smooth curve starting from the origin, \( (0,0) \).

• As seen in its graph, \( y = \sqrt{x} \) is defined only for \( x \geq 0 \) because square roots of negative numbers are not real within the context of real-valued functions.
• The graph passes through key points such as \( (0,0), (1, 1), \) and \( (4, 2) \), representing the perfect squares.
• Transformations like reflections or stretches can be easily visualized when considering changes to this base shape.

The square root function introduces its unique set of properties, serving as an excellent building block for understanding more complex graph transformations.