Problem 10 - Algebra 2

Question

Graph each function. \(y=-0.25 \sqrt{x}\)

Step-by-Step Solution

Answer

The graph starts at the origin and decreases to the right, reflecting the original square root graph across the x-axis and shrinking it vertically by a factor of 0.25. For \(x>0\), the graph only exists in the third and fourth quadrants.

Step 1: Recognize the Basic Function

Recognize that the given function is a transformation of the basic function \(y = \sqrt{x}\), which starts at the origin (0,0) and increases for all \(x > 0\). The graph only exists in the first and second quadrants.

Step 2: Identify Transformations

Identify the transformations that are applied to the basic function. The multiplication of the square root function by -0.25 flips the graph vertically (reflection across the x-axis) and squishes it (vertical shrink) by a factor of 0.25.

Step 3: Plot the Transformed Function

Plot the transformed function, \(y = -0.25 \sqrt{x}\), by first drawing the basic function \(y = \sqrt{x}\) and then applying the transformations. For each point \(x\), calculate \(-0.25 \sqrt{x}\) to get the corresponding \(y\) value and plot the point. For example, when \(x=0\), \(y=0\); when \(x=4\), \(y=-0.5\); when \(x=16\), \(y=-1\). Continue this for several values of \(x\) to get a good sketch of the graph. Note that the graph will be in the third and fourth quadrants, due to the reflection across the x-axis.

Key Concepts

Quadratic Transformation

Quadratic transformations involve changing the basic shape of a quadratic function or related functions like the square root function. When dealing with transformations, you need to identify how the original graph changes due to shifts, reflections, or rescalings.
These transformations can include:

• Translations: Shifting the graph vertically or horizontally.
• Reflections: Flipping the graph over the x-axis or y-axis, altering its direction.
• Stretching/Shrinking: Changing the graph's width, making it wider or narrower.

In our problem, the transformation applied to the square root function results in reflections and shrinking, altering its original appearance.

Vertical Reflection

A vertical reflection is a transformation that flips a graph upside down over the x-axis. This means that for every point on the original function, its reflection will have the same distance from the x-axis, just in the opposite direction.
The reflection effect is usually achieved by multiplying the function by a negative coefficient. In the exercise,

• The function is transformed from \(y = \sqrt{x}\) to \(y = -0.25 \sqrt{x}\).

This negative sign causes the graph’s orientation to flip; points that were above the x-axis are now below. This is why the transformed graph lies in the third and fourth quadrants.
This change is crucial for recognizing function behavior and is often used in graphing transformations.

Vertical Shrink

Vertical shrink refers to compressing the graph in the vertical direction. This transformation affects the height of the graph, making it lower or closer to the x-axis.
In mathematical terms, this occurs when you multiply the function by a constant between 0 and 1 (excluding 0, as this would flatten the graph entirely).
For example:

• If you have the function \(y = \sqrt{x}\), a vertical shrink turns it into \(y = 0.25 \sqrt{x}\).
• This change decreases the y-values of the points on the graph by 75%.This means that each point moves closer to the x-axis.

Vertical shrinking is handy when you need to scale down the intensity or output of a function without changing its basic shape.

Square Root Function

The square root function, represented as \(y = \sqrt{x}\), is a core mathematical function with a distinct graph shape. Its graph starts at the origin point (0,0) and moves in a gentle curve upwards to the right.
This graph exists only in the positive x-region, as square roots for negative numbers aren’t defined in real numbers.
Key Points about the Square Root Function:

• Starts at Origin: The graph begins at the point (0,0).
• Increases Continuously: The function increases as the value of x increases, but at a decreasing rate.
• Domain and Range: Original domain is \(x \geq 0\) and range is \(y \geq 0\).

The original exercise focuses on transforming this function, incorporating reflection and shrinking, which modifies how it appears on the graph. Understanding this function's basics helps in realizing the effects of such transformations.