Problem 11 - Algebra 2

Question

Graph each function. State the domain and range of each function. \(y=-4 \sqrt{x}\)

Step-by-Step Solution

Answer

Domain: \( x \geq 0 \); Range: \( y \leq 0 \).

Step 1: Identify the Type of Function

The given function is \( y = -4 \sqrt{x} \). This is a square root function scaled vertically by \(-4\). Since negative scaling flips the graph across the x-axis, it's important to note this change in the graph's orientation.

Step 2: Determine the Domain

For the square root function \( \sqrt{x} \), the domain is all non-negative values of \( x \). Therefore, for \( y = -4 \sqrt{x} \), the domain remains the same: \( x \geq 0 \).

Step 3: Determine the Range

The square root function \( \sqrt{x} \) typically has a range of \( y \geq 0 \). However, due to the negative scaling factor of \(-4\), the graph is flipped across the x-axis. Therefore, the value of \( y \) will always be less than or equal to zero. So, the range is \( y \leq 0 \).

Step 4: Plot Key Points

Select key points to better understand how the function will look when graphed. For example, calculate:- When \( x=0 \), \(y = -4\sqrt{0} = 0\).- When \( x=1 \), \(y = -4\sqrt{1} = -4\).- When \( x=4 \), \(y = -4\sqrt{4} = -8\).These points help in sketching the graph.

Step 5: Sketch the Graph

Using the key points and the knowledge that the graph is a mirrored square root curve, sketch the graph starting at the origin (0,0) and curving downwards to the right, continuously approaching negative infinity as \( x \) increases.

Step 6: Confirm Domain and Range

Once the graph is sketched, verify that it meets the domain \( x \geq 0 \) and the range \( y \leq 0 \), which confirms that the function has been graphed correctly.

Key Concepts

Square Root Function

The square root function is one of the most fundamental functions in algebra. It is usually written as \( y = \sqrt{x} \) and represents the operation of finding a number which, when squared, gives the original value under the square root.
Understanding how modifications to this function, such as scaling or shifting, affect its graph is crucial in algebra. In our exercise, the function \( y = -4 \sqrt{x} \) features a vertical scaling by \(-4\).
This negative scaling flips the graph across the x-axis, transforming the normally upward-opening square root curve into a downward-opening one. When graphing any square root function, recognizing these transformations helps in anticipating the overall shape and direction of the graph, simplifying the process of sketching it accurately.

Domain and Range

Domain and range are core concepts in functions, dictating the set of possible input values (domain) and the set of potential output values (range). Understanding them is key to solving many algebra problems.
For the square root function \( y = -4 \sqrt{x} \), the domain is restricted to non-negative \( x \)-values because you can't take the square root of a negative number and get a real result. Therefore, the domain is \( x \geq 0 \).

• Domain of \( \sqrt{x} \): \( x \geq 0 \)
• Transformation does not change domain

For the range, since the graph is flipped due to the negative scaling, the possible outcomes (or range) of the function are all non-positive numbers. This flipping means the function outputs negative values or zero, indicating the range is \( y \leq 0 \).

• Range before transformation: \( y \geq 0 \)
• After flipping: \( y \leq 0 \)

Knowing how transformations affect domain and range is critical in graphing the correct intervals.

Graphing Functions

Graphing functions entails plotting points for selected \( x \)-values and drawing a curve through these points to represent the function visually.
For \( y = -4 \sqrt{x} \), we select key values of \( x \), compute \( y \), and plot them:

• When \( x = 0 \), \( y = 0 \)
• When \( x = 1 \), \( y = -4 \)
• When \( x = 4 \), \( y = -8 \)

These key points provide a framework to sketch the curve which starts at the origin (0,0) and extends continuously downwards to the right.
The graph helps visually confirm the domain \( x \geq 0 \) as we're only using non-negative \( x \) values. It also verifies the range \( y \leq 0 \) since the curve solely resides in the negative \( y \)-values, reflecting the inverse orientation of the square root's typical behavior. Understanding these visual markers is integral in mastering the graphing of transformations.