Problem 45 - Algebra 2

Question

Identify the type of function represented by each equation. Then graph the equation. (lesson 8.5\()\) $$ y=2 \sqrt{x} $$

Step-by-Step Solution

Answer

The function is a square root function and should be graphed starting at the origin with an upward curve.

Step 1: Identify Function Type

The function given is \( y = 2 \sqrt{x} \). This is a square root function because it includes the square root of \( x \) in the equation.

Step 2: General Form of Square Root Functions

The general form of a square root function is \( y = a \sqrt{x - h} + k \), where \( a \) is a vertical stretch/compression factor, and \( h \), \( k \) are horizontal and vertical shifts, respectively. In this equation, \( a = 2 \), \( h = 0 \), and \( k = 0 \).

Step 3: Analyze the Function

For \( y = 2 \sqrt{x} \), the value of \( a = 2 \) indicates the graph will be stretched vertically by a factor of 2. Since \( h = 0 \) and \( k = 0 \), the graph is not shifted horizontally or vertically.

Step 4: Plot Key Points

To graph the function, calculate several points: begin with \( x = 0 \): \( y = 2 \cdot \sqrt{0} = 0 \); for \( x = 1 \): \( y = 2 \cdot \sqrt{1} = 2 \); for \( x = 4 \): \( y = 2 \cdot \sqrt{4} = 4 \). These points will help in sketching the graph.

Step 5: Sketch the Graph

Plot the calculated key points on a coordinate plane: \((0, 0), (1, 2), (4, 4)\). Draw a smooth curve through these points starting at the origin and moving rightwards, representing the positive values of \( x \). Since it is a square root function, the graph will only exist in the first quadrant (as \( \sqrt{x} \) is undefined for negative \( x \)).

Key Concepts

Square Root Function Insights

The square root function is a fundamental mathematical concept where the operation involves the square root of the input variable \( x \). Simply put, the square root of a number \( x \) is a value \( y \) such that \( y^2 = x \). The equation given, \( y = 2 \sqrt{x} \), is identified as a square root function because it includes \( \sqrt{x} \).

In a square root function like \( y = a \sqrt{x - h} + k \):

• \( a \) determines the vertical stretch or compression. A higher value of \( a \) means more stretching.
• \( h \) and \( k \) control the horizontal and vertical shifts of the graph, where \( h \) shifts left/right and \( k \) shifts up/down.

In this case, \( a = 2 \), meaning the function is vertically stretched by a factor of 2, while \( h = 0 \) and \( k = 0 \) imply no shifts.

Understanding Graphing Functions

Graphing functions is the visual representation of equations on a coordinate plane, providing an intuitive understanding of mathematical behavior. For the equation \( y = 2 \sqrt{x} \), the process begins by plotting key points that represent the function.

Here's how we plot the graph for this square root function:

• Pick key values for \( x \), starting with simpler numbers like 0, 1, and 4, because they make calculations easier.
• The value of \( y \) for each \( x \) is obtained by substituting \( x \) into the equation, such as \( y = 2 \sqrt{0} = 0 \), \( y = 2 \sqrt{1} = 2 \), and \( y = 2 \sqrt{4} = 4 \).

After plotting these points \( (0, 0), (1, 2), (4, 4) \) on the graph, connect them with a smooth curve starting from the origin. The graph of a square root function will typically begin at the origin and curve upwards to the right, existing only in the first quadrant due to its mathematical properties.

Equation Analysis Techniques

Equation analysis involves breaking down a mathematical equation to understand its components and real-world behavior. With \( y = 2 \sqrt{x} \), the analysis looks at several pieces of information:

• The absence of terms \( h \) and \( k \) within this function means there are no horizontal or vertical adjustments, simplifying the graphing process.
• The "2" in front of the square root affects the steepness of the graph. It doubles the output value \( y \) for any input \( x \), indicating a steeper ascent as \( x \) increases.

To analyze such functions, remember:

• Simplify complex equations by identifying patterns, like shifts, reflections, or stretches.
• Focus on how each term impacts the overall behavior, guiding how to plot and predict graph trends.

This makes equation analysis not only crucial for understanding current problems but also instrumental for problem-solving in broader mathematical contexts.