Which Graph Represents Y Startroot X Minus 4 Endroot

The graph that represents y = √(x - 4) is a curve that starts at the point (4, 0) and extends to the right, increasing as x increases. Understanding this graph requires a grasp of square root functions, transformations, and domain/range considerations. This article will delve into the characteristics of the graph, explore its relationship to other functions, and provide a comprehensive explanation to ensure a clear understanding.

Understanding the Square Root Function

The square root function, generally written as y = √x, is a fundamental concept in algebra. It essentially asks: "What number, when multiplied by itself, equals x?" The domain of this basic function is all non-negative real numbers (x ≥ 0) because you cannot take the square root of a negative number and obtain a real number result. The range is also non-negative real numbers (y ≥ 0), since the square root of a non-negative number is always non-negative.

The graph of y = √x starts at the origin (0, 0) and increases gradually as x increases. It's a curve that becomes less steep as x gets larger. This is because the rate of change of the square root function decreases as x increases.

Transformations of the Square Root Function: Horizontal Shifts

The function y = √(x - 4) is a transformation of the basic square root function, y = √x. Specifically, it's a horizontal shift. The "- 4" inside the square root affects the x-values, causing the graph to shift horizontally.

Key Concept: Whenever you have a function in the form y = f(x - h), where 'h' is a constant, the graph of y = f(x) is shifted 'h' units horizontally.

• If 'h' is positive, the graph shifts to the right.
• If 'h' is negative, the graph shifts to the left.

In our case, y = √(x - 4) has 'h' = 4, which means the graph of y = √x is shifted 4 units to the right. This is a crucial detail that defines the graph's starting point and overall position.

Characteristics of the Graph y = √(x - 4)

Now, let's examine the specific characteristics of the graph of y = √(x - 4):

1. Starting Point (Vertex): The most important feature is the starting point or vertex of the curve. For y = √x, the vertex is (0, 0). Due to the horizontal shift of 4 units to the right, the vertex of y = √(x - 4) is at (4, 0). This means the graph begins at the x-value of 4 and the y-value of 0.

2. Domain: The domain of a function is the set of all possible x-values for which the function is defined. For y = √(x - 4), the expression inside the square root (x - 4) must be non-negative. Therefore:

   x - 4 ≥ 0 x ≥ 4

   This means the domain of the function is x ≥ 4, or in interval notation, [4, ∞). The graph exists only for x-values greater than or equal to 4.

3. Range: The range of a function is the set of all possible y-values that the function can produce. Since the square root function always returns non-negative values, the range of y = √(x - 4) is the same as the basic square root function: y ≥ 0, or in interval notation, [0, ∞). The graph only exists for y-values greater than or equal to 0.

4. Increasing Function: The graph of y = √(x - 4) is an increasing function. As the x-values increase (moving to the right on the graph), the y-values also increase (moving upwards on the graph). However, the rate of increase decreases as x increases.

5. Shape: The shape of the graph is a curve, similar to the basic square root function y = √x. However, it's important to remember that it starts at (4, 0) instead of (0, 0). The curve will gradually flatten out as x increases.

Visualizing the Graph

Imagine the basic square root function, y = √x. Now, visualize grabbing that entire graph and sliding it 4 units to the right along the x-axis. That's exactly what the graph of y = √(x - 4) looks like.

• Start at (4, 0): This is your anchor point.
• Increasing Curve: From (4, 0), the graph curves upwards and to the right.
• Gradual Flattening: The curve becomes less steep as you move further to the right.

To plot a few points to further illustrate the graph:

• When x = 4, y = √(4 - 4) = √0 = 0. Point: (4, 0)
• When x = 5, y = √(5 - 4) = √1 = 1. Point: (5, 1)
• When x = 8, y = √(8 - 4) = √4 = 2. Point: (8, 2)
• When x = 13, y = √(13 - 4) = √9 = 3. Point: (13, 3)

These points, along with the understanding of the shape, allow you to accurately sketch the graph.

Comparing y = √(x - 4) to Other Transformations

Understanding how y = √(x - 4) relates to other transformations of the square root function can solidify your comprehension. Consider these variations:

1. y = √x + k (Vertical Shift): This shifts the graph of y = √x upward if k is positive and downward if k is negative. For example, y = √x + 2 shifts the graph of y = √x two units upward. The vertex would be at (0, 2).

2. y = √(x - h) + k (Horizontal and Vertical Shift): This combines both horizontal and vertical shifts. The graph of y = √(x - h) + k shifts the graph of y = √x 'h' units horizontally and 'k' units vertically. The vertex would be at (h, k). For example, y = √(x - 4) + 2 shifts the graph 4 units to the right and 2 units upward. The vertex would be at (4, 2).

3. y = a√x (Vertical Stretch/Compression): This stretches or compresses the graph vertically. If |a| > 1, the graph is stretched vertically. If 0 < |a| < 1, the graph is compressed vertically. If 'a' is negative, the graph is also reflected across the x-axis. For example, y = 2√x stretches the graph vertically, making it increase more rapidly.

4. y = √(-x) (Reflection across the y-axis): This reflects the graph of y = √x across the y-axis. The domain becomes x ≤ 0.

By comparing these transformations, you can clearly see the specific effect of the "- 4" within the square root in y = √(x - 4).

Domain and Range: A Deeper Dive

The domain and range are critical for understanding any function, and y = √(x - 4) is no exception. Let's delve a little deeper into how to determine them:

Domain:

The key is to remember that the expression inside the square root must be non-negative. This is because the square root of a negative number is not a real number. Therefore:

x - 4 ≥ 0

Solving for x:

x ≥ 4

This can be expressed in several ways:

• Inequality: x ≥ 4
• Interval Notation: [4, ∞)
• Set Notation: {x | x ∈ ℝ, x ≥ 4} (This reads: "the set of all x such that x is a real number and x is greater than or equal to 4")

Range:

The range is determined by considering the possible output values of the function. The square root function, by definition, always returns a non-negative value. Therefore, regardless of the value of x (as long as it's within the domain), √(x - 4) will always be greater than or equal to 0.

Therefore, the range is:

• Inequality: y ≥ 0
• Interval Notation: [0, ∞)
• Set Notation: {y | y ∈ ℝ, y ≥ 0}

Graphical Verification:

You can always visually verify the domain and range by looking at the graph.

• Domain: Look at the x-values that the graph covers. The graph of y = √(x - 4) starts at x = 4 and extends infinitely to the right, confirming the domain is x ≥ 4.
• Range: Look at the y-values that the graph covers. The graph of y = √(x - 4) starts at y = 0 and extends infinitely upwards, confirming the range is y ≥ 0.

Real-World Applications

While understanding the graph of y = √(x - 4) might seem purely theoretical, square root functions and their transformations have applications in various real-world scenarios:

1. Physics: The period of a simple pendulum is related to the square root of its length. Transformations of the square root function could be used to model how changes in gravity or other factors affect the pendulum's period.

2. Engineering: Square root functions can appear in calculations related to fluid dynamics, stress analysis, and other engineering problems.

3. Finance: While less direct, square root functions can be used in statistical models for calculating standard deviations and other measures of risk.

4. Computer Graphics: Square root functions can be used in algorithms for calculating distances and generating curves in computer graphics and game development.

Although the specific function y = √(x - 4) might not be directly used in these applications, understanding its properties and transformations provides a strong foundation for working with more complex mathematical models.

Common Mistakes to Avoid

When working with square root functions and their graphs, be aware of these common mistakes:

1. Forgetting the Domain Restriction: The most common mistake is forgetting that the expression inside the square root must be non-negative. This leads to incorrectly including x-values less than 4 in the domain of y = √(x - 4).

2. Incorrectly Shifting the Graph: Confusing the direction of the horizontal shift is another common error. Remember that y = √(x - 4) shifts the graph to the right, not the left.

3. Assuming Linearity: The square root function is not linear. The graph is a curve, and the rate of change is not constant.

4. Ignoring the Range: Forgetting that the square root function always returns a non-negative value can lead to errors in determining the range.

5. Misinterpreting Transformations: Confusing horizontal and vertical shifts, or stretches and compressions, can lead to incorrectly sketching the graph.

Steps to Graphing y = √(x - 4) Accurately

To ensure you can accurately graph y = √(x - 4), follow these steps:

1. Identify the Basic Function: Recognize that the function is a transformation of the basic square root function, y = √x.

2. Determine the Transformation: Identify the horizontal shift. In this case, the "- 4" indicates a shift of 4 units to the right.

3. Find the Vertex: The vertex is the starting point of the graph. For y = √(x - 4), the vertex is (4, 0).

4. Determine the Domain: Solve for x in the inequality x - 4 ≥ 0. The domain is x ≥ 4.

5. Determine the Range: The range is y ≥ 0.

6. Plot Additional Points: Choose a few x-values within the domain and calculate the corresponding y-values. Plot these points on the graph. (e.g., (5, 1), (8, 2), (13, 3)).

7. Sketch the Graph: Starting at the vertex, draw a smooth curve that passes through the plotted points and follows the general shape of the square root function. Remember that the curve becomes less steep as x increases.

8. Verify: Double-check that your graph aligns with the domain and range you determined earlier.

Conclusion

The graph of y = √(x - 4) is a transformation of the basic square root function, shifted 4 units to the right. Understanding the domain (x ≥ 4), range (y ≥ 0), vertex (4, 0), and the overall shape of the square root function allows you to accurately visualize and analyze this graph. By avoiding common mistakes and following the steps outlined above, you can confidently work with square root functions and their transformations in various mathematical and real-world contexts. Mastering this concept is a valuable step in developing a strong foundation in algebra and calculus.