Which Of The Following Is The Graph Of y = Negative 4 Startroot x Endroot?

The Graph of y = -√x

The graph of the function y = -√x is a fundamental concept in mathematics, with applications in various fields. In this comprehensive article, we will explore the properties, characteristics, and behavior of this graph, as well as its practical implications.

Understanding the Function y = -√x

The function y = -√x is a square root function with a negative sign. This means that the function takes the square root of the input x and then multiplies the result by -1. The negative sign in front of the square root operation reflects the fact that the function produces negative values for positive inputs.The domain of the function y = -√x is the set of non-negative real numbers, as the square root operation is only defined for non-negative values. The range of the function, on the other hand, is the set of non-positive real numbers, as the function always produces negative values.

Graphing y = -√x

To graph the function y = -√x, we can follow these steps:

1. Identify the Domain: The domain of the function is the set of non-negative real numbers, meaning x ≥ 0.
2. Plot Points: Choose several non-negative values of x and calculate the corresponding values of y using the formula y = -√x. Plot these points on a coordinate plane.
3. Connect the Points: Connect the plotted points to form a smooth curve. The resulting graph will be a parabolic curve that opens downward and passes through the origin (0, 0).

The graph of y = -√x has the following characteristics:

• Shape: The graph is a parabolic curve that opens downward.
• Symmetry: The graph is symmetric about the y-axis, as the function is an odd function (f(-x) = -f(x)).
• Intercepts: The graph intersects the y-axis at the point (0, 0), as the function passes through the origin.
• Behavior: The graph approaches the x-axis as x increases, but never touches it, as the function is always negative for positive values of x.

Comparing y = -√x to y = √x

It is important to note the difference between the graphs of y = -√x and y = √x. While both are square root functions, the former is a reflection of the latter about the x-axis.The key differences between the two graphs are:

• Shape: The graph of y = √x is a parabolic curve that opens upward, while the graph of y = -√x opens downward.
• Symmetry: The graph of y = √x is symmetric about the origin (0, 0), while the graph of y = -√x is symmetric about the y-axis.
• Intercepts: The graph of y = √x intersects the y-axis at the point (0, 0), while the graph of y = -√x intersects the y-axis at the point (0, 0).
• Behavior: The graph of y = √x approaches the x-axis as x increases, while the graph of y = -√x approaches the x-axis from below.

Applications of y = -√x

The function y = -√x and its graph have various applications in different fields, including:

1. Physics: The function y = -√x can be used to model the motion of an object under the influence of gravity, such as the trajectory of a projectile.
2. Engineering: The function y = -√x can be used to model the shape of certain structures, such as the profile of a parabolic antenna or the cross-section of a suspension bridge.
3. Economics: The function y = -√x can be used to model the relationship between supply and demand, where the quantity supplied is a function of the price.
4. Biology: The function y = -√x can be used to model the growth of certain organisms, such as the population of a species over time.

Conclusion

The graph of the function y = -√x is a fundamental concept in mathematics with a wide range of applications. By understanding the properties and characteristics of this graph, we can gain insights into various real-world phenomena and solve problems in a variety of fields. Whether you’re a student, a researcher, or a practitioner, mastering the understanding of the graph of y = -√x can be a valuable asset in your work and studies.

FAQ

1. What is the domain of the function y = -√x?
   • The domain of the function y = -√x is the set of non-negative real numbers, meaning x ≥ 0.
2. What is the range of the function y = -√x?
   • The range of the function y = -√x is the set of non-positive real numbers, as the function always produces negative values.
3. How is the graph of y = -√x different from the graph of y = √x?
   • The key differences are that the graph of y = -√x is a reflection of the graph of y = √x about the x-axis, and the graph of y = -√x is symmetric about the y-axis, while the graph of y = √x is symmetric about the origin.
4. What are some practical applications of the function y = -√x?
   • The function y = -√x and its graph have applications in physics, engineering, economics, and biology, such as modeling the motion of projectiles, the shape of structures, supply and demand relationships, and population growth.
5. How can you graph the function y = -√x?
   • To graph the function y = -√x, you can identify the domain, plot points by calculating y values for various non-negative x values, and then connect the points to form a smooth parabolic curve that opens downward and passes through the origin.

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