The Parabola $y=\sqrt{x-4}$ (principal Square Root) Opens:A. Down B. Up C. Right D. Left

Introduction

When it comes to parabolas, understanding their orientation is crucial in mathematics. A parabola is a type of quadratic equation that can be represented in various forms, including the standard form, vertex form, and even the principal square root form, as seen in the equation $y=\sqrt{x-4}$. In this article, we will delve into the world of parabolas and explore the orientation of the parabola $y=\sqrt{x-4}$.

Understanding Parabolas

A parabola is a U-shaped curve that can open either upwards or downwards. The orientation of a parabola is determined by its coefficient, which is the number in front of the squared term in the equation. If the coefficient is positive, the parabola opens upwards, while a negative coefficient indicates that the parabola opens downwards.

The Principal Square Root Form

The equation $y=\sqrt{x-4}$ represents a parabola in the principal square root form. This form is characterized by the presence of a square root, which indicates that the parabola is a function of the square root of the variable. In this case, the variable is $x-4$, which means that the parabola is shifted 4 units to the right.

Analyzing the Coefficient

To determine the orientation of the parabola $y=\sqrt{x-4}$, we need to analyze the coefficient of the squared term. In this case, the coefficient is 1, which is a positive number. However, we must note that the square root function is involved, which can affect the orientation of the parabola.

The Role of the Square Root Function

The square root function is a non-negative function, which means that it can only take on non-negative values. When we take the square root of a number, we are essentially finding the number that, when multiplied by itself, gives us the original number. In the case of the equation $y=\sqrt{x-4}$, the square root function is applied to the expression $x-4$.

Determining the Orientation

Since the square root function is non-negative, the parabola $y=\sqrt{x-4}$ will only take on non-negative values. This means that the parabola will open upwards, but with a twist. The parabola will only be defined for values of $x$ that are greater than or equal to 4, since the square root of a negative number is undefined.

Conclusion

In conclusion, the parabola $y=\sqrt{x-4}$ opens upwards, but with a restriction. The parabola is only defined for values of $x$ that are greater than or equal to 4, due to the presence of the square root function. This unique characteristic of the parabola $y=\sqrt{x-4}$ sets it apart from other parabolas, and highlights the importance of understanding the role of the square root function in determining the orientation of a parabola.

Final Answer

The final answer to the question of whether the parabola $y=\sqrt{x-4}$ opens up, down, right, or left is:

• A. up

This is because the parabola $y=\sqrt{x-4}$ takes on non-negative values, and is only defined for values of $x$ that are greater than or equal to 4.

Introduction

In our previous article, we explored the orientation of the parabola $y=\sqrt{x-4}$. In this article, we will delve deeper into the world of parabolas and answer some of the most frequently asked questions about the parabola $y=\sqrt{x-4}$.

Q&A

Q1: What is the vertex of the parabola $y=\sqrt{x-4}$?

A1: The vertex of the parabola $y=\sqrt{x-4}$ is at the point (4, 0). This is because the parabola is shifted 4 units to the right, and the square root function is applied to the expression $x-4$.

Q2: What is the domain of the parabola $y=\sqrt{x-4}$?

A2: The domain of the parabola $y=\sqrt{x-4}$ is all real numbers $x$ such that $x \geq 4$. This is because the square root function is non-negative, and the parabola is only defined for values of $x$ that are greater than or equal to 4.

Q3: What is the range of the parabola $y=\sqrt{x-4}$?

A3: The range of the parabola $y=\sqrt{x-4}$ is all non-negative real numbers $y$. This is because the square root function is non-negative, and the parabola takes on non-negative values.

Q4: Is the parabola $y=\sqrt{x-4}$ a function?

A4: Yes, the parabola $y=\sqrt{x-4}$ is a function. This is because each value of $x$ corresponds to a unique value of $y$, and the parabola passes the vertical line test.

Q5: Can the parabola $y=\sqrt{x-4}$ be written in the standard form?

A5: Yes, the parabola $y=\sqrt{x-4}$ can be written in the standard form $y = a(x-h)^2 + k$. However, this would require some algebraic manipulation to isolate the squared term.

Q6: What is the axis of symmetry of the parabola $y=\sqrt{x-4}$?

A6: The axis of symmetry of the parabola $y=\sqrt{x-4}$ is the vertical line $x = 4$. This is because the parabola is shifted 4 units to the right, and the axis of symmetry is the line that passes through the vertex.

Q7: Can the parabola $y=\sqrt{x-4}$ be graphed on a coordinate plane?

A7: Yes, the parabola $y=\sqrt{x-4}$ can be graphed on a coordinate plane. The graph will be a U-shaped curve that opens upwards, but with a restriction that the parabola is only defined for values of $x$ that are greater than or equal to 4.

Conclusion

In conclusion, the parabola $y=\sqrt{x-4}$ is a unique and fascinating curve that has many interesting properties. By understanding the orientation, vertex, domain, range, and axis of symmetry of the parabola, we can gain a deeper appreciation for the mathematics behind this curve.

Final Answer

The final answer to the question of whether the parabola $y=\sqrt{x-4}$ opens up, down, right, or left is:

• A. up

This is because the parabola $y=\sqrt{x-4}$ takes on non-negative values, and is only defined for values of $x$ that are greater than or equal to 4.