Graphing the Square Root Function

Oct 13, 2025 by magicalceylontours.com 57 views

Graphing the Square Root Function

To identify the true statements when graphing the square root function $f(x)=-\sqrt{x-2}+3$ , we need to understand the properties of the square root function and how it is affected by the given function.

Step 1: Understanding the Square Root Function

The square root function is defined as $f(x)=\sqrt{x}$ , where $x\geq0$ . The square root function has a few key properties:

It is an increasing function, meaning that as $x$ increases, $f(x)$ also increases.
It has a minimum value of 0, which occurs when $x=0$ .
It has a vertical asymptote at $x=0$ , meaning that the function approaches infinity as $x$ approaches 0 from the right.

Step 2: Understanding the Given Function

The given function is $f(x)=-\sqrt{x-2}+3$ . This function has a few key features:

The square root term is inside the function, which means that the function will only be defined for values of $x$ that are greater than or equal to 2.
The negative sign in front of the square root term means that the function will be decreasing as $x$ increases.
The constant term 3 is added to the function, which means that the function will be shifted up by 3 units.

Step 3: Identifying the True Statements

Now that we have a good understanding of the square root function and the given function, we can identify the true statements when graphing the function.

I. The horizontal shift is left 2 spaces.

The given function is $f(x)=-\sqrt{x-2}+3$ . The term $x-2$ inside the square root function means that the function will be shifted to the right by 2 units. This is because the function is defined for values of $x$ that are greater than or equal to 2, and the graph of the function will be shifted to the right by 2 units to reflect this.

Therefore, the statement "The horizontal shift is left 2 spaces" is false.

II. The graph of the function is a decreasing function.

The given function is $f(x)=-\sqrt{x-2}+3$ . The negative sign in front of the square root term means that the function will be decreasing as $x$ increases. This is because the square root term is always non-negative, and the negative sign in front of it means that the function will be decreasing.

Therefore, the statement "The graph of the function is a decreasing function" is true.

III. The graph of the function has a vertical asymptote at $x=2$ .

The given function is $f(x)=-\sqrt{x-2}+3$ . The term $x-2$ inside the square root function means that the function will have a vertical asymptote at $x=2$ . This is because the function is defined for values of $x$ that are greater than or equal to 2, and the graph of the function will have a vertical asymptote at $x=2$ to reflect this.

Therefore, the statement "The graph of the function has a vertical asymptote at $x=2$ " is true.

IV. The graph of the function has a minimum value of 3.

The given function is $f(x)=-\sqrt{x-2}+3$ . The constant term 3 is added to the function, which means that the function will have a minimum value of 3. This is because the square root term is always non-negative, and the negative sign in front of it means that the function will be decreasing. The constant term 3 is added to the function, which means that the function will have a minimum value of 3.

Therefore, the statement "The graph of the function has a minimum value of 3" is true.

V. The graph of the function is an increasing function.

Therefore, the statement "The graph of the function is an increasing function" is false.

Conclusion

In conclusion, the true statements when graphing the square root function $f(x)=-\sqrt{x-2}+3$ are:

The graph of the function is a decreasing function.
The graph of the function has a vertical asymptote at $x=2$ .
The graph of the function has a minimum value of 3.

The false statements are:

The horizontal shift is left 2 spaces.
The graph of the function is an increasing function.
Frequently Asked Questions (FAQs) =====================================

Q: What is the domain of the function $f(x)=-\sqrt{x-2}+3$ ?

A: The domain of the function $f(x)=-\sqrt{x-2}+3$ is all real numbers $x$ such that $x\geq2$ . This is because the square root term inside the function requires that $x-2\geq0$ , which means that $x\geq2$ .

Q: What is the range of the function $f(x)=-\sqrt{x-2}+3$ ?

A: The range of the function $f(x)=-\sqrt{x-2}+3$ is all real numbers $y$ such that $y\leq3$ . This is because the square root term inside the function is always non-negative, and the negative sign in front of it means that the function will be decreasing. The constant term 3 is added to the function, which means that the function will have a minimum value of 3.

Q: What is the x-intercept of the function $f(x)=-\sqrt{x-2}+3$ ?

A: The x-intercept of the function $f(x)=-\sqrt{x-2}+3$ is the value of $x$ such that $f(x)=0$ . This means that we need to solve the equation $-\sqrt{x-2}+3=0$ . Solving for $x$ , we get $x=7$ . Therefore, the x-intercept of the function is $x=7$ .

Q: What is the y-intercept of the function $f(x)=-\sqrt{x-2}+3$ ?

A: The y-intercept of the function $f(x)=-\sqrt{x-2}+3$ is the value of $y$ such that $f(0)=y$ . This means that we need to evaluate the function at $x=0$ . Plugging in $x=0$ into the function, we get $f(0)=-\sqrt{0-2}+3=-\sqrt{-2}+3$ . Since the square root of a negative number is not a real number, we can conclude that the y-intercept of the function is not a real number.

Q: What is the vertex of the function $f(x)=-\sqrt{x-2}+3$ ?

A: The vertex of the function $f(x)=-\sqrt{x-2}+3$ is the point on the graph of the function where the function changes from decreasing to increasing. To find the vertex, we need to find the value of $x$ such that the derivative of the function is equal to 0. The derivative of the function is given by $f'(x)=-\frac{1}{2\sqrt{x-2}}$ . Setting this equal to 0, we get $-\frac{1}{2\sqrt{x-2}}=0$ . Solving for $x$ , we get $x=2$ . Therefore, the vertex of the function is the point $(2,3)$ .

Q: What is the equation of the horizontal asymptote of the function $f(x)=-\sqrt{x-2}+3$ ?

A: The equation of the horizontal asymptote of the function $f(x)=-\sqrt{x-2}+3$ is $y=3$ . This is because the function is decreasing as $x$ increases, and the constant term 3 is added to the function, which means that the function will have a minimum value of 3.

Q: What is the equation of the vertical asymptote of the function $f(x)=-\sqrt{x-2}+3$ ?

A: The equation of the vertical asymptote of the function $f(x)=-\sqrt{x-2}+3$ is $x=2$ . This is because the square root term inside the function requires that $x-2\geq0$ , which means that $x\geq2$ . The function is not defined for values of $x$ less than 2, so the vertical asymptote is at $x=2$ .

Conclusion

In conclusion, we have answered some of the most frequently asked questions about the function $f(x)=-\sqrt{x-2}+3$ . We have discussed the domain and range of the function, the x-intercept and y-intercept, the vertex, and the horizontal and vertical asymptotes. We hope that this article has been helpful in understanding the properties of the function.