Graph The Following Function:$\[ F(x) = (x-4)^2 - 3 \\]Consider The Shape Of The Graph To Determine Whether It Is A:A. Parabola B. Line C. Circle Utilize Graphing Tools As Needed To Visualize The Function.

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Introduction

In this article, we will explore the graph of the function $f(x) = (x-4)^2 - 3$ and analyze its shape to determine whether it is a parabola, line, or circle. We will utilize graphing tools to visualize the function and gain a deeper understanding of its properties.

Understanding the Function

The given function is a quadratic function in the form of $f(x) = a(x-h)^2 + k$, where $a$, $h$, and $k$ are constants. In this case, $a = 1$, $h = 4$, and $k = -3$. This function represents a parabola that has been shifted horizontally by $4$ units to the right and vertically by $3$ units down.

Graphing the Function

To graph the function, we can start by plotting the vertex of the parabola, which is located at $(h, k) = (4, -3)$. We can then use the fact that the parabola is symmetric about the vertical line $x = h$ to plot additional points.

import numpy as np
import matplotlib.pyplot as plt

# Define the function
def f(x):
    return (x-4)**2 - 3

# Generate x values
x = np.linspace(-10, 10, 400)

# Calculate y values
y = f(x)

# Create the plot
plt.plot(x, y)
plt.title('Graph of $f(x) = (x-4)^2 - 3{{content}}#39;)
plt.xlabel('x')
plt.ylabel('y')
plt.grid(True)
plt.axhline(0, color='black')
plt.axvline(0, color='black')
plt.show()

Analyzing the Shape of the Graph

From the graph, we can see that the function represents a parabola that opens upwards. The vertex of the parabola is located at $(4, -3)$, and the parabola is symmetric about the vertical line $x = 4$. This indicates that the function has a minimum value at $x = 4$, and the value of the function decreases as we move away from the vertex.

Conclusion

In conclusion, the graph of the function $f(x) = (x-4)^2 - 3$ represents a parabola that opens upwards. The vertex of the parabola is located at $(4, -3)$, and the parabola is symmetric about the vertical line $x = 4$. This indicates that the function has a minimum value at $x = 4$, and the value of the function decreases as we move away from the vertex.

Comparison with Other Options

To determine whether the graph is a parabola, line, or circle, we can compare it with the characteristics of each option.

• Parabola: A parabola is a U-shaped curve that opens upwards or downwards. The graph of the function $f(x) = (x-4)^2 - 3$ represents a parabola that opens upwards, which matches the definition of a parabola.
• Line: A line is a straight curve that extends infinitely in both directions. The graph of the function $f(x) = (x-4)^2 - 3$ does not represent a line, as it has a U-shaped curve.
• Circle: A circle is a closed curve that is symmetric about its center. The graph of the function $f(x) = (x-4)^2 - 3$ does not represent a circle, as it has a U-shaped curve and is not symmetric about a center point.

Conclusion

In conclusion, the graph of the function $f(x) = (x-4)^2 - 3$ represents a parabola that opens upwards. The vertex of the parabola is located at $(4, -3)$, and the parabola is symmetric about the vertical line $x = 4$. This indicates that the function has a minimum value at $x = 4$, and the value of the function decreases as we move away from the vertex.

References

• Graphing Quadratic Functions
• Parabolas
• Lines
• Circles

Discussion

What do you think about the graph of the function $f(x) = (x-4)^2 - 3$? Do you think it represents a parabola, line, or circle? Why or why not? Share your thoughts in the comments below!

Related Topics

• Graphing Quadratic Functions
• Parabolas
• Lines
• Circles

Final Thoughts

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Introduction

In our previous article, we explored the graph of the function $f(x) = (x-4)^2 - 3$ and analyzed its shape to determine whether it is a parabola, line, or circle. We also provided a Python code snippet to visualize the function using matplotlib. In this article, we will answer some frequently asked questions related to graphing and analyzing the function.

Q: What is the vertex of the parabola?

A: The vertex of the parabola is located at $(4, -3)$. This is the point where the parabola changes direction and starts to open upwards.

Q: Is the parabola symmetric about the vertical line $x = 4$?

A: Yes, the parabola is symmetric about the vertical line $x = 4$. This means that if we reflect the parabola about the line $x = 4$, we will get the same parabola.

Q: What is the minimum value of the function?

A: The minimum value of the function is $-3$, which occurs at $x = 4$.

Q: Is the function a parabola, line, or circle?

A: The function is a parabola that opens upwards. It does not represent a line or circle.

Q: How can I graph the function using Python?

A: You can graph the function using the following Python code snippet:

import numpy as np
import matplotlib.pyplot as plt

# Define the function
def f(x):
    return (x-4)**2 - 3

# Generate x values
x = np.linspace(-10, 10, 400)

# Calculate y values
y = f(x)

# Create the plot
plt.plot(x, y)
plt.title('Graph of $f(x) = (x-4)^2 - 3{{content}}#39;)
plt.xlabel('x')
plt.ylabel('y')
plt.grid(True)
plt.axhline(0, color='black')
plt.axvline(0, color='black')
plt.show()

Q: Can I use other graphing tools to visualize the function?

A: Yes, you can use other graphing tools such as Desmos, GeoGebra, or Graphing Calculator to visualize the function.

Q: How can I analyze the shape of the graph?

A: You can analyze the shape of the graph by looking at the following characteristics:

• The vertex of the parabola
• The symmetry of the parabola about the vertical line $x = 4$
• The minimum value of the function
• The direction of the parabola (upwards or downwards)

Conclusion

In conclusion, the graph of the function $f(x) = (x-4)^2 - 3$ represents a parabola that opens upwards. The vertex of the parabola is located at $(4, -3)$, and the parabola is symmetric about the vertical line $x = 4$. This indicates that the function has a minimum value at $x = 4$, and the value of the function decreases as we move away from the vertex.

References

• Graphing Quadratic Functions
• Parabolas
• Lines
• Circles

Discussion

What do you think about the graph of the function $f(x) = (x-4)^2 - 3$? Do you have any questions or comments about the article? Share your thoughts in the comments below!

Related Topics

• Graphing Quadratic Functions
• Parabolas
• Lines
• Circles

Final Thoughts

In conclusion, the graph of the function $f(x) = (x-4)^2 - 3$ represents a parabola that opens upwards. The vertex of the parabola is located at $(4, -3)$, and the parabola is symmetric about the vertical line $x = 4$. This indicates that the function has a minimum value at $x = 4$, and the value of the function decreases as we move away from the vertex.