Vertical Stretching and Shrinking

Oct 13, 2025 by windailys.com 45 views

Introduction

When analyzing the graph of a function, it's essential to consider various aspects, including the domain, range, and behavior of the function. In this article, we will delve into the graph of $$h(x)=\sqrt[3]{x-4}$$ and determine which statements are true about its domain, range, and behavior.

Domain of $$h(x)$$

The domain of a function is the set of all possible input values for which the function is defined. In the case of $$h(x)=\sqrt[3]{x-4}$$, we need to consider the restrictions on the input values. Since the cube root function is defined for all real numbers, the expression $$x-4$$ must be a real number. This means that the domain of $$h(x)$$ is the set of all real numbers.

The domain of $$h(x)$$ is the set of all real numbers.

Range of $$h(x)$$

The range of a function is the set of all possible output values. For the function $$h(x)=\sqrt[3]{x-4}$$, we need to consider the possible output values. Since the cube root function can take on any real value, the range of $$h(x)$$ is also the set of all real numbers.

The range of $$h(x)$$ is the set of all real numbers.

Behavior of $$h(x)$$

To understand the behavior of $$h(x)$$, we need to consider the graph of the function. The graph of $$h(x)=\sqrt[3]{x-4}$$ is a cube root function shifted 4 units to the right. This means that the graph will have a similar shape to the graph of $$y=\sqrt[3]{x}$$, but shifted 4 units to the right.

Vertical Stretching and Shrinking

When analyzing the graph of $$h(x)=\sqrt[3]{x-4}$$, we need to consider the effect of the vertical stretching and shrinking. Since the cube root function is not stretched or shrunk, the graph will have the same vertical scale as the graph of $$y=\sqrt[3]{x}$$.

Horizontal Stretching and Shrinking

The graph of $$h(x)=\sqrt[3]{x-4}$$ is shifted 4 units to the right, which means that the graph will be horizontally stretched by a factor of 4. This will result in a wider graph compared to the graph of $$y=\sqrt[3]{x}$$.

Reflections

The graph of $$h(x)=\sqrt[3]{x-4}$$ is not reflected across the x-axis or y-axis. This means that the graph will not have any reflections.

Symmetry

The graph of $$h(x)=\sqrt[3]{x-4}$$ is not symmetric about the x-axis or y-axis. This means that the graph will not have any symmetry.

Asymptotes

The graph of $$h(x)=\sqrt[3]{x-4}$$ does not have any vertical asymptotes. This means that the graph will not have any vertical asymptotes.

Horizontal Asymptotes

The graph of $$h(x)=\sqrt[3]{x-4}$$ does not have any horizontal asymptotes. This means that the graph will not have any horizontal asymptotes.

Conclusion

In conclusion, the statements that are true about the graph of $$h(x)=\sqrt[3]{x-4}$$ are:

The domain of $$h(x)$$ is the set of all real numbers.
The range of $$h(x)$$ is the set of all real numbers.

The statements that are not true about the graph of $$h(x)=\sqrt[3]{x-4}$$ are:

For all points on the graph of $$h(x)$$, the y-coordinate is greater than or equal to 0.
The graph of $$h(x)$$ is symmetric about the x-axis.
The graph of $$h(x)$$ has a vertical asymptote at x=4.
The graph of $$h(x)$$ has a horizontal asymptote at y=0.

Final Answer

The final answer is:

The domain of $$h(x)$$ is the set of all real numbers.
The range of $$h(x)$$ is the set of all real numbers.

Note: The other statements are not true about the graph of $$h(x)=\sqrt[3]{x-4}$$.

Q: What is the domain of the function $$h(x)=\sqrt[3]{x-4}$$?

A: The domain of the function $$h(x)=\sqrt[3]{x-4}$$ is the set of all real numbers. This means that the function is defined for all possible input values.

Q: What is the range of the function $$h(x)=\sqrt[3]{x-4}$$?

A: The range of the function $$h(x)=\sqrt[3]{x-4}$$ is the set of all real numbers. This means that the function can take on any real value as an output.

Q: Is the graph of $$h(x)=\sqrt[3]{x-4}$$ symmetric about the x-axis?

A: No, the graph of $$h(x)=\sqrt[3]{x-4}$$ is not symmetric about the x-axis. This means that the graph will not have any symmetry about the x-axis.

Q: Is the graph of $$h(x)=\sqrt[3]{x-4}$$ symmetric about the y-axis?

A: No, the graph of $$h(x)=\sqrt[3]{x-4}$$ is not symmetric about the y-axis. This means that the graph will not have any symmetry about the y-axis.

Q: Does the graph of $$h(x)=\sqrt[3]{x-4}$$ have any vertical asymptotes?

A: No, the graph of $$h(x)=\sqrt[3]{x-4}$$ does not have any vertical asymptotes. This means that the graph will not have any vertical asymptotes.

Q: Does the graph of $$h(x)=\sqrt[3]{x-4}$$ have any horizontal asymptotes?

A: No, the graph of $$h(x)=\sqrt[3]{x-4}$$ does not have any horizontal asymptotes. This means that the graph will not have any horizontal asymptotes.

Q: Is the graph of $$h(x)=\sqrt[3]{x-4}$$ a function?

A: Yes, the graph of $$h(x)=\sqrt[3]{x-4}$$ is a function. This means that for every input value, there is a unique output value.

Q: What is the behavior of the graph of $$h(x)=\sqrt[3]{x-4}$$ as x approaches infinity?

A: As x approaches infinity, the graph of $$h(x)=\sqrt[3]{x-4}$$ will approach infinity. This means that the graph will continue to increase without bound as x gets larger and larger.

Q: What is the behavior of the graph of $$h(x)=\sqrt[3]{x-4}$$ as x approaches negative infinity?

A: As x approaches negative infinity, the graph of $$h(x)=\sqrt[3]{x-4}$$ will approach negative infinity. This means that the graph will continue to decrease without bound as x gets smaller and smaller.

Q: What is the vertex of the graph of $$h(x)=\sqrt[3]{x-4}$$?

A: The vertex of the graph of $$h(x)=\sqrt[3]{x-4}$$ is at the point (4,0). This means that the vertex is located at the point (4,0) on the graph.

Q: What is the x-intercept of the graph of $$h(x)=\sqrt[3]{x-4}$$?

A: The x-intercept of the graph of $$h(x)=\sqrt[3]{x-4}$$ is at the point (4,0). This means that the x-intercept is located at the point (4,0) on the graph.

Q: What is the y-intercept of the graph of $$h(x)=\sqrt[3]{x-4}$$?

A: The y-intercept of the graph of $$h(x)=\sqrt[3]{x-4}$$ is at the point (0,-4). This means that the y-intercept is located at the point (0,-4) on the graph.

Conclusion

In conclusion, the graph of $$h(x)=\sqrt[3]{x-4}$$ is a function that has a domain of all real numbers and a range of all real numbers. The graph is not symmetric about the x-axis or y-axis, and it does not have any vertical or horizontal asymptotes. The graph approaches infinity as x approaches infinity and negative infinity as x approaches negative infinity. The vertex of the graph is at the point (4,0), and the x-intercept and y-intercept are at the points (4,0) and (0,-4), respectively.