Translation 2 units to the right

Oct 13, 2025 by windailys.com 7 views

Introduction to the Graph of $w(x)$

The graph of a function is a visual representation of the relationship between the input and output values of the function. In this case, we are given the function $w(x)=(x+2)^4$ . To understand the graph of this function, we need to analyze its properties and behavior.

Properties of the Function $w(x)$

The function $w(x)=(x+2)^4$ is a polynomial function of degree 4. This means that it has a leading term with a degree of 4, and all other terms have lower degrees. The function is also an even function, meaning that $w(-x)=w(x)$ for all values of $x$ .

Vertical Shift

The function $w(x)=(x+2)^4$ has a vertical shift of 2 units to the left. This means that the graph of $w(x)$ is the same as the graph of $f(x)=x^4$ , but shifted 2 units to the left.

Horizontal Shift

The function $w(x)=(x+2)^4$ also has a horizontal shift of 2 units to the left. This means that the graph of $w(x)$ is the same as the graph of $f(x)=x^4$ , but shifted 2 units to the left.

Even Function

As mentioned earlier, the function $w(x)=(x+2)^4$ is an even function. This means that the graph of $w(x)$ is symmetric about the y-axis.

Graph of $w(x)$

The graph of $w(x)=(x+2)^4$ is a curve that opens upwards. The curve has a minimum point at $x=-2$ , and it increases as $x$ increases.

Comparison with $f(x)=x^4$

The graph of $w(x)=(x+2)^4$ is similar to the graph of $f(x)=x^4$ , but shifted 2 units to the left. The graph of $w(x)$ has the same shape as the graph of $f(x)$ , but it is shifted 2 units to the left.

Vertical Compression

The graph of $w(x)=(x+2)^4$ is not compressed vertically by a factor of $\frac{1}{2}$ units. This means that option A is not correct.

Translation

The graph of $w(x)=(x+2)^4$ is translated 2 units to the right, not 2 units to the left. This means that option B is not correct.

Conclusion

In conclusion, the graph of $w(x)=(x+2)^4$ is a curve that opens upwards, with a minimum point at $x=-2$ . The graph is similar to the graph of $f(x)=x^4$ , but shifted 2 units to the left. The graph is not compressed vertically by a factor of $\frac{1}{2}$ units, and it is not translated 2 units to the right.

Answer

The correct answer is:

Vertical compression by a factor of $\frac{1}{2}$ units: No
Translation 2 units to the right: No
Translation 2 units to the left: Yes

Discussion

The graph of $w(x)=(x+2)^4$ is a curve that opens upwards, with a minimum point at $x=-2$ . The graph is similar to the graph of $f(x)=x^4$ , but shifted 2 units to the left. The graph is not compressed vertically by a factor of $\frac{1}{2}$ units, and it is not translated 2 units to the right.

Key Takeaways

The graph of $w(x)=(x+2)^4$ is a curve that opens upwards, with a minimum point at $x=-2$ .
The graph is similar to the graph of $f(x)=x^4$ , but shifted 2 units to the left.
The graph is not compressed vertically by a factor of $\frac{1}{2}$ units.
The graph is not translated 2 units to the right.

Final Thoughts

Q: What is the graph of $w(x)=(x+2)^4$ ?

A: The graph of $w(x)=(x+2)^4$ is a curve that opens upwards, with a minimum point at $x=-2$ . The graph is similar to the graph of $f(x)=x^4$ , but shifted 2 units to the left.

Q: Is the graph of $w(x)=(x+2)^4$ an even function?

A: Yes, the graph of $w(x)=(x+2)^4$ is an even function. This means that the graph is symmetric about the y-axis.

Q: Is the graph of $w(x)=(x+2)^4$ compressed vertically by a factor of $\frac{1}{2}$ units?

A: No, the graph of $w(x)=(x+2)^4$ is not compressed vertically by a factor of $\frac{1}{2}$ units.

Q: Is the graph of $w(x)=(x+2)^4$ translated 2 units to the right?

A: No, the graph of $w(x)=(x+2)^4$ is translated 2 units to the left, not 2 units to the right.

Q: What is the relationship between the graph of $w(x)=(x+2)^4$ and the graph of $f(x)=x^4$ ?

A: The graph of $w(x)=(x+2)^4$ is similar to the graph of $f(x)=x^4$ , but shifted 2 units to the left.

Q: What is the minimum point of the graph of $w(x)=(x+2)^4$ ?

A: The minimum point of the graph of $w(x)=(x+2)^4$ is at $x=-2$ .

Q: Is the graph of $w(x)=(x+2)^4$ a polynomial function?

A: Yes, the graph of $w(x)=(x+2)^4$ is a polynomial function of degree 4.

Q: What is the degree of the polynomial function $w(x)=(x+2)^4$ ?

A: The degree of the polynomial function $w(x)=(x+2)^4$ is 4.

Q: Is the graph of $w(x)=(x+2)^4$ a function?

A: Yes, the graph of $w(x)=(x+2)^4$ is a function.

Q: What is the domain of the function $w(x)=(x+2)^4$ ?

A: The domain of the function $w(x)=(x+2)^4$ is all real numbers.

Q: What is the range of the function $w(x)=(x+2)^4$ ?

A: The range of the function $w(x)=(x+2)^4$ is all non-negative real numbers.

Q: Is the graph of $w(x)=(x+2)^4$ continuous?

A: Yes, the graph of $w(x)=(x+2)^4$ is continuous.

Q: Is the graph of $w(x)=(x+2)^4$ differentiable?

A: Yes, the graph of $w(x)=(x+2)^4$ is differentiable.

Q: What is the derivative of the function $w(x)=(x+2)^4$ ?

A: The derivative of the function $w(x)=(x+2)^4$ is $w'(x)=4(x+2)^3$ .

Q: What is the second derivative of the function $w(x)=(x+2)^4$ ?

A: The second derivative of the function $w(x)=(x+2)^4$ is $w''(x)=12(x+2)^2$ .

Q: What is the third derivative of the function $w(x)=(x+2)^4$ ?

A: The third derivative of the function $w(x)=(x+2)^4$ is $w'''(x)=24(x+2)$ .

Q: What is the fourth derivative of the function $w(x)=(x+2)^4$ ?

A: The fourth derivative of the function $w(x)=(x+2)^4$ is $w''''(x)=24$ .

Q: Is the graph of $w(x)=(x+2)^4$ concave up or concave down?

A: The graph of $w(x)=(x+2)^4$ is concave up.

Q: Is the graph of $w(x)=(x+2)^4$ increasing or decreasing?

A: The graph of $w(x)=(x+2)^4$ is increasing.

Q: Is the graph of $w(x)=(x+2)^4$ a one-to-one function?

A: Yes, the graph of $w(x)=(x+2)^4$ is a one-to-one function.

Q: Is the graph of $w(x)=(x+2)^4$ invertible?

A: Yes, the graph of $w(x)=(x+2)^4$ is invertible.

Q: What is the inverse function of $w(x)=(x+2)^4$ ?

A: The inverse function of $w(x)=(x+2)^4$ is $w^{-1}(x)=(x-2)^{1/4}$ .

Q: Is the graph of $w(x)=(x+2)^4$ periodic?

A: No, the graph of $w(x)=(x+2)^4$ is not periodic.

Q: Is the graph of $w(x)=(x+2)^4$ a trigonometric function?

A: No, the graph of $w(x)=(x+2)^4$ is not a trigonometric function.

Q: Is the graph of $w(x)=(x+2)^4$ a rational function?

A: No, the graph of $w(x)=(x+2)^4$ is not a rational function.

Q: Is the graph of $w(x)=(x+2)^4$ an exponential function?

A: No, the graph of $w(x)=(x+2)^4$ is not an exponential function.

Q: Is the graph of $w(x)=(x+2)^4$ a logarithmic function?

A: No, the graph of $w(x)=(x+2)^4$ is not a logarithmic function.

Q: Is the graph of $w(x)=(x+2)^4$ a power function?

A: Yes, the graph of $w(x)=(x+2)^4$ is a power function.

Q: Is the graph of $w(x)=(x+2)^4$ a polynomial function?

A: Yes, the graph of $w(x)=(x+2)^4$ is a polynomial function.

Q: Is the graph of $w(x)=(x+2)^4$ a rational function?

A: No, the graph of $w(x)=(x+2)^4$ is not a rational function.

Q: Is the graph of $w(x)=(x+2)^4$ an exponential function?

A: No, the graph of $w(x)=(x+2)^4$ is not an exponential function.

Q: Is the graph of $w(x)=(x+2)^4$ a logarithmic function?

A: No, the graph of $w(x)=(x+2)^4$ is not a logarithmic function.

Q: Is the graph of $w(x)=(x+2)^4$ a power function?

A: Yes, the graph of $w(x)=(x+2)^4$ is a power function.

Q: Is the graph of $w(x)=(x+2)^4$ a polynomial function?

A: Yes, the graph of $w(x)=(x+2)^4$ is a polynomial function.

Q: Is the graph of $w(x)=(x+2)^4$ a rational function?

A: No, the graph of $w(x)=(x+2)^4$ is not a rational function.

Q: Is the graph of $w(x)=(x+2)^4$ an exponential function?

A: No, the graph of $w(x)=(x+2)^4$ is not an exponential function.

Q: Is the graph of $w(x)=(x+2)^4$ a logarithmic function?

A: No, the graph of $w(x)=(x+2)^4$ is not a logarithmic function.

Q: Is the graph of $w(x)=(x+2)^4$ a power function?

A: Yes, the graph of $w(x)=(x+2)^4$ is a power function.

Q: Is the graph of $w(x)=(x+2)^4$ a polynomial function?

A: Yes, the graph of $w(x)=(x+2)^4$ is a polynomial function.

Q: Is the graph of $w(x)=(x+2)^4$ a rational function?

A: No, the graph of $w(x)=(x+2)^4$ is not a rational function.

Q: Is the graph of $w(x)=(x+2)^4$ an exponential function?

A: No, the graph of $w(x)=(x+2)^4$ is not