Suppose F ( X ) = X 2 F(x)=x^2 F ( X ) = X 2 And G ( X ) = 4 X 2 G(x)=4x^2 G ( X ) = 4 X 2 . Which Statement Best Compares The Graph Of G ( X G(x G ( X ] With The Graph Of F ( X F(x F ( X ]?A. The Graph Of G ( X G(x G ( X ] Is The Graph Of F ( X F(x F ( X ] Shifted 4 Units Left.B. The Graph Of

Introduction

When comparing the graphs of two quadratic functions, it's essential to understand the transformations that occur between them. In this discussion, we'll compare the graphs of $f(x) = x^2$ and $g(x) = 4x^2$ to determine the best statement that describes their relationship.

Understanding Quadratic Functions

A quadratic function is a polynomial function of degree two, which means the highest power of the variable is two. The general form of a quadratic function is $f(x) = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants. The graph of a quadratic function is a parabola, which is a U-shaped curve.

Comparing $f(x)$ and $g(x)$

The function $f(x) = x^2$ is a basic quadratic function with a leading coefficient of 1. The function $g(x) = 4x^2$ is also a quadratic function, but with a leading coefficient of 4. This means that the graph of $g(x)$ will be steeper than the graph of $f(x)$.

Stretching and Shrinking

When comparing the graphs of $f(x)$ and $g(x)$, we can see that the graph of $g(x)$ is stretched vertically by a factor of 4. This is because the leading coefficient of $g(x)$ is 4 times the leading coefficient of $f(x)$. As a result, the graph of $g(x)$ will be 4 times as tall as the graph of $f(x)$.

Conclusion

Based on the comparison of the graphs of $f(x)$ and $g(x)$, we can conclude that the graph of $g(x)$ is the graph of $f(x)$ stretched vertically by a factor of 4. This means that the best statement that compares the graph of $g(x)$ with the graph of $f(x)$ is:

The graph of $g(x)$ is the graph of $f(x)$ stretched vertically by a factor of 4.

Comparison of Graphs

To visualize the comparison of the graphs of $f(x)$ and $g(x)$, we can use the following graph:

	$f(x) = x^2$	$g(x) = 4x^2$
Leading Coefficient	1	4
Graph Shape	U-shaped	U-shaped
Graph Height	1 unit	4 units

As we can see from the table, the graph of $g(x)$ is stretched vertically by a factor of 4 compared to the graph of $f(x)$.

Example

Suppose we want to compare the graphs of $f(x) = x^2$ and $g(x) = 6x^2$. Using the same logic as before, we can conclude that the graph of $g(x)$ is the graph of $f(x)$ stretched vertically by a factor of 6.

Conclusion

In conclusion, the graph of $g(x) = 4x^2$ is the graph of $f(x) = x^2$ stretched vertically by a factor of 4. This means that the best statement that compares the graph of $g(x)$ with the graph of $f(x)$ is:

The graph of $g(x)$ is the graph of $f(x)$ stretched vertically by a factor of 4.

Final Answer

The final answer is:

$$$$

Introduction

In our previous discussion, we compared the graphs of $f(x) = x^2$ and $g(x) = 4x^2$ to determine the best statement that describes their relationship. We concluded that the graph of $g(x)$ is the graph of $f(x)$ stretched vertically by a factor of 4.

Q&A

Q: What is the difference between the graphs of $f(x) = x^2$ and $g(x) = 4x^2$?

A: The graph of $g(x)$ is the graph of $f(x)$ stretched vertically by a factor of 4.

Q: How do we compare the graphs of two quadratic functions?

A: To compare the graphs of two quadratic functions, we need to understand the transformations that occur between them. We can compare the leading coefficients, graph shapes, and graph heights to determine the relationship between the two graphs.

Q: What is the effect of a leading coefficient on the graph of a quadratic function?

A: A leading coefficient affects the steepness of the graph of a quadratic function. A larger leading coefficient results in a steeper graph.

Q: How do we determine the relationship between the graphs of two quadratic functions?

A: We can determine the relationship between the graphs of two quadratic functions by comparing their leading coefficients, graph shapes, and graph heights.

Q: What is the difference between the graphs of $f(x) = x^2$ and $g(x) = 6x^2$?

A: The graph of $g(x)$ is the graph of $f(x)$ stretched vertically by a factor of 6.

Q: Can we compare the graphs of two quadratic functions with different graph shapes?

A: Yes, we can compare the graphs of two quadratic functions with different graph shapes. However, we need to consider the transformations that occur between the two graphs.

Q: How do we visualize the comparison of the graphs of two quadratic functions?

A: We can use a table or a graph to visualize the comparison of the graphs of two quadratic functions.

Example Questions

Q: Compare the graphs of $f(x) = x^2$ and $g(x) = 2x^2$.

A: The graph of $g(x)$ is the graph of $f(x)$ stretched vertically by a factor of 2.

Q: Compare the graphs of $f(x) = x^2$ and $g(x) = -x^2$.

A: The graph of $g(x)$ is the graph of $f(x)$ reflected across the x-axis.

Q: Compare the graphs of $f(x) = x^2$ and $g(x) = x^2 + 2$.

A: The graph of $g(x)$ is the graph of $f(x)$ shifted vertically by 2 units.

Conclusion

In conclusion, comparing the graphs of two quadratic functions requires understanding the transformations that occur between them. We can compare the leading coefficients, graph shapes, and graph heights to determine the relationship between the two graphs. By using a table or a graph, we can visualize the comparison of the graphs of two quadratic functions.

Final Answer

The final answer is:

The graph of $g(x)$ is the graph of $f(x)$ stretched vertically by a factor of 4.