Select The Correct Answer.If Function $g(x) = X^2 + 2$ Is A Transformation Of The Quadratic Parent Function, $f(x) = X^2$, Where Is The $y$-intercept Of Function $g$ Located?A. $(0, -2$\] B. $(2,

Introduction

In mathematics, transformations of functions are essential concepts that help us understand how functions can be manipulated and represented in different ways. One of the most common types of transformations is the quadratic function, which is represented by the equation $f(x) = x^2$. In this article, we will explore the concept of transformations of quadratic functions, specifically the function $g(x) = x^2 + 2$, and determine the location of its $y$-intercept.

The Parent Function

The parent function of a quadratic function is the basic form of the function, which is $f(x) = x^2$. This function has a $y$-intercept at $(0, 0)$, meaning that when $x = 0$, the value of $y$ is also $0$. The graph of the parent function is a parabola that opens upwards, with its vertex at the origin $(0, 0)$.

The Transformation Function

The transformation function $g(x) = x^2 + 2$ is a transformation of the parent function $f(x) = x^2$. This means that the graph of $g(x)$ is a vertical shift of the graph of $f(x)$ by $2$ units upwards. In other words, for every point $(x, y)$ on the graph of $f(x)$, the corresponding point on the graph of $g(x)$ is $(x, y + 2)$.

Finding the $y$-Intercept of $g(x)$

To find the $y$-intercept of $g(x)$, we need to substitute $x = 0$ into the equation $g(x) = x^2 + 2$. This gives us:

$$g(0) = (0)^2 + 2 = 2$$

Therefore, the $y$-intercept of $g(x)$ is located at the point $(0, 2)$.

Conclusion

In conclusion, the $y$-intercept of the transformation function $g(x) = x^2 + 2$ is located at the point $(0, 2)$. This is because the graph of $g(x)$ is a vertical shift of the graph of the parent function $f(x) = x^2$ by $2$ units upwards, resulting in a $y$-intercept at $(0, 2)$.

Answer

The correct answer is:

• A. $(0, 2)$

Why is this the correct answer?

This is the correct answer because the transformation function $g(x) = x^2 + 2$ is a vertical shift of the parent function $f(x) = x^2$ by $2$ units upwards. Therefore, the $y$-intercept of $g(x)$ is located at the point $(0, 2)$, which is the correct answer.

Key Takeaways

• The parent function of a quadratic function is the basic form of the function, which is $f(x) = x^2$.
• The transformation function $g(x) = x^2 + 2$ is a vertical shift of the parent function $f(x) = x^2$ by $2$ units upwards.
• The $y$-intercept of $g(x)$ is located at the point $(0, 2)$.

Further Reading

If you want to learn more about transformations of quadratic functions, I recommend checking out the following resources:

• Khan Academy: Quadratic Functions
• Mathway: Quadratic Functions
• Wolfram Alpha: Quadratic Functions

Introduction

In our previous article, we explored the concept of transformations of quadratic functions, specifically the function $g(x) = x^2 + 2$. We determined that the $y$-intercept of $g(x)$ is located at the point $(0, 2)$. In this article, we will answer some frequently asked questions about transformations of quadratic functions.

Q: What is the parent function of a quadratic function?

A: The parent function of a quadratic function is the basic form of the function, which is $f(x) = x^2$. This function has a $y$-intercept at $(0, 0)$, meaning that when $x = 0$, the value of $y$ is also $0$.

Q: What is a transformation of a quadratic function?

A: A transformation of a quadratic function is a change in the function's graph, such as a vertical or horizontal shift, a reflection, or a dilation. For example, the function $g(x) = x^2 + 2$ is a vertical shift of the parent function $f(x) = x^2$ by $2$ units upwards.

Q: How do I find the $y$-intercept of a transformed quadratic function?

A: To find the $y$-intercept of a transformed quadratic function, you need to substitute $x = 0$ into the equation of the function. For example, to find the $y$-intercept of the function $g(x) = x^2 + 2$, you would substitute $x = 0$ into the equation, which gives you $g(0) = (0)^2 + 2 = 2$.

Q: What is the difference between a vertical shift and a horizontal shift?

A: A vertical shift is a change in the function's graph that occurs when the function is shifted up or down. For example, the function $g(x) = x^2 + 2$ is a vertical shift of the parent function $f(x) = x^2$ by $2$ units upwards. A horizontal shift, on the other hand, is a change in the function's graph that occurs when the function is shifted left or right. For example, the function $g(x) = (x - 2)^2$ is a horizontal shift of the parent function $f(x) = x^2$ by $2$ units to the right.

Q: How do I determine the type of transformation that has occurred?

A: To determine the type of transformation that has occurred, you need to examine the equation of the function and look for any changes in the $x$ or $y$ values. For example, if the equation of the function is $g(x) = x^2 + 2$, you can see that the function has been shifted up by $2$ units, which is a vertical shift.

Q: Can I have multiple transformations occur at the same time?

A: Yes, it is possible to have multiple transformations occur at the same time. For example, the function $g(x) = (x - 2)^2 + 3$ is a combination of a horizontal shift and a vertical shift. The function has been shifted $2$ units to the right and $3$ units up.

Q: How do I graph a transformed quadratic function?

A: To graph a transformed quadratic function, you need to start with the graph of the parent function and then apply the transformations. For example, to graph the function $g(x) = (x - 2)^2 + 3$, you would start with the graph of the parent function $f(x) = x^2$ and then shift it $2$ units to the right and $3$ units up.

Conclusion

In conclusion, transformations of quadratic functions are an essential concept in mathematics that help us understand how functions can be manipulated and represented in different ways. By understanding the different types of transformations and how to apply them, you can graph and analyze quadratic functions with ease.

Key Takeaways

• The parent function of a quadratic function is the basic form of the function, which is $f(x) = x^2$.
• A transformation of a quadratic function is a change in the function's graph, such as a vertical or horizontal shift, a reflection, or a dilation.
• To find the $y$-intercept of a transformed quadratic function, you need to substitute $x = 0$ into the equation of the function.
• A vertical shift is a change in the function's graph that occurs when the function is shifted up or down.
• A horizontal shift is a change in the function's graph that occurs when the function is shifted left or right.

Further Reading

If you want to learn more about transformations of quadratic functions, I recommend checking out the following resources:

• Khan Academy: Quadratic Functions
• Mathway: Quadratic Functions
• Wolfram Alpha: Quadratic Functions

These resources provide a comprehensive overview of quadratic functions and their transformations, including examples and exercises to help you practice and reinforce your understanding.