What Transformations Take The Graph Of $f(x)=x^2$ To The Graph Of $g(x)=3(x-2)^2+1$?A. Vertical Stretch By A Factor Of 3 B. Right 2 Units C. Up 1 Unit

What transformations take the graph of $f(x)=x^2$ to the graph of $g(x)=3(x-2)^2+1$?

In mathematics, transformations are used to change the graph of a function. These transformations can be vertical, horizontal, or a combination of both. In this article, we will explore the transformations that take the graph of $f(x)=x^2$ to the graph of $g(x)=3(x-2)^2+1$. We will analyze each transformation step by step and provide examples to illustrate our findings.

Vertical Stretch by a Factor of 3

The first transformation we will consider is a vertical stretch by a factor of 3. This means that the graph of $f(x)=x^2$ will be stretched vertically by a factor of 3 to obtain the graph of $g(x)=3(x-2)^2+1$. To understand this transformation, let's consider the equation of the graph of $f(x)=x^2$. The equation is in the form $y=ax^2$, where $a=1$. When we multiply the equation by 3, we get $y=3x^2$. This means that the graph of $f(x)=x^2$ will be stretched vertically by a factor of 3 to obtain the graph of $y=3x^2$.

However, the graph of $g(x)=3(x-2)^2+1$ is not just a vertical stretch of the graph of $f(x)=x^2$. It also has a horizontal shift of 2 units to the right. This means that the graph of $f(x)=x^2$ will be stretched vertically by a factor of 3 and shifted 2 units to the right to obtain the graph of $g(x)=3(x-2)^2+1$.

Right 2 Units

The second transformation we will consider is a right shift of 2 units. This means that the graph of $f(x)=x^2$ will be shifted 2 units to the right to obtain the graph of $g(x)=3(x-2)^2+1$. To understand this transformation, let's consider the equation of the graph of $f(x)=x^2$. The equation is in the form $y=ax^2$, where $a=1$. When we replace $x$ with $x-2$, we get $y=a(x-2)^2$. This means that the graph of $f(x)=x^2$ will be shifted 2 units to the right to obtain the graph of $y=a(x-2)^2$.

However, the graph of $g(x)=3(x-2)^2+1$ is not just a right shift of the graph of $f(x)=x^2$. It also has a vertical stretch by a factor of 3. This means that the graph of $f(x)=x^2$ will be shifted 2 units to the right and stretched vertically by a factor of 3 to obtain the graph of $g(x)=3(x-2)^2+1$.

Up 1 Unit

The third transformation we will consider is an upward shift of 1 unit. This means that the graph of $f(x)=x^2$ will be shifted 1 unit upward to obtain the graph of $g(x)=3(x-2)^2+1$. To understand this transformation, let's consider the equation of the graph of $f(x)=x^2$. The equation is in the form $y=ax^2$, where $a=1$. When we add 1 to the equation, we get $y=ax^2+1$. This means that the graph of $f(x)=x^2$ will be shifted 1 unit upward to obtain the graph of $y=ax^2+1$.

However, the graph of $g(x)=3(x-2)^2+1$ is not just an upward shift of the graph of $f(x)=x^2$. It also has a vertical stretch by a factor of 3 and a right shift of 2 units. This means that the graph of $f(x)=x^2$ will be shifted 1 unit upward, stretched vertically by a factor of 3, and shifted 2 units to the right to obtain the graph of $g(x)=3(x-2)^2+1$.

In conclusion, the graph of $g(x)=3(x-2)^2+1$ can be obtained from the graph of $f(x)=x^2$ by applying a combination of transformations. These transformations include a vertical stretch by a factor of 3, a right shift of 2 units, and an upward shift of 1 unit. By understanding these transformations, we can analyze the graph of $g(x)=3(x-2)^2+1$ and its relationship to the graph of $f(x)=x^2$.

• [1] "Graphing Functions" by Math Open Reference
• [2] "Transformations of Functions" by Purplemath
• [3] "Graphing Quadratic Functions" by Mathway

The graph of $g(x)=3(x-2)^2+1$ can be obtained from the graph of $f(x)=x^2$ by applying a combination of transformations. These transformations include a vertical stretch by a factor of 3, a right shift of 2 units, and an upward shift of 1 unit. By understanding these transformations, we can analyze the graph of $g(x)=3(x-2)^2+1$ and its relationship to the graph of $f(x)=x^2$.
Q&A: Transformations of the Graph of $f(x)=x^2$

In our previous article, we explored the transformations that take the graph of $f(x)=x^2$ to the graph of $g(x)=3(x-2)^2+1$. We analyzed each transformation step by step and provided examples to illustrate our findings. In this article, we will answer some frequently asked questions about the transformations of the graph of $f(x)=x^2$.

Q: What is the effect of a vertical stretch by a factor of 3 on the graph of $f(x)=x^2$?

A: A vertical stretch by a factor of 3 on the graph of $f(x)=x^2$ will result in a graph that is stretched vertically by a factor of 3. This means that the graph will be taller and wider than the original graph.

Q: What is the effect of a right shift of 2 units on the graph of $f(x)=x^2$?

A: A right shift of 2 units on the graph of $f(x)=x^2$ will result in a graph that is shifted 2 units to the right. This means that the graph will be shifted 2 units to the right of the original graph.

Q: What is the effect of an upward shift of 1 unit on the graph of $f(x)=x^2$?

A: An upward shift of 1 unit on the graph of $f(x)=x^2$ will result in a graph that is shifted 1 unit upward. This means that the graph will be shifted 1 unit above the original graph.

Q: How do the transformations of the graph of $f(x)=x^2$ affect the vertex of the graph?

A: The transformations of the graph of $f(x)=x^2$ will affect the vertex of the graph. A vertical stretch by a factor of 3 will not change the vertex of the graph, but a right shift of 2 units will shift the vertex 2 units to the right. An upward shift of 1 unit will shift the vertex 1 unit upward.

Q: How do the transformations of the graph of $f(x)=x^2$ affect the axis of symmetry of the graph?

A: The transformations of the graph of $f(x)=x^2$ will affect the axis of symmetry of the graph. A right shift of 2 units will shift the axis of symmetry 2 units to the right. An upward shift of 1 unit will not change the axis of symmetry of the graph.

Q: Can the transformations of the graph of $f(x)=x^2$ be combined?

A: Yes, the transformations of the graph of $f(x)=x^2$ can be combined. For example, a vertical stretch by a factor of 3, a right shift of 2 units, and an upward shift of 1 unit can be combined to obtain the graph of $g(x)=3(x-2)^2+1$.

Q: How can the transformations of the graph of $f(x)=x^2$ be used in real-world applications?

A: The transformations of the graph of $f(x)=x^2$ can be used in real-world applications such as modeling the motion of an object, analyzing the behavior of a system, and predicting the outcome of a situation.

In conclusion, the transformations of the graph of $f(x)=x^2$ are an important concept in mathematics. By understanding these transformations, we can analyze the graph of $g(x)=3(x-2)^2+1$ and its relationship to the graph of $f(x)=x^2$. We can also use these transformations in real-world applications such as modeling the motion of an object, analyzing the behavior of a system, and predicting the outcome of a situation.

• [1] "Graphing Functions" by Math Open Reference
• [2] "Transformations of Functions" by Purplemath
• [3] "Graphing Quadratic Functions" by Mathway

The transformations of the graph of $f(x)=x^2$ are an important concept in mathematics. By understanding these transformations, we can analyze the graph of $g(x)=3(x-2)^2+1$ and its relationship to the graph of $f(x)=x^2$. We can also use these transformations in real-world applications such as modeling the motion of an object, analyzing the behavior of a system, and predicting the outcome of a situation.