Determining the Transformation

Oct 13, 2025 by magicalceylontours.com 31 views

Transformations of Quadratic Functions: Understanding the Graph of $g(x) = 2x^2 - 28x + 3$

In mathematics, transformations of functions are essential concepts that help us understand how different operations affect the graph of a function. When we are given two functions, $f(x)$ and $g(x)$ , and we need to determine the transformation applied to the graph of $f(x)$ to produce the graph of $g(x)$ , we must analyze the coefficients and terms of the two functions. In this article, we will explore the transformation applied to the graph of $f(x) = x^2$ to produce the graph of $g(x) = 2x^2 - 28x + 3$ .

Understanding the Graph of $f(x) = x^2$

The graph of $f(x) = x^2$ is a parabola that opens upwards, with its vertex at the origin $(0, 0)$ . This function has a minimum value of 0, and its axis of symmetry is the y-axis. The graph of $f(x) = x^2$ is a simple quadratic function that can be easily visualized.

Understanding the Graph of $g(x) = 2x^2 - 28x + 3$

The graph of $g(x) = 2x^2 - 28x + 3$ is also a parabola, but it has been transformed from the graph of $f(x) = x^2$ . To understand the transformation, we need to analyze the coefficients and terms of the two functions. The graph of $g(x) = 2x^2 - 28x + 3$ has a coefficient of 2 in front of the $x^2$ term, which means that the graph has been stretched vertically by a factor of 2. The graph also has a linear term of $-28x$ , which means that the graph has been shifted to the left by some units. Finally, the graph has a constant term of 3, which means that the graph has been shifted up by some units.

Determining the Transformation

To determine the transformation applied to the graph of $f(x) = x^2$ to produce the graph of $g(x) = 2x^2 - 28x + 3$ , we need to analyze the coefficients and terms of the two functions. The coefficient of 2 in front of the $x^2$ term indicates that the graph has been stretched vertically by a factor of 2. The linear term of $-28x$ indicates that the graph has been shifted to the left by some units. To determine the exact amount of the shift, we need to divide the coefficient of the linear term by the coefficient of the $x^2$ term. In this case, the coefficient of the linear term is $-28$ , and the coefficient of the $x^2$ term is 2. Therefore, the graph has been shifted to the left by $\frac{-28}{2} = -14$ units.

In conclusion, the graph of $g(x) = 2x^2 - 28x + 3$ has been transformed from the graph of $f(x) = x^2$ by a vertical stretch by a factor of 2, a horizontal shift to the left by 14 units, and a vertical shift up by 3 units. Therefore, the correct answer is:

A. Shifted left 7 units: This is incorrect, as the graph has been shifted to the left by 14 units, not 7 units.
B. Shifted right 7 units: This is incorrect, as the graph has been shifted to the left by 14 units, not right by 7 units.
C. Shifted down 3 units: This is incorrect, as the graph has been shifted up by 3 units, not down by 3 units.
D. Shifted up 3 units: This is correct, as the graph has been shifted up by 3 units.

The final answer is D.
Transformations of Quadratic Functions: A Q&A Guide

In our previous article, we explored the transformation applied to the graph of $f(x) = x^2$ to produce the graph of $g(x) = 2x^2 - 28x + 3$ . In this article, we will provide a Q&A guide to help you understand the transformations of quadratic functions.

Q: What is a transformation of a function?

A: A transformation of a function is a change in the graph of the function that results from a change in the equation of the function. Transformations can include changes in the position, size, and orientation of the graph.

Q: What are the different types of transformations?

A: There are four main types of transformations:

Vertical Stretch or Compression: This type of transformation changes the vertical scale of the graph.
Horizontal Stretch or Compression: This type of transformation changes the horizontal scale of the graph.
Reflection: This type of transformation flips the graph over a line or a point.
Translation: This type of transformation moves the graph to a new position.

Q: How do I determine the type of transformation?

A: To determine the type of transformation, you need to analyze the coefficients and terms of the two functions. For example, if the coefficient of the $x^2$ term is changed, it may indicate a vertical stretch or compression. If the coefficient of the linear term is changed, it may indicate a horizontal shift.

Q: What is the effect of a vertical stretch or compression on the graph?

A: A vertical stretch or compression changes the vertical scale of the graph. If the coefficient of the $x^2$ term is increased, the graph will be stretched vertically. If the coefficient of the $x^2$ term is decreased, the graph will be compressed vertically.

Q: What is the effect of a horizontal stretch or compression on the graph?

A: A horizontal stretch or compression changes the horizontal scale of the graph. If the coefficient of the linear term is increased, the graph will be stretched horizontally. If the coefficient of the linear term is decreased, the graph will be compressed horizontally.

Q: What is the effect of a reflection on the graph?

A: A reflection flips the graph over a line or a point. This type of transformation can be represented by a change in the sign of the coefficient of the linear term.

Q: What is the effect of a translation on the graph?

A: A translation moves the graph to a new position. This type of transformation can be represented by a change in the constant term.

Q: How do I determine the amount of the translation?

A: To determine the amount of the translation, you need to analyze the constant term of the two functions. For example, if the constant term is increased, the graph will be shifted up. If the constant term is decreased, the graph will be shifted down.

Q: What are some common transformations of quadratic functions?

A: Some common transformations of quadratic functions include:

Vertical stretch or compression: This type of transformation changes the vertical scale of the graph.
Horizontal stretch or compression: This type of transformation changes the horizontal scale of the graph.
Reflection: This type of transformation flips the graph over a line or a point.
Translation: This type of transformation moves the graph to a new position.

In conclusion, transformations of quadratic functions are essential concepts that help us understand how different operations affect the graph of a function. By analyzing the coefficients and terms of the two functions, we can determine the type of transformation and the amount of the translation. We hope this Q&A guide has helped you understand the transformations of quadratic functions.

The final answer is that transformations of quadratic functions are essential concepts that help us understand how different operations affect the graph of a function.