Question

Write a function \(g\) whose graph represents the indicated transformation of the graph of \(f\). Use a graphing calculator to check your answer. \(f(x)=-3+|x-11| ;\) reflection in the \(y\)-axis

Short answer

The reflected function over the y-axis is \(g(x) = -3 + |x-11|\).

Step-by-step solution

Reflect Function in y-axis

To perform a reflection in the y-axis, every x in the function needs to be replaced with -x. This results in the function \(g(x)=-3+|-x-11|\).

Simplification

Next, the inside of the absolute value function can be simplified. This leads to the result \(g(x)=-3+|-x+11|\), which can be further simplified to \(g(x)=-3+|11-x|\).

Final version of the reflected function

Finally, the absolute value of a subtraction is the same regardless of the order of the operands. Therefore, the function is now in its simplest and most recognizable form as \(g(x)= -3 + |x-11|\).

Key concepts

Reflection in the Coordinate Plane

Function transformation is a significant part of Algebra 2, and reflections are among the most common types. A reflection in the y-axis involves changing the sign of the x-coordinates of a function. This means every x in the function is switched to -x, producing a mirror image of the graph. This technique is particularly useful in developing a deeper understanding of how functions behave. When you reflect the function \(f(x) = -3 + |x-11|\) across the y-axis, the transformed function becomes \(g(x) = -3 + |-x-11|\). This transformation affects how the function is oriented in the coordinate plane without altering its fundamental structure.

Utilizing a Graphing Calculator

A graphing calculator is an incredible tool in graph transformation tasks. It helps visualize complex functions and their transformations easily. Using a graphing calculator, you can plot the original function \(f(x) = -3 + |x-11|\) and its transformed version \(g(x) = -3 + |11-x|\), to observe the graphical changes after a y-axis reflection.

• Firstly, input the function in the calculator.
• Use the reflection feature to transform the graph automatically.
• Compare the original and reflected graphs.

This visualization is crucial in confirming your analytical results and understanding the practical implications of function transformations.

Understanding Absolute Value Functions

The absolute value function plays a key role in function transformation exercises. The expression \(|x-11|\) essentially measures the distance of x from 11 on a number line, regardless of direction. An absolute value function like this creates a "V"-shaped graph.

• The point \(x = 11\) is the vertex of this graph.
• Any shift left or right will adjust where the graph opens but not its V-shape.

In the context of the given problem, understanding the absolute value helps simplify the reflected equation. For \(|-x+11|\) and \(|11-x|\), the absolute value means that both are equivalent, maintaining the same basic graph shape.

Function Transformation Methods in Algebra 2

Algebra 2 introduces students to various function transformation techniques, essential for understanding complex mathematical behaviors. Function transformations include shifting, reflecting, stretching, and compressing a graph. These transformations allow you to modify a function into new forms without changing the type of the function itself.

• Reflections change the graph's orientation.
• Shifts move the graph horizontally or vertically.

By mastering these concepts in Algebra 2, students enhance their ability to tackle more advanced problems in mathematics and apply these skills in calculus and analytical geometry.