Question

Write a function \(g\) whose graph represents the indicated transformation of the graph of \(f\). Use a graphing calculator to check your answer. \(f(x)=-x+1\); reflection in the \(y\)-axis

Short answer

The function \(g(x)\) that represents the indicated transformation of the graph of \(f(x)=-x+1\) with a reflection in the \(y\)-axis is \(g(x)=x+1\).

Step-by-step solution

Identify the Original Function

The original function is given as \(f(x)=-x+1\). From this function, the transformation will be applied.

Apply the Transformation Rule

To reflect the function over the \(y\)-axis, replace \(x\) with \(-x\) in the original function. Therefore the transformed function \(g(x)\) is obtained by replacing \(x\) with \(-x\) in the expression for \(f(x)\) to get \(g(x)=-(-x)+1\).

Simplify the Transformed Function

Simplify the expression for \(g(x)\) to get \(g(x)=x+1\). This is the function after reflection in the \(y\)-axis.

Check the Result with a Graphing Calculator

Finally, plot both \(f(x)=-x+1\) and the transformed function \(g(x)=x+1\) on a graphing calculator. The graph of \(g(x)\) should look like a mirror reflection of the graph of \(f(x)\) in the \(y\)-axis.

Key concepts

Reflection Transformation

Reflection transformations are like flipping a graph over a line, such as the y-axis, or the x-axis. In this exercise, we've tackled the reflection across the y-axis.
When reflecting a function like

• Start with the original algebraic expression. For example, if you're given the function \( f(x) = -x + 1 \), you identify \( x \) as the variable to be reflected.
• To perform a reflection over the y-axis, substitute \( x \) with \( -x \). So, \( f(x) \) becomes \( g(x) = -(-x) + 1 = x + 1 \). This switches the direction of the graph horizontally, mirroring it over the y-axis.

By following these steps, the function is effectively flipped, maintaining the same shape but reversing its direction in relation to the vertical axis. The graph of the new function should appear as a mirrored version of the original function across this axis.

Graphing Calculator

A graphing calculator is a handy tool when verifying transformations like reflections. These devices allow you to depict complex graphs quickly and analyze changes visually.
Here's how you can use a graphing calculator to check your reflected graph:

• Input the original function, \( f(x) = -x + 1 \), into the calculator. Observe how it slopes downward from left to right.
• Next, input the transformed function, \( g(x) = x + 1 \). Observe how this graph rises up from left to right.
• By comparing these two graphs, you can visually confirm the reflection across the y-axis.

With a graphing calculator, transformations become clear, ensuring you understand how changes in equations influence their graphical representations.

Algebra Functions

Algebra functions define relationships between variables and often involve operations like addition, subtraction, multiplication, or division. When dealing with transformations, algebra functions serve as the backbone for these modifications.
In our scenario:

• The function \( f(x) = -x + 1 \) involves a simple linear relationship between \( x \) and the output value. The terms within a function describe its slope and intercept, determining its linear path on a graph.
• Changes to the variables or coefficients in these functions illustrate transformations. For instance, changing \( x \) to \( -x \) says, "reverse the direction across the axis," showcasing a reflection.

Mastering algebra functions is crucial because it allows you to clearly see how specific transformations, like reflections, translations, and scalings, alter the picture of the graph on a coordinate plane.