Problem 21 - Algebra 2

Question

Graph each relation and its inverse. $$ y=(2-x)^{2} $$

Step-by-Step Solution

Answer

The original equation is expressed as \( y = (2 - x)^{2} \) and its inverse as \( y = 2 - \sqrt{x} \). The graphs of these functions are generated by plotting the values obtained by substituting specific numbers for x into each equation.

Step 1: Rewrite the equation

Rewrite the given equation to make it easier to find the inverse. It's easier to find the inverse if x is expressed in terms of y. Thus, rearrange the equation to get: \( x = 2 - \sqrt{y} \).

Step 2: Find the inverse

To find the inverse of a function, interchange the positions of y and x. Therefore, the inverse will be: \( y = 2 - \sqrt{x} \).

Step 3: Graph the original function

Next, graph the original function \( y = (2-x)^{2} \). Start by creating a table of values for x and use the equation to get the corresponding y-values. Plot these points and draw a curve through the points to represent the equation graphically.

Step 4: Graph the inverse function

Now graph the inverse function \( y = 2 - \sqrt{x} \). Similar to step 3, create a table of values for x and find the corresponding y-values using the inverse function. Plot these points and draw the inverse curve.

Step 5: Checking the graph

Remember, the graphs of a function and its inverse are mirror images of each other over the line \( y = x \). The inverse function will be a flipped version across the line \( y = x \) of the original function graph.

Key Concepts

Graphing Functions

When we talk about graphing functions, we're essentially visualizing equations on a coordinate plane. A function's graph shows all the possible outputs (y-values) for each input (x-value) you provide. Visualizing helps in understanding how the function behaves.
Start by choosing a few x-values. For each x, compute the corresponding y-value using the function's equation. These pairs of (x, y) values are like plotting coordinates on a map. When you connect these coordinates smoothly, you see a picture of how the function behaves over a range.

• Choose a set of x-values.
• Calculate the y-values for each x.
• Plot these (x, y) pairs on the graph.
• Ensure the points form a continuous curve unless otherwise specified, as many functions do not have breaks.

This method works for the original function and its inverse alike. Remember, the inverse function will reflect across the line \( y = x \), which means if you fold the graph along this line, the original function will match perfectly with the inverse.

Quadratic Functions

Quadratic functions are a special type of polynomial function and have the standard form \( y = ax^2 + bx + c \). They always graph into the distinctive U-shaped curve known as a parabola.
In our problem, the function presented is slightly different, \( y = (2-x)^2 \), but it is essentially a quadratic function because it can be expanded to fit the standard form like \( y = x^2 - 4x + 4 \). The vertex of this parabola is crucial as it marks the peak or the lowest point of the curve, depending on whether the parabola opens upwards or downwards.

• If \( a > 0 \), the parabola opens upwards, making the vertex the lowest point.
• If \( a < 0 \), the parabola opens downwards, making the vertex the highest point.

Understanding how to handle these parabolas helps when graphing, especially when predicting where the function might go for additional x-values.

Function Transformations

Function transformations change the location or shape of the function graph. They include shifts, stretches, shrinks, and reflections.
In our context, the function \( y = (2-x)^2 \) has undergone several transformations from the base function \( y = x^2 \). Think of (2-x) as a horizontal transformation.

• The term \( -x \) represents a horizontal reflection, flipping the graph across the y-axis.
• The shift by 2 units to make it \( 2-x \) implies a horizontal shift to the right by 2 units.

Learners need to watch out for transformations because they can drastically change how the function appears on the graph without altering its fundamental nature. Recognizing transformations helps predict and correctly plot or solve problems involving other types of functions across various exercises.