Problem 20 - Algebra 2

Question

Graph each relation and its inverse. $$ y=(x-1)^{2} $$

Step-by-Step Solution

Answer

The inverse of the function \( y = (x-1)^2 \) is \( y = \sqrt{x} + 1 \). The graphs of the function and its inverse show the symmetry along the line y=x, with the function being a parabola and its inverse being a squareroot function.

Step 1: Inverse Function Calculation

To find the inverse of the function, first you would replace 'y' with 'x' and 'x' with 'y'. This gives \( x = (y-1)^2 \). To solve this equation for y, you first take the square root of both sides which result in \( \sqrt{x} = y-1 \). Then add 1 to both sides to fully isolate 'y', ending up with \( y = \sqrt{x} + 1 \). This is the inverse of the original function.

Step 2: Graph the original function

Graph the original function \( y = (x-1)^2 \) on the Cartesian plane. This is a parabola with a minimum point (vertex) at (1,0), opening upwards.

Step 3: Graph the inverse function

Now graph the inverse function \( y = \sqrt{x} + 1 \) on the same Cartesian plane. This graph is squareroot function that has been shifted one unit up on the y-axis

Key Concepts

Graphing Functions

When it comes to graphing functions, it's all about visualizing how a function behaves over a set of inputs. A function's graph shows the relationship between its input values (x-values) and output values (y-values).
This representation is crucial for understanding the nature and characteristics of the function.
When graphing a function and its inverse, transparency is key. You should also understand the basic properties of the original function and the inverse. Here, we dealt with a function and its inverse:

• The original function: \( y = (x-1)^2 \)
• The inverse function: \( y = \sqrt{x} + 1 \)

The graphical representation helps to identify intersections, symmetry, and other important aspects, such as the domain and the range of the functions.

Parabolas

Parabolas are U-shaped curves that can open upwards or downwards. They are graphically represented by quadratic functions of the form \( y = ax^2 + bx + c \). In our exercise, we have the function \( y = (x-1)^2 \), which is a specific quadratic equation without linear or constant terms.
The parabola has the following characteristics:

• Vertex: The turning point of the parabola. For the function \( y = (x-1)^2 \), the vertex is at (1, 0).
• Direction: The parabola opens upwards because the coefficient of the squared term is positive.
• Axis of symmetry: A vertical line that passes through the vertex, given by the equation \( x = 1 \).

Parabolas graphically illustrate symmetry and are important in many real-life applications, from physics to engineering.

Square Root Functions

Square root functions, like \( y = \sqrt{x} \), begin at a starting point and increase gradually as x increases. In our step-by-step solution, the inverse function \( y = \sqrt{x} + 1 \) is a square root function that has been translated one unit up.
Key properties of square root functions include:

• Domain: This encompasses all non-negative real numbers, since you cannot take the square root of a negative number in the real number system.
• Range: Starts from the shift on the y-axis. In this function, it's all values \( y \geq 1 \), since it is shifted up by 1 unit.
• Shape: A gradual rise from the point starting at the y-intercept.

These functions are vital for modeling growth situations where progress slows over time, such as resource depletion or diminishing returns in economics.

Cartesian Plane

The Cartesian plane, named after the mathematician René Descartes, is a two-dimensional plane crucial for graphing functions. It consists of a horizontal axis (x-axis) and a vertical axis (y-axis), intersecting at a point called the origin.
To plot functions on the Cartesian plane effectively, it's important to understand:

• Coordinates: Every point on the plane is identified by an x-value and a y-value.
• Quadrants: The plane is divided into four quadrants, each with a distinct combination of positive and negative values of x and y.
• Axes: Used to identify the intersection with the function curves, often providing insight into the function's behavior near origins and intercepts.

Understanding the Cartesian plane is foundational to visualizing how parabolas, square root, and other functions behave across different domains. It enables the comparison and analysis of the original and inverse functions in any mathematical study.