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Chapter 2: Problem 45

Use transformations to graph each function and state the domain and range. $$y=(x-1)^{2}+2$$

Short Answer

Expert verified
Domain: \( (-fty, +fty) \). Range: \( [2, +fty) \).

Step by step solution

01

Identify the parent function

The given function is a transformation of the parent function, which is the basic quadratic function, \( y = x^2 \).
02

Determine horizontal shift

The term \( (x - 1) \) inside the parenthesis indicates a horizontal shift. This means the graph of the parent function \( y = x^2 \) is shifted 1 unit to the right.
03

Determine vertical shift

The constant term \( +2 \) outside the parenthesis indicates a vertical shift. This means the graph of the parent function \( y = x^2 \) is shifted 2 units up.
04

Graph the function

Starting from the graph of \( y = x^2 \), shift the entire graph 1 unit to the right and 2 units up to obtain the graph of \( y = (x - 1)^2 + 2 \).
05

State the domain

The domain of the function \( y = (x - 1)^2 + 2 \) is all real numbers since there are no restrictions on the values that \( x \) can take. Therefore, the domain is \( (-fty, +fty) \).
06

State the range

The range of the function is determined by the vertical shift and the nature of the quadratic function. The minimum value of \( y = (x - 1)^2 + 2 \) is 2 (occurring at \( x = 1 \)), and since the parabola opens upwards, the range is \( [2, +fty) \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

horizontal shift
In a quadratic function like \( y = (x - 1)^2 + 2 \), the \text{x - 1} part inside the parenthesis shows a horizontal shift. Here, we subtract 1 from x, which means the entire graph of the standard quadratic function \( y = x^2 \) is moved to the right by 1 unit.
Think of it like this: instead of starting our graph right at the origin, we start at (1,0). This makes sense because inside the function, we want to counteract shifts to x. Adding or subtracting a value will shift the graph left or right. So if our function had been \text{x + 1} instead, it would shift left by 1.
The key takeaway:
  • Subtract inside the function - shift to the right
  • Add inside the function - shift to the left
vertical shift
Next, let's look at the constant term outside the parenthesis, which causes a vertical shift. In our function \( y = (x - 1)^2 + 2 \), the \(+2\) means we'll shift everything upwards by 2 units.
Imagine you're moving the whole graph straight up. This doesn't change the shape, just where it starts. With this transformation, the vertex of our parabola now sits at (1, 2) instead of the origin.
Remember:
  • Positive vertical shift (adding outside) - graph goes up
  • Negative vertical shift (subtracting outside) - graph goes down

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