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Chapter 2: Problem 57

Use transformations to graph each function and state the domain and range. $$y=-\sqrt{x-3}+1$$

Short Answer

Expert verified
The function \[ y = -\sqrt{x-3} + 1 \] is shifted 3 units right, reflected over the x-axis, and moved up by 1 unit. Its domain is \[ x \geq 3 \] and range is \[ y \leq 1 \].

Step by step solution

01

Identify the base function

The base function is the square root function: \[ f(x) = \sqrt{x} \]
02

Horizontal shift

Identify the horizontal shift by looking at the term inside the square root. The function is \[ y = \sqrt{x-3} \]This represents a shift 3 units to the right.
03

Reflect over x-axis

The negative sign in front of the square root indicates a reflection over the x-axis. So, \[ y = -\sqrt{x-3} \] reflects the graph of \[ y = \sqrt{x-3} \] over the x-axis.
04

Vertical shift

The +1 outside the square root indicates a vertical shift: \[ -\sqrt{x-3} + 1 \] moves the entire graph up by 1 unit.
05

Domain and Range

For the domain, set the expression inside the square root greater than or equal to zero: \[ x-3 \geq 0 \Rightarrow x \geq 3 \] For the range, after reflection and vertical shift, the range will be all values that are less than or equal to 1: \[ y \leq 1 \]

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Key Concepts

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

Horizontal Shift
A horizontal shift moves the graph of a function left or right. To visualize this, consider the function \[ f(x) = \sqrt{x} \]. If we modify this function to include \((x - 3)\), it becomes \[ f(x) = \sqrt{x-3} \]. This transformation means the entire graph shifts 3 units to the right.
Why? Because for every x in the new function, it has to be 3 units more than in the original function to produce the same output.
Imagine sliding a sheet of paper with a graph printed on it, moving it to the right by 3 units; this is a horizontal shift.

In summary:
\[ y = \sqrt{x-3} \]
shifts the graph of \[ y = \sqrt{x} \] 3 units to the right.
Vertical Shift
A vertical shift moves the graph up or down. Consider our modified function from before \[ y = \sqrt{x-3} \]. Now, we add +1 outside the square root to get \[ y = \-\sqrt{x-3}+1 \].
This +1 means the graph will move up by 1 unit. Why? Because every point on the graph will increase its y-value by 1.

Think of raising a sheet of graph paper vertically by one unit. This is a vertical shift.

Key points:
  • All outputs (y-values) of \[ y = \-\sqrt{x-3} \] are increased by 1.
  • This moves the entire graph up by 1 unit.
Reflection over x-axis
Reflection is flipping the graph over an axis. In our function, we have a negative sign before the square root: \[ y = \-\sqrt{x-3} \]. The negative sign means the graph will reflect over the x-axis.

This transformation changes the y-values to their opposites.
For each point \[ (x, y) \] on the graph of \[ \sqrt{x-3} \], the corresponding point on \[ -\sqrt{x-3} \] will be \[ (x, -y) \].

In simpler terms:
  • The graph of \[ y = \sqrt{x-3} \] flips upside down.
  • All positive y-values turn negative.

This is known as reflection over the x-axis.
Domain and Range
The domain and range describe where the function exists.

For the domain:
  • You look at allowable x-values.
  • In our function \[ y = \sqrt{x-3} \], the expression inside the square root, \((x-3)\), must be greater than or equal to 0.ull Meaning \((x \geq 3)\).

The domain is all x-values from 3 onwards, so:
  • \[ x \geq 3 \] or [3, 鈭�).

For the range:
  • We take the transformed function \[ y = -\sqrt{x-3}+1 \].
  • The result after the square root is always non-negative but the negative sign before makes this non-positive.
  • Further moved up by 1 unit, making y-values less than or equal to 1.

The range is: \[ y \leq 1 \] or (-鈭�, 1].

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