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Chapter 0: Problem 12

Graph each pair of equations on one set of axes. $$ y=\sqrt{x} \text { and } y=\sqrt{x-2} $$

Short Answer

Expert verified
The graphs of \( y = \sqrt{x} \) and \( y = \sqrt{x-2} \) do not intersect and are horizontal translations of each other.

Step by step solution

01

Understand the first equation

The first equation given is \( y = \sqrt{x} \). This is a square root function, which is defined for \( x \geq 0 \). The graph of this function is a curve that starts at the origin (0,0) and rises to the right along the positive \( x \)-axis.
02

Plot the graph of \( y = \sqrt{x} \)

To graph \( y = \sqrt{x} \), start by plotting a few key points: \( (0,0) \), \( (1,1) \), \( (4,2) \), and \( (9,3) \). Connect these points smoothly to form a curve. This graph should rise gradually, moving upwards to the right.
03

Understand the second equation

The second equation is \( y = \sqrt{x-2} \). This is a horizontal shift of the \( y = \sqrt{x} \) function, shifted to the right by 2 units. The domain for this function is \( x \geq 2 \) because the expression under the square root must be non-negative.
04

Plot the graph of \( y = \sqrt{x-2} \)

Start plotting the graph of \( y = \sqrt{x-2} \) by calculating a few points: \( (2,0) \), \( (3,1) \), \( (6,2) \), and \( (11,3) \). These are similar to the points from the first graph, but each \( x \)-value is increased by 2. Draw this graph starting at \( x = 2 \), and moving upward to the right.
05

Combine the graphs on one set of axes

On a common coordinate grid, plot both graphs. The first graph starts at \( (0,0) \) and the second at \( (2,0) \). The two curves never intersect but move alongside each other. Make sure each curve is clearly labeled for proper identification.

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Key Concepts

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

Square Root Function
A square root function is a type of function that involves the square root of an independent variable, typically denoted as \( y = \sqrt{x} \). This function is defined only for non-negative values of \( x \), meaning it cannot handle negative inputs because the square root of a negative number is not a real number. As a result, the domain for a basic square root function is \( x \geq 0 \).
The graph of a square root function has a few distinct characteristics:
  • It begins at the point \((0,0)\), known as the origin.
  • The curve only exists in the first quadrant of the coordinate grid, since both \( x \) and \( y \) are non-negative.
  • As \( x \) increases, \( y \) also increases but at a decreasing rate, creating a curve that rises gradually and smoothly.
  • The function's range is \( y \geq 0 \), spanning from zero upwards.
These attributes make the square root function unique and distinguishable from other algebraic functions.
Horizontal Shift
Horizontal shifts occur when a function's graph is moved left or right along the \( x \)-axis without altering its shape. When a square root function includes a term such as \( (x - h) \), this indicates a horizontal shift.
For example, in the equation \( y = \sqrt{x-2} \), the graph of \( y = \sqrt{x} \) is shifted 2 units to the right. This results in an updated domain of \( x \geq 2 \), allowing the starting point of the function to move from \((0,0)\) to \((2,0)\).
Key things to remember about horizontal shifts:
  • The function \( y = \sqrt{x-h} \) signifies a shift to the right by \( h \) units.
  • The function \( y = \sqrt{x+h} \) would shift to the left by \( h \) units.
  • Horizontal shifts do not affect the range or the fundamental shape of the square root curve鈥攐nly its position along the \( x \)-axis changes.
These transformations are crucial for understanding how the position of graphs can be adjusted by modifying equations.
Coordinate Grid Plotting
Coordinate grid plotting is an essential skill for visualizing mathematical functions. It's the process of marking points on a grid based on their coordinates and then connecting these points to form graphs. Let's break down this technique for square root functions:
When plotting \( y = \sqrt{x} \):
  • Start with key values like \( x = 0, 1, 4, \) and \( 9 \) to determine points \((0,0), (1,1), (4,2), (9,3)\).
  • Plot these on the grid and smoothly connect them to show the curve upward along the positive \( x \)-axis.
When plotting \( y = \sqrt{x-2} \):
  • Similarly, choose \( x = 2, 3, 6, \) and \( 11 \) resulting in \((2,0), (3,1), (6,2), (11,3)\).
  • These depict a similar curve, but starting at \( x = 2 \).
Plotting on the same grid visually demonstrates the effect of horizontal shifts, showcasing how graphs can be distinct yet related through transformations. Ensure each graph is labeled to differentiate the functions and their distinct starting points.

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