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Chapter 3: Problem 47

A function and how it is to be shifted is given. Find the shifted function, and then display the given function and the shifted function on the same screen of a graphing calculator. $$y=\sqrt{2 x+1}, \text { up } 1, \text { left } 1$$

Short Answer

Expert verified
The shifted function is \( y = \sqrt{2x + 3} + 1 \). Graph both \( y = \sqrt{2x + 1} \) and \( y = \sqrt{2x + 3} + 1 \) to see the shift.

Step by step solution

01

Understanding the Function

The original function is given as \( y = \sqrt{2x + 1} \). This is a square root function with an inside linear term \( 2x + 1 \).
02

Identify the Shifts

The problem states the shifts required: **up 1** and **left 1**. Shifting **up 1** translates the function vertically on the y-axis, and shifting **left 1** translates it horizontally on the x-axis.
03

Apply Vertical Shift

To shift the function up 1, we add 1 to the entire function. This gives us \( y = \sqrt{2x + 1} + 1 \).
04

Apply Horizontal Shift

To shift the function left 1, we increase the x variable within the square root by 1. This results in the new expression being \( y = \sqrt{2(x + 1) + 1} + 1 \). Simplifying inside the square root gives \( y = \sqrt{2x + 2 + 1} + 1 = \sqrt{2x + 3} + 1 \).
05

Graph the Functions

On a graphing calculator, input the original function \( y = \sqrt{2x + 1} \) and the shifted function \( y = \sqrt{2x + 3} + 1 \). Use the graphing feature to visualize how the function has moved up and to the left.

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Key Concepts

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

Vertical Shift
A vertical shift involves moving a graph up or down along the y-axis. When we apply a vertical shift, we are essentially adding or subtracting a fixed number from the entire function. This affects the y-values of the function without altering the x-values.
In our exercise, we have the function \(y = \sqrt{2x + 1}\) and we need to shift it up by 1 unit. This means we'll add 1 to the whole function, making it \(y = \sqrt{2x + 1} + 1\).
  • The addition of a number to the function shifts it upwards.
  • If we had subtracted a number, it would shift downwards.
Imagine each point on the graph lifting up by 1 unit. This doesn't affect the shape of the graph, only its position vertically.
Horizontal Shift
The horizontal shift moves the graph left or right along the x-axis. This is done by adjusting the x-variable within the function.
For a shift left, we increase the x-term inside the function. Rather than moving the graph by altering a constant term, we adjust the x-term directly inside the square root or other operations it affects.
In this exercise, the function \(y = \sqrt{2x+1}\) is shifted left by 1 unit. This requires us to replace \(x\) with \(x+1\), resulting in \(y = \sqrt{2(x+1) + 1} + 1\). Simplifying this gives \(y = \sqrt{2x + 3} + 1\).
  • Increasing \(x\) by a positive value moves the graph to the left.
  • If we had a negative value, it would have moved the graph right.
This shift changes where the graph starts and ends along the x-axis, without changing its overall shape.
Graphing Functions
Graphing functions allows us to visually comprehend the effects of shifts and transformations. It's a great way to see how the original function and the shifted version differ.
To graph both \(y = \sqrt{2x+1}\) and \(y = \sqrt{2x+3} + 1\), use a graphing calculator or software. This will visually stack the graphs for comparison:
  • The visual representation will clearly show how the shifted function has been moved up and left.
  • Detect the new starting point and trajectory compared to the original.
  • Note that the shape of the graph remains unchanged; only its position differs.
By graphing, you gain a clearer understanding of these transformations and can effectively visualize both simple and complex shifts.

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