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Chapter 3: Problem 56

Graph each function using the techniques of shifting, compressing, stretching, and/or reflecting. Start with the graph of the basic function (for example, \(y=x^{2}\) ) and show all the steps. Be sure to show at least three key points. Find the domain and the range of each function. $$ f(x)=-4 \sqrt{x-1} $$

Short Answer

Expert verified
Domain: \(x \geq 1\), Range: \(y \leq 0\)

Step by step solution

01

Identify the Basic Function

The basic function here is the square root function, which is written as \( y = \sqrt{x} \). This forms the foundation for the modifications we will apply.
02

Horizontal Shift

Identify the horizontal shift by examining the term inside the square root. The expression \( x-1 \) indicates a shift 1 unit to the right. So, the new function to graph is \( y = \sqrt{x-1} \).
03

Reflect Over x-axis and Vertical Stretch

Apply reflection over the x-axis and the vertical stretch. The coefficient -4 ahead of the square root means reflecting about the x-axis combined with a vertical stretch by a factor of 4. The complete transformation is \( y = -4 \sqrt{x-1} \).
04

Graph Key Points

Start plotting key points. For \( x=1 \), \( y = -4 \times \sqrt{1-1} = 0 \). For \( x=2 \), \( y = -4 \times \sqrt{2-1} = -4 \). For \( x=5 \), \( y = -4 \times \sqrt{5-1} = -8 \). Plot these points on the graph.
05

Domain and Range

Determine the domain and range. The domain of \( f(x)= -4 \sqrt{x-1} \) is \( x \geq 1 \) since the expression under the square root must be non-negative. The range of this function is \( y \leq 0 \) because the function outputs non-positive values due to the negative coefficient.

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Key Concepts

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

Understanding Horizontal Shift
A horizontal shift occurs when a function moves left or right on the graph.
In the given problem, the function is a transformation of the basic function \(y = \sqrt{x}\).
The term \(x-1\) inside the square root indicates a horizontal shift. It shifts the graph 1 unit to the right.
When we modify \(x\) to become \(x-1\), it's like saying, 'Take each x-value and subtract 1 from it.'
This means all points on the graph of \( y = \sqrt{x} \) will move 1 step to the right.
Hence, the first step to transforming given function \(f(x) = -4 \sqrt{x-1} \) is the horizontal shift, starting from \( y = \sqrt{x-1} \).
Exploring Vertical Stretch
Vertical stretching makes the graph taller or shorter by stretching or compressing it vertically.
In the problem, the term \(-4\) in front of the square root indicates a vertical stretch, combined with a reflection.
A vertical stretch by a factor of 4 means that all y-values will be multiplied by 4.
For instance, if the original y-value was 1, it will now be 4. If it was 2, it becomes 8.
Aside from stretching, there will also be a reflection which we'll discuss next.
So the Complete transformation involving a vertical stretch, considering only the stretch aspect, is depicted in the function as: \(y = 4 \sqrt{x-1}\) before reflection.
Reflection Explained
A reflection inverts the graph over a specific axis.
The minus sign in front of 4 indicates a reflection over the x-axis.
This means all positive values become negative, creating a mirror image beneath the x-axis.
For example, a point (2, 4) on the basic graph would shift to (2, -4) after reflection.
Combining vertical stretch with reflection, the function becomes: \( y = -4 \sqrt{x-1} \).
Domain and Range Clarification
The domain refers to all possible x-values, and the range refers to all possible y-values that a function can take.
For the given function \( f(x) = -4 \sqrt{x-1} \), we focus first on the domain.
The expression inside the square root \(x-1\) must be non-negative, meaning \( x \geq 1 \).
Therefore, the domain is from 1 to infinity: \([1, \infty)\).
The range, considering the vertical stretch and reflection (outputting non-positive values), becomes any value less than or equal to 0.
Hence, the range is : \((-\infty, 0] \).

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