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Chapter 2: Problem 106

The graph of \(y=\sqrt{x}\) is stretched by a factor of \(2,\) reflected in the \(x\) -axis, then translated 5 units to the right. Write the equation of the graph in its final position.

Short Answer

Expert verified
\( y = -2\sqrt{x-5} \)

Step by step solution

01

Identify the base graph

The base graph is given by the equation: \[ y = \sqrt{x} \].
02

Apply vertical stretch

To apply a vertical stretch by a factor of 2, multiply the output of the function by 2: \[ y = 2\sqrt{x} \].
03

Reflect across the x-axis

To reflect the function in the x-axis, multiply the output by -1: \[ y = -2\sqrt{x} \].
04

Translate the graph

To translate the graph 5 units to the right, replace \(x\) with \(x-5\): \[ y = -2\sqrt{x-5} \].

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Key Concepts

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

Vertical Stretch
A vertical stretch is a transformation that scales the graph of a function by a certain factor in the vertical direction. Imagine pulling or pushing the graph vertically. In a vertical stretch, each y-value of the function is multiplied by a constant factor. For example, when we take the square root function, which is given by \( y = \sqrt{x} \), and apply a vertical stretch by a factor of 2, it changes to \( y = 2 \sqrt{x} \). This means that every point on the original graph will be twice as far from the x-axis.
Reflection
A reflection is a transformation that 'flips' the graph of a function over a specific axis. To reflect a function over the x-axis, you negate the output of the function. For our base graph \( y = 2 \sqrt{x} \), reflecting it over the x-axis results in \( y = -2 \sqrt{x} \). This means all points that were above the x-axis on the positive side are now mirrored below it on the negative side.
Horizontal Translation
A horizontal translation shifts the graph of a function left or right by a certain number of units. To translate the graph of a function horizontally, you replace every instance of x in the function with \( x - h \) (for shifting right) or \( x + h \) (for shifting left). For our modified function \( y = -2 \sqrt{x} \), applying a horizontal translation 5 units to the right results in \( y = -2 \sqrt{x - 5} \). This transformation shifts the entire graph to the right without altering its shape.
Square Root Function
The square root function, given by \( y = \sqrt{x} \), is a fundamental function in mathematics. Its graph starts at the origin (0,0) and increases slowly to the right. The function is only defined for non-negative values of x (i.e., x >= 0). Transformations like vertical stretches, reflections, and translations alter the appearance and position of this base graph. In this case, the base graph goes through a series of transformations: a vertical stretch by 2, a reflection over the x-axis, and a horizontal translation 5 units to the right, resulting in the final equation \( y = -2 \sqrt{x - 5} \).

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