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Graph the function using transformations. $$y=5 \sqrt{-x}$$

Short Answer

Expert verified
The function is obtained by reflecting \( y = \sqrt{x} \) over the y-axis and stretching it vertically by a factor of 5.

Step by step solution

01

Identify the Basic Function

The given function is built on the basic square root function, which is \( y = \sqrt{x} \). Recognize that transformations will be applied to this function to obtain \( y = 5 \sqrt{-x} \).
02

Horizontal Reflection

The term \( -x \) indicates a horizontal reflection. This means that the graph of \( y = \sqrt{x} \) is reflected over the y-axis to become \( y = \sqrt{-x} \).
03

Vertical Stretch

The coefficient 5 in the function \( y = 5 \sqrt{-x} \) represents a vertical stretch. Each y-value of the function \( y = \sqrt{-x} \) is multiplied by 5, stretching the graph vertically from the x-axis.
04

Sketch the Transformed Graph

Combine the transformations: Start by reflecting the basic square root graph \( y = \sqrt{x} \) across the y-axis. Then apply the vertical stretch, multiplying each point's y-coordinate by 5. Therefore, the graph of \( y = 5 \sqrt{-x} \) is a vertically stretched version of the flipped basic square root graph.

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Key Concepts

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

Understanding the Square Root Function
The square root function is a fundamental mathematical function described by the equation \( y = \sqrt{x} \). It starts at the origin, and the value of \( y \) increases as \( x \) increases. The graph of this function forms a gentle curve upward and to the right, passing through the origin and gradually leveling off as \( x \) becomes larger.
  • The domain of the square root function is \( x \geq 0 \), meaning it only takes non-negative input values.
  • Its range is \( y \geq 0 \), producing only non-negative output values.
This basic understanding forms the cornerstone of navigating graph transformations. Once you are familiar with the nature of \( y = \sqrt{x} \), applying transformations such as reflections and stretches becomes more intuitive. These transformations adjust the shape, position, or size of the graph in a predictable way.
Knowing how the square root function behaves initially lets us anticipate how the graph will transform with each addition of negative signs or coefficients.
Applying a Horizontal Reflection
The concept of horizontal reflection involves flipping the graph over a vertical axis. In our example, the transformation is introduced by having \( -x \) inside the square root: \( y = \sqrt{-x} \). This change is visually represented by reflecting the original square root graph along the y-axis.
  • After a horizontal reflection, points that were on the right of the y-axis in the original graph of \( y = \sqrt{x} \) will now appear on the left.
  • This kind of transformation does not alter the shape of the graph, just its orientation.
The horizontal reflection inverts the direction of the graph. It's a powerful transformation, especially for re-arranging and reinterpreting how the function interacts with the coordinate plane, emphasizing symmetry and providing a new perspective.
Understanding Vertical Stretch
A vertical stretch changes the appearance of the graph by spreading it away from the x-axis. The function \( y = 5\sqrt{-x} \) includes a vertical stretch accomplished by multiplying the original function's output by 5. This transformation effectively elongates the graph vertically.
  • Here, every \( y \)-value of the square root function \( y = \sqrt{-x} \) is multiplied by 5.
  • This causes the points on the graph to stretch further away from the x-axis, altering its amplitude.
Vertical stretching does not impact the x-coordinates of the points. Instead, it enhances the vertical distance between points, making the curve appear taller or sharper. By understanding vertical stretch, one can predict how intensifying or diminishing a function exponent affects its graphical representation.

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