Chapter 1: Problem 61
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.
Knowing how the square root function behaves initially lets us anticipate how the graph will transform with each addition of negative signs or coefficients.
- 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.
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.
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.