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Chapter 7: Problem 64

Explain the effect that \(a\) has on the graph of \(y=a \sqrt{x}\)

Short Answer

Expert verified
The coefficient \( a \) in \( y = a \sqrt{x} \) scales the graph of the \( y = \sqrt{x} \) function vertically. If \( a > 1 \), the graph stretches vertically. If \( 0 < a < 1 \), the graph compresses. If \( a \) is negative, the graph is reflected on the x-axis.

Step by step solution

01

Review the basic graph

Firstly, understand how the graph of the basic square root function \( y = \sqrt{x} \) looks like. Its graph starts from the point (0, 0) and increases slowly, continuously curving upwards.
02

Understand the effect of Coefficient \( a \)

When \( a \) is multiplied with \( \sqrt{x} \), it scales the y-values of the square root function by a factor of \( 'a' \). If \( a > 1 \), the graph stretches vertically (the y-values increase for each x). If \( 0 < a < 1 \), the graph is compressed vertically (the y-values decrease for each x).
03

Notice the directions of the graph for different \( a \) values

If \( a \) is negative, the graph is reflected on the x-axis, meaning it will be upside down. If \( a \) is positive, the graph remains in the same position as the original graph.

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Key Concepts

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

Square root function
The square root function is one of the basic functions in mathematics, often denoted as \( y = \sqrt{x} \). This function only applies to non-negative values of \( x \). As \( x \) increases, \( y \) increases as well but at a decreasing rate.
  • The graph of \( y = \sqrt{x} \) starts at the origin, the point \( (0,0) \).
  • As \( x \) moves towards positive infinity, \( y \) also tends to infinity.
  • The shape of the graph resembles a gentle curve increasing upwards to the right.
This function forms the basis for various transformations, where we modify the graph to observe different mathematical behaviors.
Vertical stretch and compression
In the context of graph transformations, the term 'vertical stretch and compression' refers to how a graph alters its shape by changing its vertical dimensions. This occurs when you multiply the function by a constant.
  • For the function \( y = a \sqrt{x} \), if \( a > 1 \), each \( y \)-value is increased by the factor \( a \). This causes the graph to stretch away from the x-axis, making it look taller.
  • If \( 0 < a < 1 \), each \( y \)-value is reduced by the factor \( a \). The graph becomes compressed, appearing shorter and closer to the x-axis.
Understanding these vertical transformations helps predict how a graph will behave when its amplitude is modified without affecting the x-values themselves.
Reflection
Reflection in graph transformations refers to flipping the graph of a function across the x-axis or y-axis. In the context of the square root function, we focus on reflections across the x-axis. This is seen when \( a \) in the function \( y = a \sqrt{x} \) is negative.
  • When \( a < 0 \), the graph of \( y = a \sqrt{x} \) is inverted across the x-axis. This means every positive value that was above the x-axis is now below it.
  • This reflection results in a graph that is "upside down," compared to its positive \( a \) counterpart.
These transformations are useful in many areas of mathematics and help describe physical phenomena, like inverse relationships in natural processes.

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Most popular questions from this chapter

Find all the real square roots of each number. $$ 225 $$

Simplify. $$ \sqrt[3]{54}+\sqrt[3]{16} $$

Find all the real square roots of each number. $$ 0.0049 $$

a. The graph of \(y=\sqrt{x}\) is translated five units to the right and two units down. Write an equation of the translated function. b. The translated graph from part (a) is again translated, this time four units left and three units down. Write an equation of the translated function.

What is the inverse of \(y=x^{2}-3 ?\) $$ \begin{array}{ll}{\text { A. } y=\pm \sqrt{x}+3} & {\text { B. } y=\pm \sqrt{x}-3} \\ {\text { C. } y=\pm \sqrt{x+3}} & {\text { D. } y=\pm \sqrt{x-3}}\end{array} $$
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