/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 27 Graph each of the functions. $... [FREE SOLUTION] | 91影视

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Chapter 9: Problem 27

Graph each of the functions. $$f(x)=-2 \sqrt{x}$$

Short Answer

Expert verified
The graph is a downward curve starting at the origin with points (0,0), (1,-2), (4,-4).

Step by step solution

01

Identify the Function Type

The given function is a square root function with a negative coefficient, making it a transformation of the basic square root function \( f(x) = \sqrt{x} \). The transformation includes a vertical scaling (multiplication by \(-2\)).
02

Determine Key Points

To graph the function, start by calculating key points. Choose a set of values for \( x \) such as 0, 1, and 4, and plug them into the function to find their corresponding \( y \)-values. For example, \( f(0) = -2\sqrt{0} = 0 \), \( f(1) = -2 \sqrt{1} = -2 \), and \( f(4) = -2 \sqrt{4} = -4 \).
03

Plot Key Points

Plot the calculated key points on the graph: \((0, 0)\), \((1, -2)\), and \((4, -4)\). These points help define the shape of the graph.
04

Sketch the Graph

Draw a smooth curve through the plotted points to trace out the shape of the function \( f(x) = -2\sqrt{x} \). The graph will start at the origin \((0, 0)\), pass through \((1, -2)\), and \((4, -4)\), continuing in that path, showing that as \(x\) increases, \(f(x)\) becomes more negative.

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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 concept in mathematics and is often represented as \( f(x) = \sqrt{x} \). This function takes any non-negative number \(x\) and returns its square root. For example, \( \sqrt{4} = 2 \) because 2 times 2 is 4. Graphically, the square root function starts at the origin \(0,0\) and moves upward and to the right, forming a distinctive curve known as a parabola's side. This curve gets closer and closer to being flat as \(x\) increases, which visually shows that the growth rate of the square root function diminishes.
  • This function only applies to \(x \geq 0\), since the square root of a negative number isn't real鈥攈ence the graph is only in the first quadrant of the Cartesian plane.
  • The key points to know by heart include \( (0,0) \), \( (1,1) \), and \( (4,2) \).
  • Each point on the graph can be generated by plugging an \(x\) value into the function.
Exploring Vertical Scaling
Vertical scaling is a type of transformation that alters the height of a graph. When we apply vertical scaling to a function, we multiply its output values by a constant factor. For the given function \( f(x) = -2 \sqrt{x} \), the factor here is \ -2\.
  • Vertical scaling by -2 implies that for every point on the original curve of \( f(x) = \sqrt{x} \), we need to stretch the curve en route to making it twice its initial distance from the \( x \)-axis, and then flip it over the \( x \)-axis due to the negative sign.
  • This can radically change the shape and direction of a graph.
  • Key examples start with \( f(1) = -2 \), where the point that was originally at \( (1, 1) \) now finds itself at \( (1, -2) \).
Understanding vertical scaling helps in predicting how changes in equations affect the visuals in graphical representations.
Diving into Function Transformations
Function transformations can reposition a graph in various ways, including translations, reflections, and scalings. In our case of the function \( f(x) = -2\sqrt{x} \), we see a combination of transformations.
  • The first transformation we notice is the reflection over the x-axis. This happens because the coefficient \(-2\) is negative, flipping the graph upside down.
  • The second transformation, as covered, is vertical scaling 鈥� the graph stretches downward by a factor of 2.
These transformations can be visualized by graphing step-by-step:
  • Start with the basic graph of \( \sqrt{x} \).
  • Reflect this over the x-axis to deal with the negative sign.
  • Finally, apply vertical scaling to achieve the complete graph of \(-2\sqrt{x}\).
Overall, by understanding transformations, you can effectively graph any similar complex functions through clearly defined steps.

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Most popular questions from this chapter

The simple interest earned by a certain amount of money varies jointly as the rate of interest and the time (in years) that the money is invested. (a) If some money invested at \(7 \%\) for 2 years earns \(\$ 245\), how much would the same amount earn at \(5 \%\) for 1 year? (b) If some money invested at \(4 \%\) for 3 years earns \(\$ 273\), how much would the same amount earn at \(6 \%\) for 2 years? (c) If some money invested at \(6 \%\) for 4 years earns \(\$ 840\), how much would the same amount earn at \(8 \%\) for 2 years?

Find the constant of variation for each of the stated conditions. \(r\) varies inversely as the cube of \(t\), and \(r=\frac{1}{16}\) when \(t=4 .\)

(a) find the inverse of the given function, and (b) graph the given function and its inverse on the same set of axes. (Objective 4) $$f(x)=4 x$$

Show that \((f \circ g)(x)=x\) and \((g \circ f)\) \((x)=x\) for each pair of functions. \(f(x)=\frac{1}{2} x+\frac{3}{4}\) and \(g(x)=\frac{4 x-3}{2}\)

Show that \((f \circ g)(x)=x\) and \((g \circ f)\) \((x)=x\) for each pair of functions. \(f(x)=3 x-7\) and \(g(x)=\frac{x+7}{3}\)
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