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Ch. 1 - Functions
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 1 - FunctionsProblem 1.2.76
Chapter 1, Problem 1.2.76

Graphing


In Exercises 69–76, graph each function not by plotting points, but by starting with the graph of one of the standard functions presented in Figures 1.14–1.17 and applying an appropriate transformation.


y = (−2x)²/³

Verified step by step guidance
1
Identify the base function: The given function is y = (−2x)²/³. The base function here is y = x²/³, which is a transformation of the cube root function y = x^(1/3).
Understand the transformation: The expression (−2x)²/³ involves two transformations: a horizontal scaling and a reflection. The factor of -2 inside the function indicates a reflection across the y-axis and a horizontal compression by a factor of 1/2.
Apply the reflection: Reflect the graph of y = x²/³ across the y-axis. This means that for every point (x, y) on the graph of y = x²/³, there will be a corresponding point (-x, y) on the graph of y = (−2x)²/³.
Apply the horizontal compression: After reflecting, compress the graph horizontally by a factor of 1/2. This means that each x-coordinate of the reflected graph is multiplied by 1/2, effectively making the graph narrower.
Sketch the transformed graph: Start with the graph of y = x²/³, apply the reflection and horizontal compression, and sketch the resulting graph. Ensure that the graph is symmetric with respect to the y-axis and note the changes in the shape due to the transformations.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Standard Functions

Standard functions are basic functions that serve as building blocks for more complex functions. Examples include linear, quadratic, cubic, and absolute value functions. Understanding their shapes and properties is crucial for graphing transformations, as they provide a reference point for how transformations alter the graph.
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Introduction to Polynomial Functions

Function Transformations

Function transformations involve shifting, stretching, compressing, or reflecting a graph. For example, multiplying a function by a negative value reflects it across the x-axis, while scaling factors can stretch or compress it. Recognizing these transformations helps in graphing complex functions by modifying the graph of a standard function.
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Intro to Transformations

Fractional Exponents

Fractional exponents, such as ²/³, represent both roots and powers. The denominator indicates the root (cube root in this case), and the numerator indicates the power (squared here). Understanding how to manipulate and graph these expressions is essential for accurately representing functions with fractional exponents.
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Introduction to Exponent Rules
Related Practice
Textbook Question

Composition of Functions


A balloon’s volume V is given by V = s² + 2s + 3 cm³, where s is the ambient temperature in °C. The ambient temperature s at time t minutes is given by s = 2t − 3 °C. Write the balloon’s volume V as a function of time t.

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Textbook Question

What happens if you take B = 2π in the addition formulas? Do the results agree with something you already know?

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Textbook Question

Finding a Viewing Window


In Exercises 5–30, find an appropriate graphing software viewing window for the given function and use it to display that function’s graph. The window should give a picture of the overall behavior of the function. There is more than one choice, but incorrect choices can miss important aspects of the function.


y = 3 cos 60x

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Textbook Question

Radians and Degrees


On a circle of radius 10 m, how long is an arc that subtends a central angle of (a) 4π/5 radians? (b) 110°?

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Textbook Question

A hot-air balloon rising straight up from a level field is tracked by a range finder located 500 ft from the point of liftoff. Express the balloon’s height as a function of the angle the line from the range finder to the balloon makes with the ground.

281
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Textbook Question

Finding a Viewing Window


In Exercises 5–30, find an appropriate graphing software viewing window for the given function and use it to display that function’s graph. The window should give a picture of the overall behavior of the function. There is more than one choice, but incorrect choices can miss important aspects of the function.


f(x) = (x² − 1)/(x² + 1)

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