Table of contents

0. Functions7h 52m
1. Limits and Continuity2h 2m
2. Intro to Derivatives1h 33m
3. Techniques of Differentiation3h 18m
4. Applications of Derivatives2h 38m
5. Graphical Applications of Derivatives6h 2m
6. Derivatives of Inverse, Exponential, & Logarithmic Functions2h 37m
7. Antiderivatives & Indefinite Integrals1h 26m
8. Definite Integrals4h 44m
9. Graphical Applications of Integrals2h 27m
- Area Between Curves
  1h 18m
- Introduction to Volume & Disk Method
  1h 8m
10. Physics Applications of Integrals 3h 16m
- Kinematics
  1h 13m
- Work
  2h 3m
11. Integrals of Inverse, Exponential, & Logarithmic Functions2h 34m
12. Techniques of Integration7h 39m
13. Intro to Differential Equations2h 55m
14. Sequences & Series5h 36m
15. Power Series2h 19m
- Introduction to Power Series
  1h 9m
- Taylor Series & Taylor Polynomials
  1h 10m
16. Parametric Equations & Polar Coordinates7h 58m

0. Functions

Transformations

Calculus

0. Functions

Transformations

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4:01 minutes

Problem 1.2.43

Textbook Question

Graph the functions in Exercises 37–56.

y = (x + 1)²/³

Verified step by step guidance

Identify the type of function: The given function is y = (x + 1)^(2/3). This is a power function with a fractional exponent, which indicates a root function.

Determine the domain: Since the base of the power (x + 1) is raised to a fractional power with an even numerator, the expression is defined for all real numbers x. Therefore, the domain is all real numbers.

Find the critical points: To find critical points, take the derivative of the function with respect to x. Use the chain rule to differentiate y = (x + 1)^(2/3).

Analyze the behavior at critical points: Evaluate the derivative to find where it is zero or undefined. These points will help determine local maxima, minima, or points of inflection.

Sketch the graph: Use the information from the derivative and critical points to sketch the graph. Consider the symmetry, intercepts, and end behavior of the function to complete the graph.

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Key Concepts

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

Graphing Functions

Graphing functions involves plotting points on a coordinate plane to visualize the behavior of a function. It requires understanding the function's domain, range, and any transformations applied to the basic function. For y = (x + 1)²/³, identifying key features like intercepts, asymptotes, and symmetry helps in sketching an accurate graph.

Fractional Exponents

Fractional exponents, such as ²/³, indicate roots and powers. The expression (x + 1)²/³ can be interpreted as the cube root of (x + 1) squared. Understanding fractional exponents is crucial for determining the function's behavior, especially near critical points where the base might be zero or negative.

Transformations of Functions

Transformations involve shifting, stretching, or compressing the graph of a function. The expression y = (x + 1)²/³ includes a horizontal shift left by 1 unit due to the (x + 1) term. Recognizing transformations helps in predicting how the graph will move or change shape compared to the parent function y = x²/³.

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Textbook Question

Shifting Graphs

Exercises 27–36 tell how many units and in what directions the graphs of the given equations are to be shifted. Give an equation for the shifted graph. Then sketch the original and shifted graphs together, labeling each graph with its equation.

y = x³ Left 1, down 1

Textbook Question

Shifting Graphs

y = −√x Right 3

Textbook Question

Shifting Graphs

y = (1/2)(x + 1) + 5 Down 5, right 1

Textbook Question

Graph the functions in Exercises 37–56.

y = |x − 2|

Textbook Question

Graph the functions in Exercises 37–56.

y = (x + 2)³/² + 1

Textbook Question

Graph the functions in Exercises 37–56.

y = (1/x²) − 1

Textbook Question

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 = (1/2x) − 1

Textbook Question

Graphing

y = (−2x)²/³

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