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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2:43 minutes

Problem 1.2.23

Textbook Question

Shifting Graphs

The accompanying figure shows the graph of y = −x² shifted to two new positions. Write equations for the new graphs.

<IMAGE>

Verified step by step guidance

Identify the original function: The original function given is y = -x². This is a downward-opening parabola centered at the origin (0,0).

Understand graph shifts: A graph can be shifted horizontally and/or vertically. A horizontal shift involves adding or subtracting a constant from the x-variable, while a vertical shift involves adding or subtracting a constant from the entire function.

Determine horizontal shifts: If the graph is shifted to the right by 'h' units, the equation becomes y = -(x-h)². If shifted to the left by 'h' units, it becomes y = -(x+h)².

Determine vertical shifts: If the graph is shifted upwards by 'k' units, the equation becomes y = -x² + k. If shifted downwards by 'k' units, it becomes y = -x² - k.

Combine shifts for new equations: If the graph is shifted both horizontally by 'h' units and vertically by 'k' units, the new equation is y = -(x-h)² + k. Use the information from the image to determine the specific values of 'h' and 'k' for each new graph position.

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Key Concepts

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

Graph Shifting

Graph shifting involves translating a function's graph horizontally or vertically without altering its shape. A vertical shift occurs when a constant is added or subtracted from the function, while a horizontal shift involves adding or subtracting from the input variable. For example, the function y = -x² shifted up by 3 units becomes y = -x² + 3.

Quadratic Functions

Quadratic functions are polynomial functions of degree two, typically expressed in the form y = ax² + bx + c. The graph of a quadratic function is a parabola, which can open upwards or downwards depending on the sign of 'a'. Understanding the standard form helps in identifying the vertex and the direction of the parabola, which is crucial when shifting the graph.

Transformation of Functions

Transformations of functions include various operations that change the position or shape of the graph. These transformations can be translations, reflections, stretches, or compressions. In the context of the given question, recognizing how to apply vertical and horizontal shifts to the original quadratic function is essential for writing the equations of the new graphs.

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Related Practice

Textbook Question

Describe how each graph is obtained from the graph of 𝔂 = ƒ(x).

a. 𝔂 = ƒ(x - 5)

Textbook Question

Describe how each graph is obtained from the graph of 𝔂 = ƒ(x).

d. 𝔂 = ƒ(2x + 1)

Textbook Question

Describe how each graph is obtained from the graph of 𝔂 = ƒ(x).

e. 𝔂 = ƒ( x ) - 4

Textbook Question

In Exercises 55–58, graph each function, not by plotting points, but by starting with the graph of one of the standard functions presented in Figures 1.15–1.17, and applying an appropriate transformation.

y = - √(1 + x/2)

Textbook Question

Shifting Graphs

The accompanying figure shows the graph of y = −x² shifted to four new positions. Write an equation for each new graph.

<IMAGE>

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

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