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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 1.2.26
Chapter 1, Problem 1.2.26

Shifting Graphs


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


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Verified step by step guidance
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Identify the original function: The original function given is y = -x². This is a downward-opening parabola centered at the origin (0,0).
Understand the types of shifts: Graphs can be shifted vertically, horizontally, or both. A vertical shift involves adding or subtracting a constant to the function, while a horizontal shift involves adding or subtracting a constant inside the function's argument.
Determine the vertical shifts: If the graph is moved up or down, this is a vertical shift. For example, if the graph is moved up by 'k' units, the new equation becomes y = -x² + k. If moved down by 'k' units, it becomes y = -x² - k.
Determine the horizontal shifts: If the graph is moved left or right, this is a horizontal shift. For example, if the graph is moved right by 'h' units, the new equation becomes y = -(x-h)². If moved left by 'h' units, it becomes y = -(x+h)².
Combine shifts if necessary: If the graph is shifted both horizontally and vertically, combine the transformations. For example, if the graph is shifted right by 'h' units and up by 'k' units, the equation becomes y = -(x-h)² + k.

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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 horizontal shift is achieved by adding or subtracting a value from the input variable (x), while a vertical shift is done by adding or subtracting a value from the output variable (y). For example, shifting the graph of y = -x² to the right by 3 units results in the equation y = -(x - 3)².
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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'. In the case of y = -x², the parabola opens downwards, and its vertex is at the origin (0,0). Understanding the standard form helps in identifying how shifts affect the graph.
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Vertex Form of a Quadratic

The vertex form of a quadratic function is given by y = a(x - h)² + k, where (h, k) is the vertex of the parabola. This form is particularly useful for graphing and understanding transformations, as it clearly shows how the graph shifts based on the values of h and k. For instance, if h is positive, the graph shifts to the right, and if k is positive, it shifts upward.
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Related Practice
Textbook Question

In Exercises 19–32, find the (a) domain and (b) range.


𝔂 = tan(2x - π)

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Textbook Question

In Exercises 9–16, determine whether the function is even, odd, or neither.


𝔂 = x⁵ - x³ - x

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Textbook Question

Functions and Graphs


Find the natural domain and graph the functions in Exercises 15–20.


g(x) = √−x

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Textbook Question

In Exercises 37 and 38, write a piecewise formula for the function.


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Textbook Question

In Exercises 19–32, find the (a) domain and (b) range.

____

𝔂 = -2 + √1 - x

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Textbook Question

In Exercises 19–32, find the (a) domain and (b) range.


𝔂 = 3 cos x + 4 sin x (Hint: A trig identity is required.)

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