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Ch. 1 - Functions
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 1 - FunctionsProblem 1.2.10
Chapter 1, Problem 1.2.10

How do you obtain the graph of y=−3f(x)y=-3f\(\left\)(x\(\right\)) from the graph of y=f(x)y=f\(\left\)(x\(\right\))?

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Start with the graph of y = f(x). This is your original function graph.
The transformation y = -3f(x) involves two operations: a vertical reflection and a vertical stretch.
First, apply a vertical reflection across the x-axis. This changes the graph of y = f(x) to y = -f(x), flipping it upside down.
Next, apply a vertical stretch by a factor of 3. This means you multiply all y-values of the graph by 3, making the graph taller by stretching it away from the x-axis.
Combine these transformations to obtain the final graph of y = -3f(x), which is the vertically stretched and reflected version of the original graph y = f(x).

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Key Concepts

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

Function Transformation

Function transformation refers to the process of altering the graph of a function through various operations, such as shifting, stretching, or reflecting. In this case, the transformation involves multiplying the function f(x) by -3, which results in a vertical stretch by a factor of 3 and a reflection across the x-axis.
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Intro to Transformations

Vertical Stretch and Reflection

A vertical stretch occurs when the output values of a function are multiplied by a factor greater than 1, making the graph taller. A reflection across the x-axis flips the graph upside down. The transformation y = -3f(x) combines both effects: it stretches the graph of f(x) vertically by a factor of 3 and reflects it over the x-axis.
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Foci and Vertices of an Ellipse

Graphing Techniques

Graphing techniques involve methods for accurately plotting the transformations of functions. To graph y = -3f(x), one can start with the original graph of y = f(x), apply the vertical stretch by multiplying the y-values by 3, and then reflect the resulting points across the x-axis to obtain the final graph.
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Related Practice
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Find the inverse of the function ƒ(x) = 2x. Verify that ƒ(ƒ⁻¹(x)) = x and ƒ⁻¹(ƒ(x)) = x .

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Convert the following expressions to the indicated base.


a1lnaa^{\(\frac{1}{\ln a}\)} using basa e, for a>0a > 0 and a≠1a ≠ 1

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Simplify the difference quotient (ƒ(x)-ƒ(a)) / (x-a) for the following functions.

ƒ(x) = 4 - 4x + x²

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Let ƒ(x) = 1/ (x³+1).

Compute ƒ(2) and ƒ(y²).

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Textbook Question

Inverse sines and cosines Evaluate or simplify the following expressions without using a calculator.


cos⁻¹ (- 1/2 )

Textbook Question

Simplify the difference quotient (ƒ(x)-ƒ(a)) / (x-a) for the following functions.

ƒ(x) = x⁴