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

Ch. 1 - Functions

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch. 1 - Functions

Problem 1.2.10

Chapter 1, Problem 1.2.10

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

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.

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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