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Chapter 1: Problem 8

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

Short Answer

Expert verified
Answer: The two transformations are vertical stretching by a factor of 3 and reflecting the graph across the x-axis (flipping).

Step by step solution

01

Analyze the effect of multiplying by -3

To go from the graph of \(y=f(x)\) to \(y=-3f(x)\), we are multiplying the function by a constant -3. This constant -3 affects the graph in two ways: it results in vertical stretching by a factor of 3 and reflects the graph across the x-axis (flips it).
02

Vertical stretching by a factor of 3

To perform vertical stretching, we multiply the y-coordinate of every point in the original graph (graph of \(f(x)\)) by the magnitude of the constant, which is 3 in this case. In other words, we transform a point \((x, f(x))\) into \((x, 3f(x))\) for every x-value. This stretches the graph vertically by making it 3 times taller.
03

Reflect the graph across the x-axis (flip)

To reflect the graph across the x-axis, we multiply the y-coordinate of every point on the vertical-stretched graph (graph of \(3f(x)\)) by -1. In other words, we transform a point \((x, 3f(x))\) into \((x, -3f(x))\) for every x-value. This flips the graph across the x-axis so that the graph of \(y=-3f(x)\) is obtained. Now, by combining the vertical stretching by a factor of 3 (Step 2) and reflecting the graph across the x-axis (Step 3), we have obtained the graph of \(y=-3f(x)\) from the graph of \(y=f(x)\).

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