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Chapter 2: Problem 43

Write the equation of each graph after the indicated transformation\((s)\) The graph of \(y=|x|\) is reflected in the \(x\) -axis, stretched by a factor of \(3,\) then translated seven units to the right and nine units upward.

Short Answer

Expert verified
The equation after the transformations is \(y = -3|x-7| + 9\).

Step by step solution

01

- Reflect in the x-axis

Start by reflecting the graph of the given function in the x-axis. Reflecting in the x-axis changes the function from \(y = |x|\) to \(y = -|x|\).
02

- Stretch by a factor of 3

Next, stretch the function vertically by a factor of 3. This means multiplying the entire function by 3. The new function becomes \(y = -3|x|\).
03

- Translate seven units to the right

Translate the function seven units to the right. This involves replacing \(x\) with \(x-7\) in the function. The modified function will now be \(y = -3|x-7|\).
04

- Translate nine units upwards

Finally, translate the function nine units upwards. This means adding 9 to the function. The final transformed function is \(y = -3|x-7| + 9\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Reflection in the x-axis
Reflection in the x-axis is a transformation that flips the graph of a function over the x-axis. This transformation negates the y-values of the function. For the given function, which is the absolute value function represented by \(y = |x|\), reflecting it in the x-axis changes every y-coordinate to its opposite. So, the function \(y = |x|\) becomes \(y = -|x|\) after this reflection. This step inverts the values while keeping the same x-coordinates.
Vertical Stretching
Vertical stretching (or compression) changes the steepness of the graph. This is done by multiplying the entire function by a factor. If the factor is greater than 1, the graph stretches, making it steeper. If it's between 0 and 1, the graph compresses. For example, multiplying the function \(y = -|x|\) by 3, which is the stretching factor, yields \(y = -3|x|\). This makes the graph steeper, as each y-coordinate becomes three times its original value. The shape remains the same, only stretched vertically.
Horizontal Translation
Horizontal translation shifts the graph left or right. This transformation changes the x-coordinates but not the y-coordinates of the function. To translate the function to the right, replace every instance of \(x\) with \(x - h\), where \(h\) is the number of units to shift. For our function, we move it 7 units to the right. Thus, we replace \(x\) with \(x-7\), resulting in the new function \(y = -3|x-7|\). This moves the shape of the graph without changing its form.
Vertical Translation
Vertical translation shifts the graph up or down. This transformation changes the y-coordinates by adding or subtracting a value. For moving the graph upwards, add the required units to the function. In our example, we move the function 9 units up. Adding 9 to the function \(y = -3|x-7|\) results in \(y = -3|x-7| + 9\). This final step adjusts the height of the graph, positioning it correctly while keeping the same shape from the previous transformations.

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