Question Video: Combining Function Transformations | Nagwa Question Video: Combining Function Transformations | Nagwa

# Question Video: Combining Function Transformations Mathematics • Second Year of Secondary School

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Use the shown graph of 𝑦 = 𝑓(𝑥) to find 𝑎, 𝑏, and 𝑐 so that 𝑓(𝑥) = 𝑎|𝑥 − 𝑏| + 𝑐.

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### Video Transcript

Use the shown graph of 𝑦 equals 𝑓 of 𝑥 to find 𝑎, 𝑏, and 𝑐 so that 𝑓 of 𝑥 equals 𝑎 times the absolute value of 𝑥 minus 𝑏 plus 𝑐.

By visual inspection, we can see that 𝑓 of 𝑥 is some kind of transformed absolute value function. Sketching in 𝑔 of 𝑥, the standard absolute value function, will help us to see what transformations have occurred. First of all, we can see that the slope of the graph has changed. It has been stretched upwards or squashed inwards, which amounts to the same thing.

Looking at the graphs, we can see that the standard absolute value function has a slope of plus or minus one. This has changed to a slope of plus or minus three for 𝑓 of 𝑥. Second, by looking at the vertices of the graphs, we can see that 𝑓 has been shifted three units to the left. Finally, we can see that the graph of 𝑓 of 𝑥 has shifted four units down.

The geometric transformation changing the slope of the graph from one to three can be thought of as a vertical dilation with a scale factor of three. Algebraically, this corresponds to multiplying the absolute value function by three.

Notice that this transformation can also be thought of as a horizontal dilation with a scale factor of one-third, that is to say, squashing the graph horizontally rather than stretching it vertically. Algebraically, this corresponds to multiplying the variable 𝑥 by the reciprocal of one-third, that is, by three. Because the absolute value function is a linear function passing through the origin, this amounts to the same transformation.

The next transformation is a translation in the 𝑥-direction by three units to the left. Algebraically, this corresponds to adding three units to the variable 𝑥. The final transformation is a vertical translation by four units down. Algebraically, this corresponds to subtracting four from the whole function.

We have arrived at our transformed function. 𝑓 of 𝑥 equals three times the absolute value of 𝑥 plus three, or 𝑥 minus negative three, all minus four. Thus, 𝑎 equals three, 𝑏 equals negative three, and 𝑐 equals negative four.

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