Question

Explain how to identify linear reflections and translations.

Answer 1

Explanation:

In the context of functions, the transformations we're often concerned with are vertical and horizontal scaling (sometimes involving reflections), and translation.

Scale factors multiply x or y. A negative scale factor represents a reflection across one of the coordinate axes.

Translations add something to x or y.

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More detail

A function is scaled vertically by multiplying its value by the scale factor. If the scale factor is k, then the scaled function is ...

  y = k·f(x)

If k is negative, it represents a reflection across the x-axis. (Each y-value is replaced by its opposite, so what was up is now down.)

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A function is scaled horizontally by dividing the independent variable by the scale factor. If the scale factor is k, then the scaled function is ...

  y = f(x/k)

If k is negative, it represents a reflection across the y-axis. (Each x-value is replaced by its opposite, so that what was right is now left.)

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A function is translated vertically by adding the translation amount to the function value. If the translation upward is by k units, the translated function is ...

  y = f(x) +k

If k is negative, the translation is downward.

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A function is translated horizontally by subtracting the translation amount from the independent variable. If the translation to the right is by k units, the translated function is ...

  y = f(x -k)

If k is negative, the translation is to the left.

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Example

Suppose we want the equation of an arch that has the shape of a parabola. It will be 600 feet wide at the base and 350 feet high (flatter than the St. Louis Gateway Arch). The left end of the arch will be 100 feet to the left of the "zero" horizontal reference point.

The parent function for a parabola is y=x². We can invert it and give it a height of 1 unit (and a width of 2 units) by multiplying by -1 and adding 1. The first attachment shows this function: y = 1-x².

The horizontal width of this prototype is 2 units (from -1 to +1). We want it to have a width of 600 units, so we need to scale it horizontally by a factor of 600/2 = 300. The horizontally scaled function will be ...

  y = f(x/300) = 1-(x/300)²

We want to vertically scale the function by a factor of 350, so we multiply it by 350:

  y = 350f(x) = 350(1 -(x/300)²)

This will give an equation for an arch centered on the y-axis. We want to move it 200 units to the right, so we translate this function by subtracting 200 from x:

  y = f(x-200) = 350(1 -((x-200)/300)²)

The equation for our arch is ...

  f(x) = 350(1 -((x-200)/300)²) . . . . as shown in the second attachment

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Hopefully, going through the process of writing a scaled, translated, and reflected function can help you see what to look for in any function you might run across.