Which function is the result of vertically stretching f(x) = x ^ 2 - 7 by a factor of 2 and translating downward 5 units?

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26-06-2022
Mathematics

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Which function is the result of vertically stretching f(x) = x ^ 2 - 7 by a factor of 2 and translating downward 5 units?

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ap8997154 ap8997154 · Answer 1 · 2022-06-28T08:20:36+02:00

The function that is the result of vertically stretching f(x) = x²-7 by a factor of 2 and translating downward 5 units is 2x² - 19.

How does the transformation of a function happen?

The transformation of a function may involve any change.

Usually, these can be shifted horizontally (by transforming inputs) or vertically (by transforming output), stretched (multiplying outputs or inputs) etc.

If the original function is y = f(x), assuming the horizontal axis is the input axis and the vertical is for outputs, then:

Horizontal shift (also called phase shift):

Left shift by c units, y=f(x+c) (same output, but c units earlier)
Right shift by c units, y=f(x-c)(same output, but c units late)

Vertical shift:

Up by d units: y = f(x) + d
Down by d units: y = f(x) - d

Stretching:

Vertical stretch by a factor k: y = k × f(x)
Horizontal stretch by a factor k: y = f(x/k)

The given function f(x) is needed to be stretched vertically by a factor of 2, therefore, the function can be written as,

g(x) = 2(x² - 7) = 2x² - 14

Now, the function is needed to be translated by 5 units in the downward direction, therefore,

p(x) = g(x)-5 = 2x² - 14 -5

p(x) = 2x² - 19

Hence, the function that is the result of vertically stretching f(x) = x²-7 by a factor of 2 and translating downward 5 units is 2x² - 19.

Learn more about Transforming functions:

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