Question

Describe the transformation to get from f(x) to g(x)

g(x)=f(x)-4

g(x)=f(x+6)

g(x)=1/2f(-x)

g(x)=-f(5x)

g(x)=3f(x+4)-6

g(x)=-f(3/2x)-5

Answer 1

The transformation of the function f(x) into g(x) are obtained as follows;

10) f(x) is translated 4 units downwards to get g(x)

11) f(x) is translated 6 units to the left to get g(x)

12) f(x) is shrunk vertically by a factor of (1/2) and it is also reflected across the y-axis to get g(x)

13) g(x) is obtained by a reflection of f(x) across the x-axis and a shrinking of f(x) by a factor 5

14) To get g(x), f(x) is dilated by a factor of 3 and translated 4 units to the left followed by a vertical translation of 6 units downwards

15) The function f(x) is reflected across the x-axis and shrunk horizontally by a factor of (3/2), and also translated 5 units downwards to get g(x)

What is a transformation of a function in mathematics?

A function is transformed by changing its location to the left, right, upwards or downwards or by vertical or horizontal shrinking.

The transformations of a function, y = f(x) are presented as follows;

The parent function is; y = f(x),

The transformed function is; y = a·f(b·(x + c)) + d

Where a, b, c, d are real numbers that serve as representation of the transformation of the function, f(x) as follows;

A variable located outside the parenthesis of the function, f, represent vertical transformations, while horizontal transformation are produced by the variables inside the parenthesis

When a variable is added or subtracted, the function is translated but maintains its shape, while when the operation is a multiplication or a division, it represents a dilation

The use of a minus signifies a reflection

In the equation, y = a·f(b·(x + c)) + d, the effects of the variables are;

• a; Dilation of the function vertically

• b; Dilation of the function horizontally

• c; Horizontal translation of the function

• d; Vertical translation of the function

10) g(x) = f(x) - 4

The transformation of the function f(x) to get g(x) is a vertical translation 4 units down

11) g(x) = f(x + 6)

The transformation in the above function is c = 6, which represents an horizontal translation.

The positive sign indicates that the function is translated to the left and the number of units of translation is 6 units

• Therefore, the function, f(x) is translated 6 units to the left to get f(x)

12. g(x) = (1/2)·f(-x)

The transformation variables are, a = (1/2), and b = -1

The multiplication indicates that the function, f(x) is dilated, and the value of 0 < a < 1 indicates that the graph shrinks by a factor of a = (1/2)

The variable b = -1 indicates that the function f(x) is reflected across the y-axis

• Therefore, the transformation of f(x) to get g(x) = (1/2)·f(-x) is a vertical shrinking by a factor of (1/2) followed by a reflection across the y-axis

13. g(x) = -f(5·x)

The transformation variables are; a = -1 and b = 5

• Therefore, the graph is reflected across the x-axis followed by an horizontal shrinking by a factor of 5

14. g(x) =3·f(x + 4) - 6

The transformation variables are a = 3, and c = 4 and d = -6

• The function f(x) is therefore dilated by a factor of 3 followed by a horizontal translation of 4 units to the left and a vertical translation of 6 units down to get g(x)

15. g(x) = -f((3/2)·x) - 5

In the above transformation of f(x) to g(x), we have;

a = -1, b = (3/2), and d = -5, therefore;

• The function f(x) is reflected with respect to the x-axis, it shrinks horizontally by a factor of (3/2) and is translated 5 units downwards to get the graph of g(x)

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