Question

Let the graph of g be a vertical stretch by a factor of 3 and a reflection in the y-axis, followed by a translation 2 units left of the graph of f(x)=x^2−2x+1. Write a rule for g.

Answer 1

• The rule for g is g(x) = 3x² + 18x + 27

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The given function f(x) = x² - 2x + 1 can be transformed into the function g(x) as follows, g(x) is obtained by:

• First, reflecting the graph of f(x) in the y-axis,
• Then, by vertically stretching the reflected graph by a factor of 3, and,
• Finally, by translating the stretched graph 2 units to the left.

So the rule for g(x) can be obtained by performing these operations on the rule for f(x).

Reflection in the y-axis can be achieved by replacing x with -x in f(x).

This gives:

• f(-x) = (-x)² - 2(-x) + 1 = x² + 2x + 1

Next, we need to vertically stretch the graph by a factor of 3. To achieve this, we can multiply f(-x) by 3.

This gives:

• 3f(-x) = 3(x² + 2x + 1)

Finally, we need to translate the stretched graph 2 units to the left.

To achieve this, we can replace x with (x + 2) in 3f(-x).

This gives:

• g(x) = 3f(-(x + 2)) = 3((x + 2)² + 2(x + 2) + 1)

Simplifying this expression, we get:

• g(x) = 3(x² + 6x + 9)

Therefore, the rule for g is g(x) = 3x² + 18x + 27.