2x5−x3+x2+4; reflection in the y-axis and a vertical stretch by a factor of 3, followed by a translation 1 unit down

Answer 1

Starting with the function f(x) = 2x⁵ - x³ + x² + 4

Reflection in the y-axis ⇒ f(-x)

Vertical stretch by factor of 3 ⇒ 3f(-x)

Translation 1 unit down ⇒ 3f(-x) - 1

Plugging all that into the original function:

3{2(-x)⁵ - (-x)³ + (-x)² + 4} - 1

= 3{-2x⁵ + x³ + x² + 4} - 1

= -6x⁵ + 3x³ + 3x² + 12 - 1

= -6x⁵ + 3x³ + 3x² + 11

Answer 2

The required polynomial is .

The given question solved in the following steps.

1. Apply vertical stretch by a factor 3. You get then the polynomial .

     

1. Reflection in the y-axis is the change of x to (-x).

    So, after reflection, the new polynomial is 

      

1.  Translation 2 units left is the change of x by (x+2) in the polynomial.

      So, the new polynomial g(x)     

      

The required value of g(x) is .