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^2−3x+1. Write a rule for g.

Answer 1

From my perspective the best way to do this is to rewrite f(x) in vertex form (which is a standard transformation form).

f(x) = x² - 3x + 1 then complete the square:

f(x) = (x² - 3x + 9/4) - 9/4 + 1

f(x) = (x - 3/2)² - 5/4

Apply a vertical stretch factor of 3 to get y = 3(x - 3/2)² - 5/4

Reflect it over the y-axis: y = 3(-(x + 3/2))² - 5/4

Translate left by 2: y = 3(-(x + 3/2 + 2))² - 5/4

Simplify: g(x) = 3(x + 7/2)² - 5/4

Put it back into standard form as needed:

g(x) = 3(x² + 7x + 49/4) - 5/4

g(x) = 3x² + 21x + 71/2