SOLUTION: Using the graph of f(x)= x^2 as a guide, describe the transformations,g(x)=-3(x-2)^2+9

Question 391302: Using the graph of f(x)= x^2 as a guide, describe the transformations,g(x)=-3(x-2)^2+9
Answer by haileytucki(390) (Show Source): You can put this solution on YOUR website!
f(x)=x^(2)_g(x)=-3(x-2)^(2)+9
Squaring an expression is the same as multiplying the expression by itself 2 times.
f(x)=x^(2)_g(x)=(-3*(x-2)(x-2))+9_f(g(x))
Multiply each term in the first group by each term in the second group using the FOIL method. FOIL stands for First Outer Inner Last, and is a method of multiplying two binomials. First, multiply the first two terms in each binomial group. Next, multiply the outer terms in each group, followed by the inner terms. Finally, multiply the last two terms in each group.
f(x)=x^(2)_g(x)=(-3(x*x+x*-2-2*x-2*-2))+9_f(g(x))
Simplify the FOIL expression by multiplying and combining all like terms.
f(x)=x^(2)_g(x)=(-3(x^(2)-4x+4))+9_f(g(x))
Multiply -3 by each term inside the parentheses.
f(x)=x^(2)_g(x)=(-3x^(2)+12x-12)+9_f(g(x))
Add 9 to -12 to get -3.
f(x)=x^(2)_g(x)=-3x^(2)+12x-3_f(g(x))
Setup the composite result function. The fog notation is interpreted as f(g(x)).
f(g(x))
Evaluate f(g(x)) by substituting in the value of g into f.
f(-3x^(2)+12x-3)=(-3x^(2)+12x-3)^(2)
Squaring an expression is the same as multiplying the expression by itself 2 times.
f(-3x^(2)+12x-3)=(-3x^(2)+12x-3)(-3x^(2)+12x-3)
Multiply each term in the first polynomial by each term in the second polynomial.
f(-3x^(2)+12x-3)=(-3x^(2)*-3x^(2)-3x^(2)*12x-3x^(2)*-3+12x*-3x^(2)+12x*12x+12x*-3-3*-3x^(2)-3*12x-3*-3)
Multiply each term in the first polynomial by each term in the second polynomial.
f(-3x^(2)+12x-3)=(9x^(4)-72x^(3)+162x^(2)-72x+9)
Remove the parentheses around the expression 9x^(4)-72x^(3)+162x^(2)-72x+9.
f(-3x^(2)+12x-3)=9x^(4)-72x^(3)+162x^(2)-72x+9
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