Solve problems related to transformations of functions with confidence, thanks to expert examples and equations.

Graph the function by hand, not by plotting points but by starting with the graph of one of the standard functions then apply the appropriate transformations. y= -x²-4.

Find a linear mapping G that maps ${\displaystyle [0,1]\times [0,1]}$ to the parallelogram in the xy-plane spanned by the vectors $(2,3)$ and ${\displaystyle (4,1)}$.

find a formula for the quadratic function whose graph has the given y-intercept of y=-1 and a vertex at (-1,-2)

State the parent function that must be transformed to create the graph of each of the following functions. Then describe the transformations that must be applied to the parent function.
a) ${\displaystyle y=\left(\frac{5}{4}\right)x^4+3}$
b) $y=3x-4$
c) ${\displaystyle y={(3x+4)}^3-7}$
d) ${\displaystyle y=-{(x+8)}^4}$
e) $y=-4.8(x-3)(x-3)$
f) ${\displaystyle 2{(\left(\frac{1}{5}\right)x+7)}^3-4}$

The following question consider the Gompertz equation, a modification for logistic growth, which is often used for modeling cancer growth, specifically the number of tumor cells. [T] The Gompertz equation has been used to model tumor growth in the human body. Starting from one tumor cell on day 1 and assuming \(α=0.1\) and a carrying capacity of 10 million cells, how long does it take to reach “detection” stage at 5 million cells?

Describe the transformations that were applied to $y=x^3$to create each of the following functions.
a) ${\displaystyle y=12{(x-9)}^3-7}$
b) ${\displaystyle y=\left(\frac{7}{8}(x+1)\right)3+3}$
c) ${\displaystyle y=-2{(x-6)}^3-8}$
d) $y=(x+9)(x+9)(x+9)$
e) ${\displaystyle y=-2{(-3(x-4))}^3-5}$
f) ${\displaystyle y={\left(\frac{3}{4}(x-10)\right)}^3}$