Quadratic Equations and Applications

Suppose $${f(x)=x^2}$$, where $$x$$ can be any real number. 

Using a graphing calculator or other graphing technology, graph the function $$f$$ and each of the transformations to function $$f$$ shown below. For each graph, describe how the graph of each transformation compares to function $$f$$.

a.  $$a(x)=f(x)+2$$

b.  $$b(x)=-2f(x)$$

c.  $$c(x)={1\over2} f(x)$$

d.  $$d(x)=f(x+2)$$

e.  $$e(x)=-f(x-2)-2$$

Graphs of functions $${{{f(x)}}}$$  and $${{{{g(x)}} }}$$ are shown below. In each pair of graphs, the graph of function $${{{{g(x)}} }}$$ represents a transformation of function $${{{f(x)}}}$$.

1. For each pair of graphs, describe the transformation that maps the graph of $${{{f(x)}}}$$ to the graph of $${{g(x)}}$$.
2. Let $${{g(x)}}=a(x-h)^2+k$$. For each graph of $${{g(x)}}$$, identify the values of $$a$$, $$h$$, and $$k$$ to make the equation true. 

A quadratic function $${{f(x)}}$$ is vertically stretched by a factor of $$3$$ and translated right by $$6$$ units.

Write an equation for $${h(x)}$$ to represent the transformation of the graph of function $${{f(x)}}$$ if

a.  $${{f(x)}}=x^2$$

b.  $${{f(x)}}=x^2-4$$

c.  $${{f(x)}}=-(x-1)^2+2 $$

Function $$f$$ is given by the equation $$f(x)=(x-2)^2+3$$.

Function $$g$$ is given by the equation $$g(x)=-(x+1)^2-1$$.

Describe the transformations that take the graph of $$f(x)$$ to the graph of $$g(x)$$.