Mastering Quadratics: Transform Your Function with Our Examples

Find the linear equations that can be used to convert an (x, y) equation to a (x, v) equation using the given angle of rotation $\theta$ . $\theta ={70}^{\circ }$

g is related to one of the parent functions described in Section 1.6. Describe the sequence of transformations from f to g. ${\displaystyle g\left(x\right)=-\left|x\right|-2}$

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) ${\displaystyle y={\left(\frac{3}{4}(x-10)\right)}^3}$

For ${\displaystyle y=-\log 3}$
x a. Use transformations of the graphs of ${\displaystyle y=\log 2x}$
x and ${\displaystyle y=\log 3x}$
x to graph the given functions. b. Write the domain and range in interval notation. c. Write an equation of the asymptote.

${\displaystyle {\sec }^2\left(x\right)-1=0}$

Let L(Rm,Rn) be the set of all linear transformations from Rm to Rn. Is L(Rm,Rn), a subspace of F(Rm,Rn), the space of all functions from Rm to Rn? Justify your answer carefully.

Explain how transformations, families of functions, and parent functions are related.

h varies jointly as f and g. Find h when f = 20, g = 15, and k = 4.

h is related to one of the six parent functions. (a) Identify the parent function f. (b) Describe the sequence of transformations from f to h. (c) Sketch the graph of h by hand. (d) Use function notation to write h in terms of the parent function f. $h(x)=f(x-1)+4$

Sketch a graph of the function. Use transformations of functions when ever possible. ${\displaystyle f\left(x\right)=\frac{1}{3}(x-5),2\le x\le 8}$

Begin by graphing ${\displaystyle f\left(x\right)={\log }_{2}x}$ Then use transformations of this graph to graph the given function. What is the graphs

For ${\displaystyle y=-{\log }_{2}x}$.
a) Use transformations of the graphs of ${\displaystyle y={\log }_{2}x}$ and ${\displaystyle y={\log }_{3}x}$ o graph the given functions.
b) Write the domain and range in interval notation.
c) Write an equation of the asymptote.

g is related to one of the six parent functions. (a) Identify the parent function f. (b) Describe the sequence of transformations from f to g. (c) Sketch the graph of g by hand. (d) Use function notation to write g in terms of the parent function f.${\displaystyle g\left(x\right)=\frac{1}{3}{(x-2)}^3}$

Sketch a graph of the function. Use transformations of functions when ever possible. ${\displaystyle f\left(x\right)=1-\sqrt{x+2}}$

Sketch a graph of the function. Use transformations of functions when ever possible. ${\displaystyle f\left(x\right)=\frac{1}{{(x-1)}^3}}$

Begin by graphing ${\displaystyle f\left(x\right)={\log }_{2}x}$ Then use transformations of this graph to graph the given function. What is the vertical asymptote? Use the graphs to determine each functions domain and range. ${\displaystyle g\left(x\right)=-2{\log }_{2}x}$

h is related to one of the six parent functions.
a) Identify the parent function f.
b) Describe the sequence of transformations from f to h.
c) Sketch the graph of h by hand.
d) Use function notation to write h in terms of the parent function f.
${\displaystyle h\left(x\right)={(-x)}^2-8}$

Begin by graphing
${\displaystyle f\left(x\right)=2^2}$
Then use transformations of this graph to graph the given function. Be sure to graph and give equations of the asymptotes. Use the graphs to determine each functions

g is related to one of the six parent functions.
a) Identify the parent function f.
b) Describe the sequence of transformations from f to g.
c) Sketch the graph of g by hand.
d) Use function notation to write g in terms of the parent function f.
${\displaystyle g\left(x\right)=-2|x-1|-4}$

Sketch a graph of the function. Use transformations of functions when ever possible. ${\displaystyle f\left(x\right)=-\frac{1}{x^2}}$