Question

Determine which sequences of transformations could be applied to the parent function f(x) = x to obtain the graph of g.

Answer 1

Answer:

Reflect over the y-axis, vertically stretch by a factor of 3, and then shift down 1 unit

Step-by-step explanation:

Parent function:  $f(x)=x$

The graph of the parent function is a straight line graph that intersects the axes at the origin (0, 0) and has a positive slope of 1 unit.

To determine the sequence of transformations, find the equation of the transformed function in slope--intercept form.

Slope-intercept form of a linear function:  $f(x)=mx+b$

(where m is the slope and b is the y-intercept)

To calculate the slope of the transformed function, choose two points on the line and use the slope formula:

• Let (x₁, y₁) = (0, -1)
• Let (x₂, y₂) = (1, -4)

$\implies \sf slope\:(m)=\dfrac{y_2-y_1}{x_2-x_1}=\dfrac{-4-(-1)}{1-0}=-3$

The y-intercept (where the line crosses the y-axis) of the transformed function is (0, -1).

Therefore the equation of the transformed function is:

$g(x)=-3x-1$

Translations

For a > 0

$f(x+a) \implies f(x) \: \textsf{translated}\:a\:\textsf{units left}$

$f(x-a) \implies f(x) \: \textsf{translated}\:a\:\textsf{units right}$

$f(x)+a \implies f(x) \: \textsf{translated}\:a\:\textsf{units up}$

$f(x)-a \implies f(x) \: \textsf{translated}\:a\:\textsf{units down}$

$y=a\:f(x) \implies f(x) \: \textsf{stretched parallel to the y-axis (vertically) by a factor of}\:a$

$y=f(ax) \implies f(x) \: \textsf{stretched parallel to the x-axis (horizontally) by a factor of} \: \dfrac{1}{a}$

$y=-f(x) \implies f(x) \: \textsf{reflected in the} \: x \textsf{-axis}$

$y=f(-x) \implies f(x) \: \textsf{reflected in the} \: y \textsf{-axis}$

Comparing the transformed function's equation with the parent function:

$\begin{cases}f(x)=x\\g(x)=-3x-1\end{cases}$

Transformations

1.  Reflection in the y-axis:

    $f(-x)=-x$

2.  Vertically stretched by a factor of 3:

    $3f(-x)=-3x$

3.  Shifted 1 unit down:

    $3f(-x)-1=-3x-1$

Summary

Reflect over the y-axis, vertically stretch by a factor of 3, and then shift down 1 unit

Learn more about translations here:

https://brainly.com/question/27815602

https://brainly.com/question/27845947