Let f(x)=8x. The graph of f(x) is transformed into the graph of g(x) by a vertical stretch of 4 and a translation of 7 units down. What is an equation for g(x)?

Mar 4, 2017

$g \left(x\right) = 32 x - 7$

Explanation:

In $y = a f b \left(x + c\right) + d$:

$a$ represents the vertical stretch

$b$ represents the horizontal stretch

$c$ represents the horizontal translation

$d$ represents the vertical translation

We are given that $g \left(x\right)$ has a vertical stretch and a vertical translation (down and up are on the y-axis, hence it's a vertical translation).

Therefore, the equation of $g \left(x\right)$ is

$g \left(x\right) = 4 \left(8 x\right) - 7 = 32 x - 7$

Practice Exercises

1. Consider the function $f \left(x\right) = {x}^{3}$. $f \left(x\right)$ undergoes a vertical transformation of $5$ units down and a horizontal translation of $2$ units right. It is reflected over the x-axis. What is $f \left(x\right)$'s new equation?

2. The point $\left(6 , - 4\right)$ lies on the graph of function $g \left(x\right)$. What are the coordinates of the point that will lie on $2 g \left(3 x - 9\right) + 1$?

Solutions

1. $f \left(x\right) = - {\left(x - 2\right)}^{3} + 2$
2. $\left(5 , - 7\right)$

Hopefully this helps!