How are the graphs #f(x)=x^3# and #g(x)=0.75(x+1)^3# related?

Question

Elijah Allen · Accepted Answer

See below

#f(x)# is what we call the parent function. Let's see what #x^3# looks like: graph{x^3 [-10, 10, -5, 5]} Now let's see what #(x+1)^3# looks like: graph{(x+1)^3 [-10, 10, -5, 5]} You should see that it's just a shifted version of the parent function. In this case, it's a horizontal shift #1# unit to the left. Now, let's see what the constant in the function does. Since it's hard to see what's going on if we use #0.75#, let's graph another function that will tell us what a constant THAT is LESS THAN #1# does. Let's look at #0.1(x+1)^3# graph{0.1(x+1)^3 [-10, 10, -5, 5]} You should see that the constant is the #y#-intercept - the value when it crosses the #x# axis.

Madelyn Anderson · Answer

The graph of #g(x)# can be described as a transformation of the graph of #f(x)#.

First, take a look at the graph of #f(x)#. graph{x^3 [-10, 10, -5, 5]} Now look at the graph of #g(x)#. graph{0.75(x+1)^3 [-10, 10, -5, 5]} The graph of #f(x)# is vertically shrunk by a factor of #0.75# and translated left 1 unit to obtain the graph of #g(x)#.

Lincoln Anderson · Answer

undefined The graphs of $ f(x) = x^3 $ and $ g(x) = 0.75(x+1)^3 $ are related by a vertical compression and a horizontal translation to the left by one unit. The graph of $ g(x) $ is obtained from the graph of $ f(x) $ by compressing it vertically by a factor of $ 0.75 $, which means that the y-coordinates of all points on the graph are multiplied by $ 0.75 $. Additionally, the graph of $ g(x) $ is shifted to the left by one unit compared to the graph of $ f(x) $, as indicated by the $ +1 $ inside the parentheses.