Answer:
g(x) is flipped across the x-axis and translated -3 units
Step-by-step explanation:
BACKGROUND:
The rules of functions of the type you're asking for follow this structure (note that terms I use may not abide by the textbook definition):
f(x) = <em>a</em>(x - <em>b</em>)^<em>n</em> + <em>c</em>
<em>a</em> - vertical stretch or compression. If negative it will also flip the function across the x-axis.
<em>b</em> - horizontal translation. Note that if the number representing <em>b</em> is positive, there will be a negative translation and if negative there will be a positive translation.
- ex: (x + 3) will move the function 3 units to the <em>left </em>or in the <em>negative </em>direction on the graph
<em>c </em>- vertical translation of the function. Does not abide by the same rule as the horizontal in terms of having the opposite effect on the graph compared to what's implied.
- ex: (x + 3) - 4 will move 4 units down just like what's implied by the negative sign
<em>n </em>- there are a few special rules you can use to figure out the end behavior (EB) of functions like this one. If the exponent is positive, no matter how big the number it will have the same end behavior as a quadratic function. A function with an odd exponent will have the same end behavior as a linear function. (not useful in this case but still noteworthy)
- ex: f(x) = x^2 has the same end behavior as g(x) = x^6
WORK:
In this case, if abiding by these rules, g(x) = -(x + 3)^4 is:
- flipped across the x axis because of the negative value for <em>a</em>.
- translated 3 units to the left because of the value of <em>b</em>.
- has the same end behavior as the function f(x) = -x^2 (EB isn't useful in this case but may be in the future.)
Note: please comment with questions because the explanation I'm giving is highly visual and based on what I was taught, so I'm not a master at this type of math either. I'd recommend the free app Desmos to check out how changing a function impacts its graph.
