Given the pair of functions f and g, sketch the graph of...

Given the pair of functions $f$ and $g,$ sketch the graph of $y=g(x)$ by starting with the graph of $y=f(x)$ and using transformations. Track at least three points of your choice through the transformations. State the domain and range of $g$.
$$f(x)=x^{3}, g(x)=(x+2)^{3}+1$$

Key Concepts

Cubic Functions

Cubic functions, defined by expressions such as x^3, are polynomial functions of degree three. They are characterized by their smooth, continuous curves that extend indefinitely in both the positive and negative directions, inherently having domains and ranges that include all real numbers.

Domain and Range

The domain and range describe all possible input values and output values, respectively, for a function. For functions like cubic functions that are defined for all real numbers, translations generally do not restrict these sets, meaning the domain and range often remain as all real numbers.

Function Transformations

Function transformations involve operations like translations, stretches, compressions, and reflections that move or change the shape of the graph of a function. These operations help us understand how modifications to the function's formula affect its graph.

Horizontal Translation

A horizontal translation shifts the entire graph of a function left or right. In the expression (x + 2)^3, the '+2' results in moving the graph 2 units to the left, because the input values must be decreased by 2 to achieve the same output as the original function.

Vertical Translation

A vertical translation moves the graph up or down. The addition of '+1' outside the function, as seen in (x + 2)^3 + 1, shifts the graph upward by 1 unit, effectively increasing each function output by 1.

Transcript

00:01 So for this problem, we are asked to graph f of x and the transform function g of x, determine the domain and range of g of x, and also kind of track a few points along the way.

00:14 So here i have f of x graphed, and we're going to use the parent function, point zero, 1 ,1, and negative 1 ,1.

00:29 Now, with g of x, inside the parentheses tells me that this graph is going to move to the left one, or i'm sorry, to the right one.

00:39 Since it's a negative one, you need to switch your sign, so it's actually going to move to the right.

00:45 The negative three in front of the parentheses is going to tell me that it reflects over the x -axis, and that it's vertically stretched by three.

01:08 And then the two is actually my k -value, even though it's in the front, you could really put it in the back, but that's going to tell me that this function shifted up to.

01:18 So what i'm going to do is list the changes that need to be made to the x and y to get my new points so for my x all that i have to do is add one to all of my xes since it's shifting to the right one now to each of these ys i need to multiply them by negative three and then i also need to add two so whatever number is in front of the parentheses you're going to multiply by that y and then your k value you're either going to or subtract that value depending on if it's positive or negative.

02:00 I'm going to go ahead and do that through with all these points.

02:03 So if i add 0 plus 1, i get 1.

02:07 0 times negative 3 is 0 plus 2 gives me 2...

Please give Ace some feedback