Question

Find a formula for a function describing the given situation. Graph the function and give a domain that makes sense for the problem. Recall that with constant speed. distance \(=\) speed \(\cdot\) time elapsed or \(d=v t\) A function \(y=f(x)\) such that \(y\) is 1 less than the cube of \(x\)

Short answer

Answer: The explicit formula for the function is \(y = x^3 - 1\), and its domain is all real numbers, represented as \((-\infty, \infty)\).

Step-by-step solution

Find the formula for the function

To find the formula for the function \(y=f(x)\), we need to express \(y\) in terms of \(x\). Since \(y\) is 1 less than the cube of \(x\), we can write this as: \(y = x^3 - 1\).

Determine the domain for the function

Since there are no restrictions mentioned in the problem, the domain for the cubic function is all real numbers. In interval notation, this can be represented as: \((-\infty, \infty)\).

Graph the function

To graph the function \(y = x^3 - 1\), we can plot some points for different values of \(x\). Some points we can use are: - When \(x=-2\), \(y=(-2)^3 - 1= -9\) - When \(x=-1\), \(y=(-1)^3 - 1 = -2\) - When \(x=0\), \(y=(0)^3 - 1 = -1\) - When \(x=1\), \(y=(1)^3 - 1 = 0\) - When \(x=2\), \(y=(2)^3 - 1 = 7\) Now, we can plot these points on the coordinate plane and connect them with a smooth curve. The graph of the function \(f(x) = x^3 - 1\) is a cubic function that starts at the point \((-\infty, -\infty)\), passes through the points listed above, and extends to \((\infty, \infty)\). As \(x\) increases, the function increases even more rapidly.

Key concepts

Function Graphing

Graphing functions helps us visualize how they behave. Let's dive into graphing our specific function: \(y = x^3 - 1\). For a cubic function like this, the graph will have a specific shape called a cubic curve.
When we graph \(y = x^3 - 1\), we're looking at how the cube of a number shifts down by one. To graph it:

• Choose a range of x-values, both positive and negative.
• Calculate the corresponding y-values using the function formula \(y = x^3 - 1\).
• Plot these points on a coordinate plane.
• Connect the points with a smooth curve, following the pattern of a cubic function.

Cubic graphs usually have an S-shape, starting from the bottom on the left, curving up in the middle, and rising steeply on the right. In our specific function, this will be moved one unit down due to the "minus 1" in the formula. This shift is visible in how the graph positions compared to the basic cubic \(y = x^3\).

Domain of a Function

The domain of a function consists of all possible input values for which the function is defined. In simpler terms, it's all the x-values you can plug into the function without any problems occurring. For the cubic function \(y = x^3 - 1\), there are no restrictions.
This means that any real number can be used as an x-value without causing undefined behavior, such as division by zero or square roots of negative numbers, which aren't concerns in cubic functions. Therefore, the domain of our function is all real numbers.
In interval notation, the domain is written as \((-fty, fty)\), indicating that the function can accept any number from negative infinity to positive infinity. You can plug in any number, and the function will produce a corresponding output.

Function Transformation

Function transformation involves changing the graph of a basic function in specific ways. In our problem, the basic function is a cubic function \(y = x^3\), which we then transform by subtracting 1 from the entire function.

• The subtraction of 1 from \(x^3\) (i.e., \(x^3 - 1\)) translates the graph downwards by 1 unit.
• This form of vertical translation means that every point on the basic cubic graph shifts down by the same amount.

Another common form of transformation is horizontal shifting, which involves adding or subtracting a number inside the function's variable term, but that doesn't apply here. Instead, look at how the simpler polynomial \(x^3\) shifts from the points we calculated, each consistently lowered by 1 in their y-values. Understanding these transformations helps anticipate and sketch graphs quickly, based on simple modifications to standard functions.