Problem 54 - Thinking Mathematically

Question

Evaluate \(f(x)\) for the given values of \(x\). Then use the ordered pairs \((x, f(x))\) from your table to graph the function. $$ \begin{aligned} &f(x)=(x+1)^{3}\\\ &\begin{array}{|r|c|} \hline{}{\underline{\phantom{xx}}}{\boldsymbol{c}} & \boldsymbol{f}(\boldsymbol{x})=(\boldsymbol{x}+\mathbf{1})^{\mathbf{3}} \\ \hline-3 & \\ \hline-2 & \\ \hline-1 & \\ \hline 0 & \\ \hline 1 & \\ \hline \end{array} \end{aligned} $$

Step-by-Step Solution

Answer

The ordered pairs that will be used to graph the function are (-3, -8), (-2, -1), (-1, 0), (0, 1) and (1, 8).

Step 1: Calculate f(x) for each x value

Insert the given x values into the function \(f(x) = (x+1)^3\) and calculate the resulting y values.\n\n When x is -3, \(f(-3) = (-3 + 1)^3 = (-2)^3 = -8\). \n When x is -2, \(f(-2) = (-2 + 1)^3 = (-1)^3 = -1\). \n When x is -1, \(f(-1) = (-1 + 1)^3 = (0)^3 = 0\). \n When x is 0, \(f(0) = (0 + 1)^3 = 1^3 = 1\). \n When x is 1, \(f(1) = (1 + 1)^3 = 2^3 = 8\).

Step 2: Write down the ordered pairs

The ordered pairs are (-3, -8), (-2, -1), (-1, 0), (0, 1), (1, 8)

Step 3: Graph the function

Plot the ordered pairs on a set of coordinate axes. Connect the points with a smooth curve to graph the function f(x).

Key Concepts

Evaluating Functions

Evaluating functions is a fundamental skill in algebra that involves finding the output of a function given an input value. It requires substituting a specific value for the variable in the function's formula and performing the necessary arithmetic operations.

For example, take the function \( f(x) = (x+1)^3 \). To evaluate this function for \( x = -3 \), substitute -3 for every instance of \( x \) in the formula, resulting in \( f(-3) = (-2)^3 = -8 \). Repeat this process for other values of \( x \) to build a list of outputs, or \( y \)-values, which can be paired with their corresponding inputs as ordered pairs to graph the function.

Plotting Ordered Pairs

Plotting ordered pairs entails placing points on a coordinate plane based on their \( x \) and \( y \) values, which are derived from evaluating a function. Each ordered pair is of the form (x, y), where \( x \) is the horizontal position, and \( y \) is the vertical position.

Using the evaluations from our function \( f(x) = (x+1)^3 \), we get pairs such as (-3, -8), (-2, -1), (-1, 0), (0, 1), and (1, 8). Start by locating the \( x \)-value on the horizontal axis, then move vertically to the \( y \)-value, and mark that spot. Once all pairs are plotted, you can connect the dots to visualize the shape of the function.

Cubic Functions

Cubic functions are a type of polynomial function where the highest power of the variable is three, often represented as \( ax^3 + bx^2 + cx + d \), where \( a \), \( b \), \( c \), and \( d \) are constants, and \( a \) is non-zero. These functions produce a characteristic 'S' shaped curve called a cubic curve.

The function \( f(x) = (x+1)^3 \), from our example, is a cubic function. The graph of a cubic function has distinct features such as one or two turning points. It can intersect the x-axis at a maximum of three points and the y-axis at exactly one point, reflecting its odd-powered nature, which allows it to have both positive and negative outputs.

Function Transformations

Function transformations involve altering the basic graph of a function in various ways, including shifting, stretching, compressing, or reflecting. A transformation affects the visual presentation of a graph but does not change its fundamental behavior or properties.

For the function \( f(x) = (x+1)^3 \), adding 1 inside the parentheses shifts the graph of the parent function \( x^3 \) horizontally left by one unit. If we were to apply other transformations, like multiplying the function by a constant, the shape of the graph would stretch vertically (if the constant is greater than one) or compress (if the constant is between zero and one). Understanding these transformations helps in sketching or predicting the shape of the function's graph.