Problem 56 - Algebra A Combined Function

Question

Fill in each table so that each ordered pair is a solution of the given function. $$ \begin{aligned} &f(x)=2 x^{2}\\\ &\begin{array}{|r|r|} \hline \boldsymbol{x} & \boldsymbol{y} \\ \hline 0 & \\ \hline 1 & \\ \hline-1 & \\ \hline 2 & \\ \hline-2 & \\ \hline \end{array} \end{aligned} $$

Step-by-Step Solution

Answer

The completed table is: \((0, 0), (1, 2), (-1, 2), (2, 8), (-2, 8)\).

Step 1: Understand the Function

The function given is \( f(x) = 2x^2 \). This is a quadratic function where we will substitute various values of \( x \) to find the corresponding \( y \) values.

Step 2: Substitute \( x = 0 \)

To find \( y \), substitute \( x = 0 \) into the function: \( f(0) = 2(0)^2 = 0 \). Thus, the ordered pair is \((0, 0)\).

Step 3: Substitute \( x = 1 \)

Substitute \( x = 1 \) into the function: \( f(1) = 2(1)^2 = 2 \). Thus, the ordered pair is \((1, 2)\).

Step 4: Substitute \( x = -1 \)

Substitute \( x = -1 \) into the function: \( f(-1) = 2(-1)^2 = 2 \). Thus, the ordered pair is \((-1, 2)\).

Step 5: Substitute \( x = 2 \)

Substitute \( x = 2 \) into the function: \( f(2) = 2(2)^2 = 8 \). Thus, the ordered pair is \((2, 8)\).

Step 6: Substitute \( x = -2 \)

Substitute \( x = -2 \) into the function: \( f(-2) = 2(-2)^2 = 8 \). Thus, the ordered pair is \((-2, 8)\).

Key Concepts

Ordered Pairs

Ordered pairs are simply pairs of numbers that have a specific order, typically written as \(x, y\). The first number in an ordered pair is known as the x-coordinate, and the second number is the y-coordinate.
These pairs are crucial in the context of functions, as they represent the input-output relationship within a function.In the exercise provided, each x-value has a corresponding y-value, creating several ordered pairs:

• For \(x = 0\), the ordered pair is \( (0, 0) \).
• For \(x = 1\), the ordered pair is \( (1, 2) \).
• For \(x = -1\), the ordered pair is \( (-1, 2) \).
• For \(x = 2\), the ordered pair is \( (2, 8) \).
• For \(x = -2\), the ordered pair is \( (-2, 8) \).

Each ordered pair arises from substituting a specific x-value into the quadratic function to find the corresponding y-value.

Function Evaluation

Function evaluation involves determining the output of a function for a specific input. In simpler terms, when we evaluate a function, we are essentially asking, "What is the value of the function when x is some specific number?"
For the quadratic function \(f(x) = 2x^2\), we evaluate the function by substituting the x-values provided in the problem.For instance, when evaluating for \(x = 1\):

• Plug \(1\) into the function: \(f(1) = 2(1)^2\).
• This simplifies to \(f(1) = 2 \), giving the y-value.

Through function evaluation, we see how the quadratic expression \(2x^2\) transforms based on different inputs to yield specific ordered pairs.

Substitution Method

The substitution method is a fundamental concept for solving equations and evaluating functions. This technique involves replacing a variable with a specified value to find an answer.
In the context of quadratic functions, it's especially useful for determining the y-value given an x-value.For the exercise, we used the substitution method step-by-step:

• Start with the function \(f(x) = 2x^2\).
• Substitute each given x-value into the function, one at a time.
• Calculate the expression to find the y-value.

For example, when \(x = -1\):

• Substitute to get \(f(-1) = 2(-1)^2 = 2\).
• This calculation indicates that y is \(2\) when x is \(-1\).

Using substitution, we systematically generate the ordered pairs needed to evaluate the function under specific conditions.

Table of Values

A table of values is a strategic way to organize the function evaluation results systematically. It helps compare different x-values and their corresponding y-values at a glance.
Creating a table of values provides insight into the function's behavior across different inputs.For our quadratic function \(f(x) = 2x^2\), the table of values consists of the following:

• \(x = 0\), \(y = 0\)
• \(x = 1\), \(y = 2\)
• \(x = -1\), \(y = 2\)
• \(x = 2\), \(y = 8\)
• \(x = -2\), \(y = 8\)

By compiling the results in a table format, we can easily see the patterns, such as symmetry in quadratic functions where, for instance, \(x = 2\) and \(x = -2\) produce the same y-value. This visualization aids in better understanding the characteristics and structure of quadratic functions.