Problem 55 - Algebra A Combined Function

Question

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

Step-by-Step Solution

Answer

0, 1, 1, 4, 4

Step 1: Understanding the Function

We are given the function \( f(x) = x^2 \). This function squares the value of \( x \) to find \( y \). Our task is to find \( y \) for each provided \( x \) value in the table.

Step 2: Calculate y when x = 0

Substitute \( x = 0 \) into the function: \( f(0) = 0^2 = 0 \). Therefore, \( y = 0 \) for \( x = 0 \).

Step 3: Calculate y when x = 1

Substitute \( x = 1 \) into the function: \( f(1) = 1^2 = 1 \). Thus, \( y = 1 \) for \( x = 1 \).

Step 4: Calculate y when x = -1

Substitute \( x = -1 \) into the function: \( f(-1) = (-1)^2 = 1 \). Therefore, \( y = 1 \) for \( x = -1 \).

Step 5: Calculate y when x = 2

Substitute \( x = 2 \) into the function: \( f(2) = 2^2 = 4 \). Hence, \( y = 4 \) for \( x = 2 \).

Step 6: Calculate y when x = -2

Substitute \( x = -2 \) into the function: \( f(-2) = (-2)^2 = 4 \). Thus, \( y = 4 \) for \( x = -2 \).

Key Concepts

Ordered Pairs

An ordered pair is a mathematical concept used to represent a pair of values. These values are arranged in a specific order within parentheses, typically written as \((x, y)\). An ordered pair can represent different types of relationships, such as coordinates on a graph, solutions to equations, or inputs and outputs of functions.
In the context of functions, each ordered pair consists of:

• The first component: This is usually the input or independent variable, often denoted as \(x\).
• The second component: This is usually the output or dependent variable, denoted as \(y\). In a function, \(y\) is typically written as \(f(x)\).

By listing ordered pairs, we can clearly illustrate how changing the input \(x\) affects the output \(y\). Each ordered pair can be plotted as a point on a two-dimensional plane, providing a visual representation of the function's behavior.

Function Evaluation

To evaluate a function means to determine the output value \(y\) for a given input value \(x\). This process involves substituting the input \(x\) into the function's equation and then performing the necessary calculations.
For our quadratic function \(f(x) = x^2\):

• Substitute the specific value of \(x\) into the function.
• Perform the arithmetic operation indicated (squaring the \(x\) value).
• Obtain the corresponding \(y\) value, which is \(f(x)\).

Let's see how this works with an example: If we substitute \(x = 2\), the function will be evaluated as follows: \[ f(2) = 2^2 = 4 \] The function evaluation shows that when \(x\) is 2, \(y\) equals 4. By repeating this process for various values of \(x\), we can see how the function behaves for each particular input.

Table of Values

A table of values is a simple yet powerful tool that helps us understand the relationship between two variables in a function. This table lists several \(x\) values along with their corresponding \(y\) values, which are determined through function evaluation.
In this exercise, the given function is \(f(x) = x^2\). To complete the table:

• Start with each \(x\) listed in the table.
• Calculate \(y\) by applying the function \(f(x) = x^2\) to each \(x\).
• Fill in the table with the resulting \(y\) values.

The completed table for the function \(f(x) = x^2\) might look like this in your mind:- For \(x = 0\), \(y = 0^2 = 0\)- For \(x = 1\), \(y = 1^2 = 1\)- For \(x = -1\), \(y = (-1)^2 = 1\)- For \(x = 2\), \(y = 2^2 = 4\)- For \(x = -2\), \(y = (-2)^2 = 4\)These calculated values offer more insight into the nature of the function as they showcase symmetry and specific behavior at each point you considered. Meanwhile, plotting these points on a graph helps visualize the quadratic curve, enhancing comprehension.