Problem 56 Fill in each table so that each ... [FREE SOLUTION]

Chapter 11: Problem 56

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

Short Answer

Expert verified

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

Step by step solution

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.

Substitute $ x = 0 $

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

Substitute $ x = 1 $

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

Substitute $ x = -1 $

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

Substitute $ x = 2 $

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

Substitute $ x = -2 $

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

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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91影视

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

Short Answer

Step by step solution

Understand the Function

Substitute \( x = 0 \)

Substitute \( x = 1 \)

Substitute \( x = -1 \)

Substitute \( x = 2 \)

Substitute \( x = -2 \)

Key Concepts

Ordered Pairs

Function Evaluation

Substitution Method

Table of Values

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