Problem 57 Fill in each table so that each ... [FREE SOLUTION]

Chapter 11: Problem 57

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

Short Answer

Expert verified

The table values are (0, 0), (1, -1), (-1, -1), (2, -4), and (-2, -4).

Step by step solution

Understand the Function

The function given is $ f(x) = -x^2 $. This is a quadratic function where the output $ y $ is the negative of the square of the input $ x $. This means for any value of $ x $, you calculate $ x^2 $ and then take the negative of that result to find $ y $.

Calculate for $ x = 0 $

Substitute $ x = 0 $ into the function: $ f(0) = -(0)^2 = 0 $. So the ordered pair is $ (0, 0) $.

Calculate for $ x = 1 $

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

Calculate for $ x = -1 $

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

Calculate for $ x = 2 $

Substitute $ x = 2 $ into the function: $ f(2) = -(2)^2 = -4 $. So the ordered pair is $ (2, -4) $.

Calculate for $ x = -2 $

Substitute $ x = -2 $ into the function: $ f(-2) = -(-2)^2 = -4 $. So the ordered pair is $ (-2, -4) $.

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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 written in a specific order:

The first number represents the value of the independent variable, often labeled as $ x $.
The second number represents the value of the dependent variable, which we find using a function, often labeled as $ y $. For example, in the ordered pair $ (x, y) $, $ x $ is the input and $ y $ is the output derived from $ x $.

Understanding ordered pairs is crucial when working with functions, as they help establish the relationship between input and output. By determining the output $ y $ for each input $ x $ using the function provided, you can create ordered pairs that represent this relation. Thus, ordered pairs are not only foundations of functions but also essential in graphing points on a coordinate plane.

Substitution Method

The substitution method is a fundamental mathematical technique. It's used to find outputs of a function by substituting values for the input variables. Here's how it works:

First, identify the function you are dealing with. In our case, it is $ f(x) = -x^2 $.
Substitute the given value from the ordered pair's $ x $ position into the function.
Calculate to find the corresponding $ y $ value.

Each calculation provides a new ordered pair, showing the relationship between $ x $ and $ y $. For example, substituting $ x = 0 $ into the function gives $ f(0) = -(0)^2 = 0 $, yielding the ordered pair $ (0, 0) $. This simple substitution process is essential for evaluating functions and understanding their behavior.

Function Evaluation

Function evaluation involves determining the output of a function for a given input. Using the same function, $ f(x) = -x^2 $, each input $ x $ is placed into the function to solve for the output $ y $. Here鈥檚 a step-by-step of how function evaluation works:

Take an input value, say $ x = 1 $.
Insert it into the equation as $ f(1) = -(1)^2 $.
Calculate the result: $ -(1)^2 = -1 $.
The result $ -1 $ is your function output, thus $ y = -1 $.

These calculations are performed for each input value to produce a set of results. More specifically, each evaluation provides a clarity by showing how different input values affect the behavior of the quadratic function.

Negative Quadratic

A negative quadratic function like $ f(x) = -x^2 $ has distinctive features. This type of function is called 'negative' because the coefficient of the $ x^2 $ term is negative. The basic characteristics of negative quadratics include:

The graph of the function is a downward-opening parabola, unlike the upward parabola of a positive quadratic.
This results in a highest point called the 'vertex', rather than a lowest point.
For each positive $ x $ value, the output $ y $ will be negative, which influences the ordered pairs negatively.

Recognizing the properties of negative quadratics is important for understanding their impacts on mathematical modeling and graphical representations. Thus, evaluating a negative quadratic helps in predicting and analyzing its outcomes efficiently.

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91影视

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

Short Answer

Step by step solution

Understand the Function

Calculate for \( x = 0 \)

Calculate for \( x = 1 \)

Calculate for \( x = -1 \)

Calculate for \( x = 2 \)

Calculate for \( x = -2 \)

Key Concepts

Ordered Pairs

Substitution Method

Function Evaluation

Negative Quadratic

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