Problem 57 - 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

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

Step 1: 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 \( y \).

Step 2: Calculate for \( x = 0 \)

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

Step 3: Calculate for \( x = 1 \)

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

Step 4: Calculate for \( x = -1 \)

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

Step 5: Calculate for \( x = 2 \)

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

Step 6: Calculate for \( x = -2 \)

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

Key Concepts

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’s 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.