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Chapter 5: Problem 94

Find seven solutions in your table of values for each equation by using integers for \(x\) starting with \(-3\) and ending with 3. $$y=9-x^{2}$$

Short Answer

Expert verified
The seven pairs of \(x\) and \(y\) values, when \(x\) varies from -3 to 3, are (-3, 0), (-2, 5), (-1, 8), (0, 9), (1, 8), (2, 5), (3, 0).

Step by step solution

01

Understand the Problem

We are given the equation \(y=9-x^{2}\) and asked to find the values of \(y\) for \(x\) in the range -3 to 3. The result of this process should be a table with seven pairs of \(x\) and \(y\) values.
02

Substitute \(x = -3\) into the equation

We start with the lowest suggested value for \(x\), which is -3. Substituting \(x = -3\) into the equation gives us \(y = 9 - (-3)^{2} = 9 - 9 = 0\). So, when \(x = -3\), \(y = 0\).
03

Substitute \(x = -2\) into the equation

Next, substitute \(x = -2\) into the equation. This gives us \(y = 9 - (-2)^{2} = 9 - 4 = 5\). So, when \(x = -2\), \(y = 5\).
04

Substitute \(x = -1\) into the equation

Then, substitute \(x = -1\) into the equation. This gives us \(y = 9 - (-1)^{2} = 9 - 1 = 8\). So, when \(x = -1\), \(y = 8\).
05

Substitute \(x = 0\) into the equation

Next, substitute \(x = 0\) into the equation. This gives us \(y = 9 - 0^{2} = 9\). So, when \(x = 0\), \(y = 9\).
06

Substitute \(x = 1\) into the equation

Then, substitute \(x = 1\) into the equation. This gives us \(y = 9 - 1^{2} = 9 - 1 = 8\). So, when \(x = 1\), \(y = 8\).
07

Substitute \(x = 2\) into the equation

Next, substitute \(x = 2\) into the equation. This gives us \(y = 9 - 2^{2} = 9 - 4 = 5\). So, when \(x = 2\), \(y = 5\).
08

Substitute \(x = 3\) into the equation

Finally, substitute \(x = 3\) into the equation. This gives us \(y = 9 - 3^{2} = 9 - 9 = 0\). So, when \(x = 3\), \(y = 0\).

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Key Concepts

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

Table of Values
A table of values is a helpful tool when working with functions, making it easier to see how different input values affect the output. In our example, we're using the equation \( y = 9 - x^2 \). This is a quadratic function due to the \( x^2 \) term. The task involves evaluating this function for each integer \( x \) from \(-3\) to \(3\), and recording each corresponding \( y \) value.
Creating a table of values is a systematic approach to visualize this relation:
  • Start by listing each integer value of \( x \) in the first column.
  • Calculate the corresponding \( y \) value using the given function, and place it in the second column.
  • This results in a table with pairs \((x, y)\).
This organized format not only helps with solving the problem but also provides insights into how the function behaves.
Substitution Method
The substitution method is crucial for solving the problem and evaluating the function for specific values of \( x \). It involves replacing the variable \( x \) with its given value in the equation to find the corresponding \( y \) value. Here's how it's done for our quadratic function:
  • Take the value \( x = -3 \).
  • Replace \( x \) in the equation \( y = 9 - x^2 \) with \(-3\).
  • This gives us \( y = 9 - (-3)^2 = 9 - 9 = 0 \).
Repeat this process for each integer value of \( x \) from \(-3\) to \(3\).
By substituting and calculating each \( y \), you gain a practical understanding of function evaluation. It's like checking off each box in a list to confirm the result for every \( x \) value.
Solving Equations
Though not always discussed explicitly in connection to tables of values, solving equations is a key skill presented here in the form of substitution. Solving the equation \( y = 9 - x^2 \) for specific values of \( x \) does not mean to find zeros of the function, but to determine \( y \) values given \( x \) values. It鈥檚 a different form of problem-solving which:
  • Helps illustrate the nature of quadratic functions.
  • Demonstrates symmetry, as seen here where \( y \) values repeat for symmetrical \( x \) values, e.g., \(-2\) and \(2\) both yield \( y = 5 \).
  • Emphasizes calculations as integral in evaluating expressions.
Understanding this connection gives a broader view of what it means to "solve" beyond the typical zero-finding, highlighting the relationship between inputs and outputs in quadratic functions.

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