Which equation has the following pairs: (-7, 8), (-5, -8), (-3, 8)?

Steps to Solve

1. List Given Points The points from the table are:
   $(-7, 8)$, $(-5, -8)$, and $(-3, 8)$.

2. Determine the Relationship To find a possible equation, let's look for patterns in the $y$ values relative to the $x$ values. We can start by comparing the pairs:

   • For $x = -7$, $y = 8$
   • For $x = -5$, $y = -8$
   • For $x = -3$, $y = 8$

3. Examine Changes in $y$ Identify how $y$ changes with $x$:

   • From $-7$ to $-5$, $y$ changes from $8$ to $-8$, a decrease of $16$.
   • From $-5$ to $-3$, $y$ changes from $-8$ to $8$, an increase of $16$.

4. Analyze Possible Equation Types Since there is significant fluctuation, consider that the relationship might be a piecewise function or an oscillating function.

5. Test Possible Functions One likely candidate could be a quadratic equation or piecewise functions since $y$ is the same again at $x = -3$.
   Trying the quadratic form $y = ax^2 + bx + c$ is one way to explore this, but given the pairs, we could also try simpler linear equations with the points.

6. Test Example Linear Equations

   • For line segments between points, analyze them:
     The slope from $(-7, 8)$ to $(-5, -8)$ is calculated as: $$ m = \frac{-8 - 8}{-5 + 7} = \frac{-16}{2} = -8.$$
   • This could help define segments of a piecewise function.

7. Construct Final Equation Based on coordinates and changes of sign in $y$, you can hypothesize relationships that reflect these changes.