Each table shows solution points for one of the following...

Each table shows solution points for one of the following equations.
(i) $y=k x+5$
(ii) $y=x^{2}+k$
(iii) $y=k x^{3 / 2}$
(iv) $x y=k$
Match each equation with the correct table and find $k$. Explain your reasoning.

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Key Concepts

Substitution and Parameter Identification

This concept involves plugging in given values of variables into an equation to solve for unknown constants or parameters. It is a fundamental technique in algebra that ensures the equation exactly fits the provided data or conditions.

Functional Forms and Graphical Behavior

Understanding the characteristics of different types of functions—such as linear functions with constant slopes, quadratic functions with parabolic curves, power functions with specific rate changes, and equations representing constant products—is crucial. This knowledge allows one to predict how the output changes as input values vary and to recognize the general shape or pattern of the graph.

Matching Equations to Data Patterns

This involves comparing the theoretical behavior of various function types with the observed behavior in a set of data points. By analyzing the patterns in the data, such as constant increments, curvature, or reciprocal relationships, one can determine which equation form correctly models the data, and consequently identify the correct parameter values.

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