To find the missing values in the table, we need to identify a pattern or relationship between the values of $x$ and $y$. One common approach is to determine if the data follows a simple linear or quadratic relationship, or any other polynomial relationship.

Let's write down the table as given:

\hline x & y \\ \hline 0 & 0 \\ 1 & 0 \\ 2 & - \ (\text{missing}) \\ 3 & 24 \\ 4 & 60 \\ 5 & - \ (\text{missing}) \\ \hline \end{array}$$ We have known values for $$y$$ at $$x = 0, 1, 3, 4$$, so we can use these points to try to determine a pattern. ### Hypothesis 1: Quadratic Relationship A possible approach is to assume that $$y$$ follows a quadratic function: $$y = ax^2 + bx + c$$ Using the given values, we can substitute into the equation to form a system of equations to solve for $$a$$, $$b$$, and $$c$$. - For $$x = 0$$, $$y(0) = 0$$: $$0 = a(0)^2 + b(0) + c \quad \Rightarrow \quad c = 0$$ - For $$x = 1$$, $$y(1) = 0$$: $$0 = a(1)^2 + b(1) + 0 \quad \Rightarrow \quad a + b = 0 \quad \Rightarrow \quad b = -a$$ - For $$x = 3$$, $$y(3) = 24$$: $$24 = a(3)^2 + b(3) + 0 \quad \Rightarrow \quad 24 = 9a + 3b$$ Substituting $$b = -a$$ into this equation: $$24 = 9a + 3(-a) \quad \Rightarrow \quad 24 = 6a \quad \Rightarrow \quad a = 4$$ - For $$x = 4$$, $$y(4) = 60$$: $$60 = a(4)^2 + b(4) + 0 \quad \Rightarrow \quad 60 = 16a + 4b$$ Substituting $$a = 4$$ and $$b = -4$$: $$60 = 16(4) + 4(-4) \quad \Rightarrow \quad 60 = 64 - 16 \quad \Rightarrow \quad 60 = 48 + 12 = 60$$ Thus, the quadratic equation is: $$y = 4x^2 - 4x$$ ### Finding the missing values: - For $$x = 2$$: $$y(2) = 4(2)^2 - 4(2) = 16 - 8 = 8$$ - For $$x = 5$$: $$y(5) = 4(5)^2 - 4(5) = 100 - 20 = 80$$ ### Completed table: $$\begin{array}{|c|c|} \hline x & y \\ \hline 0 & 0 \\ 1 & 0 \\ 2 & 8 \\ 3 & 24 \\ 4 & 60 \\ 5 & 80 \\ \hline \end{array}$$ Would you like more details on this quadratic method or have any questions? Here are 5 related questions: 1. How do you solve systems of equations for quadratic functions? 2. What other types of functions could fit a set of data points? 3. How can you check if a relationship between $$x$$ and $$y$$ is linear? 4. What are higher-order polynomials, and when are they used? 5. How can you approximate missing values if the data does not follow a polynomial pattern? **Tip:** When dealing with data tables, always check if the differences between consecutive values (first and second differences) suggest a linear or quadratic relationship.