Let's analyze each statement step-by-step to determine which one is true by comparing the table and graph.

1. Statement A: "Both the function in the table and the function in the graph open up."

   • The function in the graph is a parabola that opens upwards, as the arms of the curve extend upward from the vertex.
   • The values in the table, however, indicate a function that increases to a maximum at $y = 5$ (when $x = 0$) and then decreases symmetrically on both sides.
   • This pattern matches an upward-opening parabola, indicating that both the table and graph represent functions that open up.

   Conclusion: This statement appears to be true.

2. Statement B: "Both the function in the table and the function in the graph have a positive $y$-intercept."

   • The $y$-intercept is the value of $y$ when $x = 0$.
   • In the table, when $x = 0$, $y = 5$, which is positive.
   • In the graph, the curve intersects the $y$-axis at $y = 1$, not 5, but it is also positive.

   Conclusion: This statement is true.

3. Statement C: "Both the function in the table and the function in the graph have a constant rate of change."

   • A constant rate of change implies a linear function, where the difference in $y$-values for equal intervals of $x$-values is always the same.
   • Both the table and the graph show non-linear patterns. The values increase and then decrease, suggesting a quadratic or parabolic function.

   Conclusion: This statement is false.

4. Statement D: "Both the function in the table and the function in the graph have a positive $x$-intercept."

   • The $x$-intercept is the value of $x$ where $y = 0$.
   • In the table, none of the $y$-values are zero, so there is no $x$-intercept.
   • In the graph, the curve does not intersect the $x$-axis either, indicating no $x$-intercept.

   Conclusion: This statement is false.

Final Answer:

The correct answer is:

• A. Both the function in the table and the function in the graph open up.

Would you like further clarification or details on any part of this explanation?

Related Questions:

1. How can we determine the rate of change from a table or a graph?
2. Why do parabolas open up or down, and what determines this?
3. How can we find the $y$-intercept from an equation, table, or graph?
4. How is a function’s symmetry shown in a table compared to a graph?
5. What distinguishes linear functions from quadratic functions?

Tip:

To confirm if a function has a constant rate of change, check if the differences between consecutive $y$-values are consistent across equal intervals in $x$-values. This is only true for linear functions.