https://file.aiquickdraw.com/mathbot/file/17306142768033qonsb5z.jpg

Compare the table and graph.

Which statement is true?

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

B. Both the function in the table and the function in the graph have a positive 𝑦-intercept.

C. Both the function in the table and the function in the graph have a constant rate of

change.

D. Both the function in the table and the function in the graph have a positive 𝑥-intercept.

Explain it step by step.

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.

Compare the table and graph.

Which statement is true?

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

B. Both the function in the table and the function in the graph have a positive 𝑦-intercept.

C. Both the function in the table and the function in the graph have a constant rate of change.

D. Both the function in the table and the function in the graph have a positive 𝑥-intercept.

Explain it step by step.

Analyze the properties of a quadratic function based on a table and a graph. Determine which statements are true about the graph opening direction, y-intercept, rate of change, and x-intercept. Ideal for students in Grades 7-9 studying quadratic equations and functions.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation