Math Problem Statement
Which options correctly represent the relationship between the x-values and the y-values in
the table? 0-4 2-16 4-28 6-40 10-64
The y-intercept is
. The slope or constant rate of change is
Solution
To determine the relationship between the -values and -values in the table, we calculate the differences and see if they form a consistent relationship (e.g., linear or otherwise).
Table:
x &: 0, 2, 4, 6, 10 \\ y &: 4, 16, 28, 40, 64 \end{aligned}$$ ### Step 1: Verify the relationship type (linear or non-linear) We calculate the differences in $$y$$-values for consecutive $$x$$-values: 1. When $$x$$ increases from 0 to 2: $$y_{\text{change}} = 16 - 4 = 12$$ 2. When $$x$$ increases from 2 to 4: $$y_{\text{change}} = 28 - 16 = 12$$ 3. When $$x$$ increases from 4 to 6: $$y_{\text{change}} = 40 - 28 = 12$$ 4. When $$x$$ increases from 6 to 10: $$y_{\text{change}} = 64 - 40 = 24$$ Since the change in $$y$$-values is not constant (12, 12, 12, 24), the relationship is **non-linear**. --- ### Step 2: Check if the slope or constant rate of change exists For linear functions, the slope ($$m$$) would remain constant: $$m = \frac{\Delta y}{\Delta x}$$ Since $$\Delta y / \Delta x$$ is inconsistent, there is no constant rate of change. --- ### Step 3: Determine the equation type To identify the equation type, examine the $$x$$ and $$y$$ pairs more closely: It appears that $$y$$ is related to $$x^2$$ or a quadratic term, because the $$y$$-values grow faster as $$x$$ increases. Testing $$y = a \cdot x^2 + bx + c$$ might work. ### **Answers** 1. **The y-intercept ($$c$$)**: When $$x = 0$$, $$y = 4$$. Thus, the $$y$$-intercept is: $$c = 4$$ 2. **The slope (rate of change)**: There is no constant rate of change because the relationship is non-linear. Let me know if you want me to derive the exact quadratic equation or explore further! 😊 --- ### Related Questions 1. Could the relationship be quadratic or exponential? 2. How do we calculate the equation of the curve from the table values? 3. What is the practical significance of the $$y$$-intercept in this context? 4. How do higher differences in $$y$$-values indicate the type of relationship? 5. What is the general form of non-linear functions, and how are they analyzed? **Tip**: For non-linear relationships, consider looking at the second differences to identify if a quadratic pattern exists.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Relationships between variables
Non-linear functions
Quadratic equations
Formulas
y = ax^2 + bx + c
Slope formula: m = Δy / Δx
Theorems
Quadratic relationships identification
Suitable Grade Level
Grades 8-10