To determine the initial value (y-intercept) and the rate of change (slope) of the linear function represented by the line in the image, we can follow these steps:
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Finding the Slope (Rate of Change): The slope \( m \) can be calculated using the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Using the points provided: \[ (x_1, y_1) = (-4, -\frac{10}{3}) \quad \text{and} \quad (x_2, y_2) = (-1, -\frac{4}{3}) \] Substitute the coordinates into the slope formula: \[ m = \frac{-\frac{4}{3} - (-\frac{10}{3})}{-1 - (-4)} = \frac{-\frac{4}{3} + \frac{10}{3}}{-1 + 4} = \frac{\frac{6}{3}}{3} = \frac{2}{3} \]
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Finding the Initial Value (Y-intercept): The y-intercept occurs when \( x = 0 \). We can use the point-slope form of the equation of a line: \[ y - y_1 = m(x - x_1) \] Using one of the points \((-4, -\frac{10}{3})\): \[ y - (-\frac{10}{3}) = \frac{2}{3}(x + 4) \] Simplifying this, we can find the equation of the line. To find the y-intercept (when \( x = 0 \)): \[ y + \frac{10}{3} = \frac{2}{3}(0 + 4) \] \[ y + \frac{10}{3} = \frac{8}{3} \] \[ y = \frac{8}{3} - \frac{10}{3} = -\frac{2}{3} \]
So, the initial value (y-intercept) is \( -\frac{2}{3} \), and the rate of change (slope) is \( \frac{2}{3} \).
Final Answer: The correct option is: The initial value is negative Start Fraction 2 over 3 End Fraction, and the rate of change is Start Fraction 2 over 3 End Fraction.