To analyze the linear function represented in the graph, we need to determine the initial value (which typically corresponds to the y-intercept) and the rate of change (which corresponds to the slope of the line).
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Finding the Initial Value (y-intercept): The initial value (y-intercept) is the value of y when x = 0. According to the points given, when \( x = 0 \), \( y = 3 \). So the initial value is: \[ \text{Initial Value} = 3 \]
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Finding the Rate of Change (slope): The slope (rate of change) of a linear function can be calculated using the formula: \[ \text{slope} = \frac{y_2 - y_1}{x_2 - x_1} \] We can use any two points on the line to calculate the slope. Let's use the points \( (-3, 4) \) and \( (0, 3) \).
\[ y_2 = 3, \quad y_1 = 4, \quad x_2 = 0, \quad x_1 = -3 \] \[ \text{slope} = \frac{3 - 4}{0 - (-3)} = \frac{-1}{3} = -\frac{1}{3} \]
Thus, we find:
- The initial value (y-intercept) is \( 3 \).
- The rate of change (slope) is \( -\frac{1}{3} \) or approximately \( -0.333 \) when rounded to three decimal places.
The final answers are:
- The initial value is 3, and the rate of change is -0.333.