To sketch a graph of a linear function with an initial value of -1 and a slope of 3, follow these steps:
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Identify the Initial Value: The initial value, typically the y-intercept (where the line crosses the y-axis), is -1. This means the point (0, -1) is on the graph.
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Determine the Slope: The slope of the function is given as 3. This means that for every unit increase in the x-direction, the y-value increases by 3 units. Mathematically, this can be expressed as: \[ m = \frac{\Delta y}{\Delta x} = 3 \quad \text{(slope)} \]
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Plot the Initial Point: Start your graph by plotting the point (0, -1), which is your y-intercept.
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Use the Slope to Find Another Point: From (0, -1), move 1 unit to the right (in the positive x-direction) and 3 units up (since the slope is 3). This will take you to the point (1, 2).
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Draw the Line: Draw a straight line through the points (0, -1) and (1, 2). This line will extend infinitely in both directions.
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Qualitative Features: The graph will be a straight line that rises from left to right, given the positive slope. It will continually increase as x increases.
The resulting sketch should clearly show a line going through the points (0, -1) and (1, 2), emphasizing the upward trend that represents the positive slope of 3.