To find the appropriate value of \( k \) that transforms the function \( f(x) = x + 2 \) into \( y = f(kx) \), we need to understand how the transformation affects the graph.
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Identify the Original Function: The function given is \( f(x) = x + 2 \).
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Transformation: The function \( y = f(kx) \) can be rewritten as: \[ y = kx + 2 \] In this transformation, the slope of the line (which corresponds to the value of \( k \)) determines how steep the line becomes.
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Analyzing the Points: You provided the following points:
- (-2, -4)
- (-1, -1)
- (0, 2)
- (1, 5)
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Finding the Slope from Points: To find the multiplier \( k \), we'll calculate the slope of the transformed function using selected points \( (x_1, y_1) \) and \( (x_2, y_2) \):
- From the points \( (0, 2) \) and \( (1, 5) \), the slope \( m \) can be calculated as: \[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{5 - 2}{1 - 0} = \frac{3}{1} = 3 \]
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Setting the Slope: In our transformed function \( y = kx + 2 \), the slope \( k \) would need to equal the calculated slope: \[ k = 3 \]
Therefore, the value of \( k \) that transforms \( f(x) = x + 2 \) into a function whose graph includes the provided points is \( k = 3 \).