To determine the value of \( k \) that would create the function \( y = f(kx) \) based on the function \( f(x) = x + 2 \), we need to analyze how the transformation affects the function.
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The function \( f(x) = x + 2 \) has a y-intercept of 2.
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When we transform this function to \( f(kx) \), we are changing the input of the function. To find how \( f(kx) \) behaves, we can substitute \( kx \) into the original function:
\[ f(kx) = kx + 2 \]
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From the points given on the graph, we can see the original points from \( f(x) \) at specific \( x \) values:
- For \( x = 0 \), \( y = 2 \)
- For \( x = 1 \), \( y = 3 \) (which is not in the given points)
- For \( x = 2 \), \( y = 4 \) (also not in the given points)
- For \( x = -1 \), \( y = 1 \) (again not given)
- For \( x = -2 \), \( y = 0 \) (not included)
However, we are focused particularly on how scaling \( x \) affects these output values.
- The function \( f(kx) = kx + 2 \) scales the input along the x-axis. We require that the transformation matches the behavior of the plotted points.
To see how to transform \( y = x + 2 \) so we end up with values similar to the points \((2,8)\), \((1,5)\), \((0,2)\), \((-1,-1)\), and \((-2,-4)\).
- For example, we see that at \( x = 2 \), \( y \) should equal 8, i.e.: \[ k(2) + 2 = 8 \] \[ 2k + 2 = 8 \] \[ 2k = 6 \] \[ k = 3 \]
- You could check from any other point too, e.g. at \( x = 1 \): \[ k(1) + 2 = 5 \] \[ k + 2 = 5 \] \[ k = 3 \]
All points consistently give the same value.
Thus, the value of \( k \) that transforms \( f(x) = x + 2 \) into \( f(kx) \) such that it matches the plotted points is:
\[ \boxed{3} \]
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