To find the equation of the cubic graph in the desired form \( y = k (x + a)(x - b)(x - c) \), we first identify the roots and the necessary parameters from the information given.
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The line cuts the graph at \( x = -3 \) and \( x = 2 \). This implies that the cubic graph has roots \( -3 \) and \( 2 \). Since it is a cubic equation, we also need a third root. Let’s denote the third root as \( r \).
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The graph cuts the y-axis at \( y = 12 \). This means that when \( x = 0 \), the output of the equation should equal \( 12 \).
Using these points, we can set up our equation. Given the two roots we have, we can start writing the equation:
\[ y = k(x + 3)(x - 2)(x - r) \]
Now we need to find the value of \( k \) and the third root \( r \). To do that, we evaluate the equation at \( x = 0 \):
\[ y(0) = k(0 + 3)(0 - 2)(0 - r) = k(3)(-2)(-r) = 6kr \]
Since we want \( y(0) = 12 \), we set up the equation:
\[ 6kr = 12 \]
This simplifies to:
\[ kr = 2 \]
Now we can handle the value of \( r \). Since we don't have the exact value for the third root but we still need to find \( k \), we can express \( k \) in terms of \( r \):
\[ k = \frac{2}{r} \]
At this point, we can choose any arbitrary value for \( r \). For simplicity, let's assume \( r = 1 \).
Therefore:
\[ k = \frac{2}{1} = 2 \]
With \( r = 1 \), our equation then becomes:
\[ y = 2(x + 3)(x - 2)(x - 1) \]
Thus, our final equation in the required form is:
\[ y = 2(x + 3)(x - 2)(x - 1) \]
If a different value for \( r \) is desired, one just needs to adjust \( k \) accordingly while maintaining the \( kr = 2 \) relationship.