10 A cubic function was found to have the following factors: (x-1),(x+2),& (x-3) Identify the associated graph.

To identify the graph of a cubic function given its factors, we first need to construct the function from these factors. The factors provided are \( (x-1) \), \( (x+2) \), and \( (x-3) \). Therefore, the cubic function can be expressed as:<br /><br />\[ f(x) = a(x-1)(x+2)(x-3) \]<br /><br />where \( a \) is a constant that affects the vertical stretch or compression and the orientation of the graph.<br /><br />### Key Features of the Graph:<br />1. **Roots/Zeros**: The roots of the function are the values of \( x \) that make each factor zero. From the factors, the roots are:<br /> - \( x = 1 \)<br /> - \( x = -2 \)<br /> - \( x = 3 \)<br /><br />2. **End Behavior**: Since this is a cubic function, it will have the following end behavior:<br /> - As \( x \to -\infty \), \( f(x) \to -\infty \) if \( a > 0 \) or \( f(x) \to \infty \) if \( a < 0 \).<br /> - As \( x \to \infty \), \( f(x) \to \infty \) if \( a > 0 \) or \( f(x) \to -\infty \) if \( a < 0 \).<br /><br />3. **Turning Points**: A cubic function can have up to two turning points. These occur between the roots.<br /><br />4. **Y-intercept**: This occurs when \( x = 0 \). Substitute \( x = 0 \) into the function to find the y-intercept:<br /> \[ f(0) = a(0-1)(0+2)(0-3) = a(-1)(2)(-3) = 6a \]<br /><br />### Sketching the Graph:<br />- Plot the roots on the x-axis at \( x = -2 \), \( x = 1 \), and \( x = 3 \).<br />- Determine the y-intercept at \( y = 6a \).<br />- Consider the end behavior based on the sign of \( a \). If \( a > 0 \), the graph starts in the third quadrant and ends in the first quadrant. If \( a < 0 \), it starts in the second quadrant and ends in the fourth quadrant.<br />- Draw a smooth curve through the roots, ensuring it passes through the y-intercept and follows the determined end behavior.<br /><br />Without additional information about the value of \( a \), we cannot specify the exact orientation or vertical stretch/compression of the graph. However, the general shape will follow the characteristics outlined above.