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A curve with 2 turning points is plotted on a grid and is labeled q left parenthesis x right parenthesis. The plot starts in quadrant 3, rises to a turning point in quadrant 2, drops to a turning point in quadrant 3, and rises through quadrants 4 and 1.

Find the missing values in the factored form of q(x)
, given its graph.

(1 point)
q(x)=
(x+4)
(x+
)(x−
​​​​)​​

1 answer

Based on the description of the curve \( q(x) \), let's analyze it to find the missing values:

  1. The curve has two turning points, and it starts in quadrant 3, indicating that it begins below the x-axis and then moves upward to reach the first turning point in quadrant 2 (above the x-axis). This suggests that the function has at least one root in quadrant 3 (to the left of the turning point) and potentially another right around the turning point.

  2. After reaching its first turning point, the curve drops back down to another turning point in quadrant 3, indicating that there is likely a root to the right of the first turning point.

  3. Finally, the curve rises through quadrants 4 and 1, indicating that it crosses the x-axis again in quadrant 4 (having a root also there).

From this information, we can make some deductions about the factored form of \( q(x) \):

  • We already know there is a root at \( x = -4 \) (since the function is moving from negative to positive in quadrant 2).
  • There should also be another negative root (as the curve drops again after the first turning point in quadrant 3), which we can denote as \( x = r_1 \).
  • There must be another root towards the right in quadrant 4, which we can denote as \( x = r_2 \).

To fill in the missing values, it appears we are missing two values in the factored form.

The factored form can be represented as: \[ q(x) = (x + 4)(x + r_1)(x - r_2) \]

Given the context of the question, if the turning points are known or can be approximated, we can give explicit values for \( r_1 \) and \( r_2 \). However, without specific numerical coordinates or an identifiable pattern in the turning points from the graph described, we cannot definitively determine \( r_1 \) and \( r_2 \).

For example, if we assume \( r_1 = -2 \) (closer to zero than -4) and \( r_2 = 1 \) (a simple positive value), we can express it as: \[ q(x) = (x + 4)(x + 2)(x - 1) \]

However, please adjust the values of \( r_1 \) and \( r_2 \) based on your observations of the graph for accurate representation.

So to finalize: \[ q(x) = (x + 4)(x + 2)(x - 1) \] (Assuming \( r_1 = -2 \) and \( r_2 = 1 \))