Question

select all the correct locations on the image.

identify which functions have complex roots by selecting the function names on the provided coordinate plane.

Explanation:

Analyze the degree and real roots of each polynomial function

• Red function (bottom-left): This is a parabola (degree 2) opening downwards with its vertex at \((-4, -5)\). It does not cross the x-axis, meaning it has 0 real roots and 2 complex roots.
• Black function (middle): This is a cubic-like curve (degree 3) that crosses the x-axis at three distinct real points: near \(x = -3\), \(x = 1\), and \(x = 3\). Since a degree 3 polynomial has 3 real roots, it has 0 complex roots.
• Blue function (top-right): This is a quartic-like curve (degree 4) with two local minima and one local maximum. It crosses the x-axis at two distinct real points: near \(x = -1\) and \(x = 2\). Since a degree 4 polynomial must have 4 roots in total, and only 2 are real, it has 2 complex roots (corresponding to the local minimum that stays entirely above the x-axis near \(x = 4\)).
• Grey function (bottom-right): This is a cubic-like curve (degree 3) that crosses the x-axis at only one real point near \(x = 5\). Since a degree 3 polynomial has 3 roots in total and only 1 is real, it must have 2 complex roots (corresponding to the local maximum near \(x = 7\) that stays entirely below the x-axis).

Identify the labels corresponding to functions with complex roots

• Red function (bottom-left): Has complex roots. The selection box is located near \((-7, -7)\).
• Blue function (top-right): Has complex roots. The selection box is located near \((7, 8)\).
• Grey function (bottom-right): Has complex roots. The selection box is located near \((8, -6)\).
• Black function (middle): Has only real roots. The selection boxes near \((-4, 8)\), \((7, 4)\), and \((1, -8)\) correspond to functions or parts with only real roots.

Answer:

The functions that have complex roots are:

• The red function (bottom-left, with the selection box near \((-7, -7)\)) (Correct answer)
• The blue function (top-right, with the selection box near \((7, 8)\)) (Correct answer)
• The grey function (bottom-right, with the selection box near \((8, -6)\)) (Correct answer)
• The black function (middle, with selection boxes near \((-4, 8)\), \((7, 4)\), and \((1, -8)\))