Cubic and Cube Root Functions — Test

Question 14 • Algebraic Reasoning

Which is the graph of the function f of x is equal to x cubed plus 6 x squared plus 11 x plus 6 ?

Answer

Image with description A coordinate plane has both x and y axes labeled from negative 8 to 8. A curve is plotted that passes through the point negative 4 comma negative 4, intersects the x-axis at the point negative 3.5 comma 0, increases to a relative maximum near the point negative 2.5 comma 2.4, decreases to a relative minimum near the point negative 1.4 comma 1.8, and increases through the point 0 comma 8.

Image with description A coordinate plane has both x and y axes labeled from negative 8 to 8. A curve is plotted that passes through the point negative 4 comma negative 6, intersects the x-axis at the point negative 3 comma 0, increases to a relative maximum near the point negative 2.5 comma 0.4, decreases to a relative minimum near the point negative 1.4 comma negative 0.4, and increases through the point 0 comma 6.

Image with description A coordinate plane has both x and y axes labeled from negative 8 to 8. A curve is plotted that passes through the point negative 6 comma negative 6, intersects the x-axis at the point negative 5 comma 0, increases to a relative maximum near the point negative 4.5 comma 0.4, decreases to a relative minimum near the point negative 3.4 comma negative 0.4, and increases through the point negative 2 comma 6.

Image with description A coordinate plane has both x and y axes labeled from negative 8 to 8. A curve is plotted that passes through the point negative 4 comma negative 8, increases to a relative maximum near the point negative 2.5 comma negative 1.6, decreases to a relative minimum near the point negative 1.4 comma negative 2.4, and increases through the point 0 comma 4.

Explanation

Factor the polynomial: f(x) = (x + 1)(x + 2)(x + 3), so its x-intercepts are −3, −2, and −1. Also, f(0) = 6 and f(−4) = −6. Choice C matches these values and has turning points near x = −2.5 and x = −1.4, as expected for this cubic.

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