Cubic and Cube Root Functions — Test

Question 12 • Algebraic Reasoning

Which is the graph of the function f of x is equal to 2 x cubed minus 7 x squared plus 2 x plus 3 ?

Answer

Image with description A coordinate plane shows a cubic polynomial function. The x-axis is labeled from negative 8 to 8 in increments of 2, and the y-axis is labeled from negative 8 to 8 in increments of 2. The graph has a local maximum at the point 2 comma 3 and a local minimum at the point 4 comma negative 5. From the local maximum, the curve moves downwards and to the left, crossing the x-axis near 1 comma 0 and continuing downwards to an arrowhead pointing near 1 comma negative 7. From the local minimum, the curve moves upwards and to the right, crossing the x-axis near 5 comma 0 and continuing upwards to an arrowhead pointing near 5 comma 8.

Image with description A coordinate plane shows a cubic polynomial function. The x-axis is labeled from negative 8 to 8 in increments of 2, and the y-axis is labeled from negative 8 to 8 in increments of 2. The graph has a local maximum at the point negative 4 comma 3 and a local minimum at the point negative 2 comma negative 5. From the local maximum, the curve moves downwards and to the left, crossing the x-axis near negative 5 comma 0 and continuing downwards to an arrowhead pointing near negative 5 comma negative 7. From the local minimum, the curve moves upwards and to the right, crossing the x-axis near negative 1 comma 0 and continuing upwards to an arrowhead pointing near negative 1 comma 8.

Image with description A coordinate plane shows a cubic polynomial function. The x-axis is labeled from negative 8 to 8 in increments of 2, and the y-axis is labeled from negative 8 to 8 in increments of 2. The graph has a local maximum on the y-axis at the point 0 comma negative 1 and a local minimum at the point 2 comma negative 8. From the local maximum, the curve moves downwards and to the left, continuing downwards to an arrowhead pointing near negative 1 comma negative 8. From the local minimum, the curve moves upwards and to the right, crossing the x-axis near 3 comma 0 and continuing upwards to an arrowhead pointing near 4 comma 8.

Image with description A coordinate plane shows a cubic polynomial function. The x-axis is labeled from negative 8 to 8 in increments of 2, and the y-axis is labeled from negative 8 to 8 in increments of 2. The graph has a local maximum on the y-axis at the point 0 comma 3 and a local minimum at the point 2 comma negative 5. From the local maximum, the curve moves downwards and to the left, crossing the x-axis near negative 1 comma 0 and continuing downwards to an arrowhead pointing near negative 1 comma negative 7. From the local minimum, the curve moves upwards and to the right, crossing the x-axis near 3 comma 0 and continuing upwards to an arrowhead pointing near 3 comma 8.

Explanation

The function has y-intercept f(0) = 3, so its graph passes through (0, 3). Its derivative is 6x² − 14x + 2, which gives turning points near x = 0.15 and x = 2.18; the graph has a local maximum near (0, 3) and a local minimum near (2, −5). The function’s zeros are x = −0.5, 1, and 3, and its positive leading coefficient means it falls to the left and rises to the right. These features match choice B.

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