Practice Test

Question 8 • Algebra II

Which is the graph of the function f of x is equal to x cubed minus x squared minus 17 x minus 15 ?

Answer

Image with description A coordinate plane. The horizontal axis is labeled from negative 5 to 5 with increments of 1, and the vertical axis is labeled from negative 50 to 50 with increments of 10. The curve has a local maximum vertex located at the coordinate points negative 2 comma 16 and a local minimum vertex located at 2 point 5 comma negative 38. The curve crosses the vertical axis at a y-intercept of 0 comma negative 6 and intersects the horizontal axis at three x-intercepts located at negative 3 point 5 comma 0, negative 0 point 5 comma 0, and 4 point 8 comma 0. The left end of the curve extends downward into the third quadrant with an arrow at negative 4 point 5 comma negative 50, while the right end extends upward into the first quadrant with an arrow at 5 comma 10.

Image with description A coordinate plane. The horizontal axis is labeled from negative 5 to 5 with increments of 1, and the vertical axis is labeled from negative 50 to 50 with increments of 10. The curve represents a polynomial function with a local maximum vertex located at the coordinate points negative 2 comma 26 and a local minimum vertex located at 2 point 5 comma negative 27. Moving along the curve, it crosses the vertical axis at a y-intercept of 0 comma 5 and intersects the horizontal axis at three x-intercepts located at negative 4 comma 0, 0 point 5 comma 0, and 4 point 5 comma 0. The left end of the curve extends downward into the third quadrant with an arrow at negative 4 point 8 comma negative 50, while the right end extends upward into the first quadrant with an arrow at 5 comma 25.

Image with description A coordinate plane. The horizontal axis is labeled from negative 5 to 5 with increments of 1, and the vertical axis is labeled from negative 50 to 50 with increments of 10. The curve represents a polynomial function with a local maximum vertex located at the coordinate points negative 2 comma 36 and a local minimum vertex located at 2 comma negative 16. Moving along the curve, it crosses the vertical axis at a y-intercept of 0 comma 15 and intersects the horizontal axis at three x-intercepts located at negative 4 comma 0, 1 comma 0, and 4 comma 0. The left end of the curve extends downward into the third quadrant with an arrow at negative 4 point 5 comma negative 50, while the right end extends upward into the first quadrant with an arrow at 5 comma 35.

Image with description A coordinate plane. The horizontal axis is labeled from negative 5 to 5 with increments of 1, and the vertical axis is labeled from negative 50 to 50 with increments of 10. The curve represents a polynomial function with a local maximum vertex located at the coordinate points negative 2 comma 7 and a local minimum vertex located at 2 comma negative 47. Moving along the curve, it crosses the vertical axis at a y-intercept of 0 comma negative 15 and intersects the horizontal axis at three x-intercepts located at negative 3 point 2 comma 0, minus 1 comma 0, and 4 point 6 comma 0. The left end of the curve extends downward into the third quadrant with an arrow at negative 4 point 5 comma negative 50, while the right end extends upward into the first quadrant with an arrow at 5 comma 0.

Explanation

First check the y-intercept: f(0) = -15, matching choice D. Factoring gives x³ − x² − 17x − 15 = (x + 1)(x + 3)(x − 5), so the x-intercepts are −3, −1, and 5. The derivative, 3x² − 2x − 17, gives turning points near x = −2.1 and x = 2.7, with a local maximum near 7 and a local minimum near −48. Those features match the graph described in choice D.

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