Transformations of Functions — Quiz

Question 3 • Mathematics III

Which of the following is the graph of y is equal to negative open paren x minus 2 close paren cubed minus 5 ?

Answer

Image with description On a coordinate plane, the scale on the x axis range from negative 5 to 5 in increments of 1, and the scale on the y axis range from negative 10 to 0 in increments of 1. The curve lies in the 3rd quadrant and passes through points (negative 3 point 5 comma negative 3) and (negative 1 comma negative 6). It has the point of inflection at (negative 2 comma negative 5).

Image with description On a coordinate plane, the scale on the x axis ranges from negative 5 to 5 in increments of 1, and the scale on the y axis ranges from negative 10 to 0 in increments of 1. The curve lies in the 4th quadrant and passes through points (1 comma negative 3) and (4 comma negative 9). It has the point of inflection at (2 comma negative 5).

Image with description On a coordinate plane, the scale on the x axis ranges from negative 5 to 5 in increments of 1, and the scale on the y axis ranges from negative 10 to 0 in increments of 1. The curve lies in the 4th quadrant and passes through points (3 point 5 comma negative 2) and (1 comma negative 7). It has the point of inflection at (2 comma negative 5).

Image with description On a coordinate plane, the scale on the x axis ranges from negative 5 to 5 in increments of 1, and the scale on the y axis ranges from negative 10 to 0 in increments of 1. The curve lies in the 3rd quadrant and passes through points (negative 1 comma negative 3) and (negative 4 comma negative 10). It has the point of inflection at (negative 2 comma negative 5).

Explanation

The graph of y = −(x − 2)³ − 5 is the basic cubic shifted right 2 and down 5, then reflected so it decreases from left to right. Its point of inflection is therefore (2, −5); for example, the function gives y = −4 at x = 1 and y = −6 at x = 3. Choice B has the inflection point at (2, −5) and shows a decreasing curve, matching these features.

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