Practice Exam

Question 21 • Mathematics III

Which of the following shows the graph of y is equal to negative 2 to the x th power minus 1 ?

Answer

Image with description On a coordinate plane, the curve approaches x axis for negative values of x in the third quadrant. It passes through (negative 1 comma negative 1), (0 comma negative 2), (1 comma negative 4), and decreases rapidly in the fourth quadrant.

Image with description A coordinate plane with a horizontal x-axis and a vertical y-axis, both labeled from negative 10 to 10 in increments of 2. A curve extends horizontally to the left along the horizontal line y = 1 with an arrowhead pointing left. From left to right, the curve passes through 0 comma 0, 1 comma negative 1, 2 comma negative 3, 3 comma negative 7, and curves steeply downward into the fourth quadrant with an arrowhead pointing down.

Image with description A coordinate plane with a horizontal x-axis and a vertical y-axis, both labeled from negative 10 to 10 in increments of 2. A curve extends horizontally to the left just below the x-axis with an arrowhead pointing left. From left to right, the curve passes through 0 comma negative 0.5, 1 comma negative 1, 2 comma negative 2, 3 comma negative 4, and curves steeply downward into the fourth quadrant with an arrowhead pointing down.

Image with description On a coordinate plane, the curve approaches the line y equals negative 1 for negative values of x in the third quadrant. It passes through (0 comma negative 2), (1 comma negative 3), (2 comma negative 5), and decreases rapidly in the fourth quadrant.

Explanation

For y = −2^x − 1, first calculate the values: at x = 0, y = −1 − 1 = −2; at x = 1, y = −2 − 1 = −3; and at x = 2, y = −4 − 1 = −5. As x becomes very negative, 2^x approaches 0, so the graph approaches y = −1. Choice C has these points and this horizontal asymptote.

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