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Maximum, Minimum, and Analyzing the Vertex

9.11.4 Maximum, Minimum, and Analyzing the Vertex

9.11.4 • Maximum, Minimum, and Analyzing the Vertex

Cool Down

1.

Without graphing, find the vertex (of a graph) of the graph of a quadratic function defined by f(x)=−x2−14x−60f(x)=−x2−14x−60. Show your reasoning.

Compare your answer:

The vertex is at (−7,−11)(−7,−11).

f(x)=−(x2+14x+60)f(x)=−(x2+14x+60)

f(x)=−(x2+14x+49+11)f(x)=−(x2+14x+49+11)

f(x)=−((x+7)2+11)f(x)=−((x+7)2+11)

f(x)=−(x+7)2−11f(x)=−(x+7)2−11

2.

Does the yy-coordinate of the vertex correspond to a maximum or a minimum value of the function? Explain how you know.

Compare your answer:

A maximum value. The squared term has a coefficient of -1. The squared term will be 0 when xx is -7. For all other values of xx, the squared term will be subtracted from -11, resulting in outputs that are less than -11.

Note: If the vertex isn't asked for and the "value of the maximum or minimum" is asked for, the maximum or minimum value occurs at the yy-value of the vertex because that is the value of the coordinate that represents the vertical height of the point.