Find the minimum or maximum value

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Presentation transcript:

Find the minimum or maximum value EXAMPLE 3 Find the minimum or maximum value Tell whether the function f(x) = – 3x2 – 12x + 10 has a minimum value or a maximum value. Then find the minimum or maximum value. SOLUTION Because a = – 3 and – 3 < 0, the parabola opens down and the function has a maximum value. To find the maximum value, find the vertex. x = – = – = – 2 b 2a – 12 2(– 3) The x-coordinate is – b 2a f(–2) = – 3(–2)2 – 12(–2) + 10 = 22 Substitute –2 for x. Then simplify.

EXAMPLE 3 Find the minimum or maximum value ANSWER The maximum value of the function is f(– 2) = 22.

EXAMPLE 4 Find the minimum value of a function SUSPENSION BRIDGES The suspension cables between the two towers of the Mackinac Bridge in Michigan form a parabola that can be modeled by the graph of y = 0.000097x2 – 0.37x + 549 where x and y are measured in feet. What is the height of the cable above the water at its lowest point?

Find the minimum value of a function EXAMPLE 4 Find the minimum value of a function SOLUTION The lowest point of the cable is at the vertex of the parabola. Find the x-coordinate of the vertex. Use a = 0.000097 and b = – 0.37. x = – = – ≈ 1910 b 2a – 0.37 2(0.000097) Use a calculator. Substitute 1910 for x in the equation to find the y-coordinate of the vertex. y ≈ 0.000097(1910)2 – 0.37(1910) + 549 ≈ 196

EXAMPLE 4 Find the minimum value of a function ANSWER The cable is about 196 feet above the water at its lowest point.

GUIDED PRACTICE for Examples 3 and 4 3. Tell whether the function f(x) = 6x2 + 18x + 13 has a minimum value or a maximum value. Then find the minimum or maximum value.  1 2 Minimum value; ANSWER

GUIDED PRACTICE for Examples 3 and 4 SUSPENSION BRIDGES 4. The cables between the two towers of the Takoma Narrows Bridge form a parabola that can be modeled by the graph of the equation y = 0.00014x2 – 0.4x + 507 where x and y are measured in feet. What is the height of the cable above the water at its lowest point? Round your answer to the nearest foot. ANSWER 221 feet