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2.11 Warm Up Graph the functions & compare to the parent function, y = x². Find the vertex, axis of symmetry, domain & range. 1. y = x² - 2 2. y = 2x² 3. y = 1/3x² + 3

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2.11 Graph y = ax 2 + bx + c

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Vocabulary If a>0, then the y coordinate of the vertex is the minimum value. If a<0, the y coordinate of the vertex is the maximum value. The x coordinate of the vertex is x = b 2a –

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EXAMPLE 1 Find the axis of symmetry and the vertex 12 2(– 2) x = – - b 2a = = 3= 3 Substitute – 2 for a and 12 for b. Then simplify. For the function y = –2x 2 + 12x – 7 a. b. The x- coordinate of the vertex is, or 3. b 2a – y = –2(3) 2 + 12(3) – 7 = 11 Substitute 3 for x. Then simplify. ANSWER The vertex is (3, 11). a = 2 and b = 12.

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EXAMPLE 2 Graph y = ax 2 + bx + c Graph y = 3x 2 – 6x + 2. Determine whether the parabola opens up or down. Because a > 0, the parabola opens up. STEP 1 STEP 2 Find and plot the vertex.

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EXAMPLE 2 To find the y- coordinate, substitute 1 for x in the function and simplify. y = 3(1) 2 – 6(1) + 2 = – 1 So, the vertex is (1, – 1). STEP 3 Plot two points. Choose two x- values one on each side of the vertex. Then find the corresponding y- values. Graph y = ax 2 + b x + c The x- coordinate of the vertex is b 2a2a, or 1. –

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EXAMPLE 2 Standardized Test Practice x 0 2 y 22 STEP 4 Draw a parabola through the plotted points. STEP 5 Find the axis of symmetry.

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GUIDED PRACTICE for Examples 1 and 2 1. Find the axis of symmetry and vertex of the graph of the function y = x 2 – 2x – 3. ANSWER x = 1, (1, –4). 2. Graph the function y = 3x 2 + 12x – 1. Label the vertex and axis of symmetry. ANSWER

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EXAMPLE 3 Find the minimum or maximum value Tell whether the function f ( x ) = – 3x 2 – 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 2a2a – 12 2(– 3) f(–2) = – 3(–2) 2 – 12(–2) + 10 = 22 Substitute –2 for x. Then simplify. The x- coordinate is – b 2a2a The maximum value of the function is f ( – 2 ) = 22.

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Find the minimum value of a function EXAMPLE 4 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 = 0.000097x 2 – 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? SUSPENSION BRIDGES

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Find the minimum value of a function EXAMPLE 4 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 2a2a – 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 The cable is about 196 feet above the water at its lowest point.

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GUIDED PRACTICE for Examples 3 and 4 3. Tell whether the function f ( x ) = 6x 2 + 18x + 13 has a minimum value or a maximum value. Then find the minimum or maximum value. 1 2 Minimum value; ANSWER 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 = 0.00014x 2 – 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

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