EXAMPLE 4 Find the minimum or maximum value Tell whether the function y = 3x 2 – 18x + 20 has a minimum value or a maximum value. Then find the minimum.

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EXAMPLE 4 Find the minimum or maximum value Tell whether the function y = 3x 2 – 18x + 20 has a minimum value or a maximum value. Then find the minimum or maximum value. SOLUTION Because a > 0, the function has a minimum value. To find it, calculate the coordinates of the vertex. x = – b2a b2a = – (– 18) 2a = 3 y = 3(3) 2 – 18(3) + 20 = –7 ANSWER The minimum value is y = –7. You can check the answer on a graphing calculator.

EXAMPLE 5 Solve a multi-step problem Go - Carts A go-cart track has about 380 racers per week and charges each racer $35 to race. The owner estimates that there will be 20 more racers per week for every $1 reduction in the price per racer. How can the owner of the go-cart track maximize weekly revenue ?

EXAMPLE 5 Solve a multi-step problem SOLUTION STEP 1 Define the variables. Let x represent the price reduction and R(x) represent the weekly revenue. STEP 2 Write a verbal model. Then write and simplify a quadratic function. R(x) = 13, x – 380x – 20x 2 R(x) = – 20x x + 13,300

EXAMPLE 5 Solve a multi-step problem STEP 3 Find the coordinates (x, R(x)) of the vertex. x = – b2a b2a = – 320 2(– 20) = 8 Find x - coordinate. R(8) = – 20(8) (8) + 13,300 = 14,580 Evaluate R(8). ANSWER The vertex is (8, 14,580), which means the owner should reduce the price per racer by $8 to increase the weekly revenue to $14,580.

GUIDED PRACTICE for Examples 4 and 5 7. Find the minimum value of y = 4x x – 3. SOLUTION Because a > 0, the function has a minimum value. To find it, calculate the coordinates of the vertex. x = – b2a b2a = – 16 2a = – 2 y = 4(– 2) (– 2) – 3 = – 19 ANSWER The minimum value is y = – 19. You can check the answer on a graphing calculator.

GUIDED PRACTICE for Examples 4 and 5 8. What If ? In Example 5, suppose each $1 reduction in the price per racer brings in 40 more racers per week. How can weekly revenue be maximized? STEP 1 Define the variables. Let x represent the price reduction and R(x) represent the weekly revenue. SOLUTION

GUIDED PRACTICE for Examples 4 and 5 STEP 2 Write a verbal model. Then write and simplify a quadratic function. R(x) = – 20x x + 13,300

GUIDED PRACTICE for Examples 4 and 5 STEP 3 Find the coordinates (x, R(x)) of the vertex. x = – b2a b2a = – 1020x 2(– 40) = 12.5 Find x - coordinate. R(12.75) = – 40(12.75) (12.75) + 13,300 = Evaluate R(12.75). ANSWER The vertex is (12.75, 19,802.5), which means the owner should reduce the price per racer by $12.75 to increase the weekly revenue to $19,