FINDING MINIMUM AND MAXIMUM VALUES OF QUADRATIC FUNCTIONS WORKSHEET

Find the maximum or minimum value of the following function.

Problem 1 :

y = x² - 4x + 1

Solution

Problem 2 :

y = -x² - x + 1

Solution

Problem 3 :

y = 5x² - 3

Solution

Problem 4 :

y = (1/2)x² - x - 4

Solution

Problem 5 :

The cables between the two towers of the Tacoma Narrows Bridge in Washington form a parabola that can be modeled by

y = 0.00016x² − 0.46x + 507

where x and y are measured in feet. What is the height of the cable above the water at its lowest point?

Solution

Problem 6 :

The parabola shows the path of your first golf shot, where x is the horizontal distance (in yards) and y is the corresponding height (in yards). The path of your second shot can be modeled by the function f(x) = −0.02x(x − 80). Which shot travels farther before hitting the ground? Which travels higher?

Solution

Find the maximum or minimum value of the following function.

Problem 1 :

y = –x² + 2x + 3

Solution

Problem 2 :

y = 2x² + 4x – 3

Solution

Problem 3 :

y = –3x² + 4x 

Solution

Problem 4 :

x is the horizontal distance (in feet) and y is the vertical distance (in feet). Find and interpret the coordinates of the vertex.

a) The path of a basketball thrown at an angle of 45° can be modeled by

y = −0.02x² + x + 6

b) The path of a shot put released at an angle of 35° can be modeled by

y = −0.01x² + 0.7x + 6

Solution

Problem 5 :

The engine torque y (in foot-pounds) of one model of car is given by

y = −3.75x² + 23.2x + 38.8

where x is the speed (in thousands of revolutions per minute) of the engine.

a. Find the engine speed that maximizes torque. What is the maximum torque?

b. Explain what happens to the engine torque as the speed of the engine increases.

Solution

Problem 1 :

Determine the equation of a quadratic function that has a minimum at (-2, -3) and passes through (-1, 1).

Solution

Problem 2 :

Determine the equation of a quadratic function that has a maximum at (2, 10) and passes through (1, 8).

Solution

Problem 3 :

Find the minimum of the parabola: y = 2x² + 8x + 9

(a) (-2, 1)     (b) (2, 33)     (c) (2, 17)

(d) (-2, -17)     (e) None of these

Solution

Problem 4 :

Find the maximum of the parabola: y = -3x² + 12x + 1

(a) (6, -5)     (b) (-2, -19)     (c) (2, 13)

(d) (1, 14)     (e) None of these

Solution

Problem 5 :

The graph of y = x² is shown in the standard (x, y) coordinate plane below. For which of the following equations is the graph of the parabola shifted 3 units to the right and 2 units down?

A) y = (x + 3)² - 2                 B) y = (x - 3)² - 2

C) y = (x + 3)² + 2                D) y = (x - 3)² + 2

Solution

Problem 6 :

The height of a bridge is given by the equation y = -3x² + 12x, where y is the height of the bridge (in miles) and x is the number of miles from the base of the bridge.

i. How far from the base of the bridge does the maximum height occur?

ii. What is the maximum height of the bridge?

A) i. 2 miles               ii. 12 miles

B) i. -2 miles              ii. 12 miles

C) i. 3 miles               ii. 9 miles

D) i. 3 miles              ii. 6 miles

Solution