Applications of Vertex Form
1. You are a pyrotechnician. You have just launched a firework which can be modeled by the equation , where is the height of the firework in meters, and is the time, in seconds.
a) What is the vertex in this situation, and what does it represent?
b) What was the initial height of the firework?
c) Oh no! Your second firework, with the equation was a dud! [A dud is an explosive that fails to detonate] What time did it end up hitting the ground?
2. A fireworks display is being set up to be launched off of a hill so they will be displayed over the lake. The path of the “Big Boom” firework is modeled by the quadratic relation and the path of the “Sparkle Sensation” firework is modeled by the quadratic relation, . In both relations, represents the height of the firework above the water, in meters, after seconds.
a) How high above water level is the hill where the “Big Boom” firework was launched?
b) The “Sparkle Sensation” firework is designed to go off at a lower height than the “Big Boom” firework. What is the difference in the heights of the two fireworks?
c) The organizers want both fireworks to go off at the same time. Which firework should be set off first? How long after the first one is launched, should the second one be launched?
3. A skateboarding video game uses a quadratic relation to model the path of a skateboarder when they leave the ramp. One of the players left following a path of modeled by the quadratic relation where is the vertical height above the ground and is the horizontal distance from the end of the ramp. All distances are measured in meters.
a) How high is the ramp above the ground?
b) How far does the skateboarder land from the end of the ramp?
c) If the skateboarder was trying to jump over an obstacle that was 3.6 m tall, would they make it.
4. Danny wants to build a bridge for his ants to climb on. Since ants are tiny, Danny’s bridge is not very large, and can be modeled by the equation , where is the height above the bottom of the cage and is the horizontal distance. Both are in centimeters.
a) What is the total length of Danny’s bridge?
b) How tall is Danny’s bridge?
c) If Danny reduced the height of his bridge to 5 cm, what would the new equation be?
5. The vertical distance d (in cm) of the end of a robot arm above a conveyor belt is given by where is the time in seconds.
a) What height does the robot arm start at, and when does it return again to its starting height?
b) What is its minimum height?
Homework: p. 271 # 9, 10, 14, 16, 19*, p. 294 # 12*