a What is the highest point on a parabola that opens downward?
B
b What is the highest point on a parabola that opens downward?
C
c Since x and y represent units of measure, they cannot be negative.
A
aMoisture Content: about 14.1 %
Popping Volume: about 55.5cm^3/g
B
bMoisture Content: about 13.6 %
Popping Volume: about 44.1cm^3/g
C
cHot-Air Popping Domain: 5.52 ≤ x ≤ 22.6 Hot-Air Popping Range: 0≤ y ≤ 55.5
Hot-Oil Popping Domain: 5.35 ≤ x ≤ 21.8 Hot-Oil Popping Range:0≤ y ≤ 44.1
Explanation: See solution.
Practice makes perfect
a We are given two quadratic functions that model the popping volume of popcorn.
Hot-air popping
Hot-oil popping
f(x) = -0.761(x-5.52)(x-22.6)
g(x) = -0.652(x-5.35)(x-21.8)
The moisture content that maximizes the popping volume corresponds to the x-coordinate of the vertex, and the maximum volume corresponds to the y-coordinate. Let's determine the vertex of y= f(x).
Function
Vertex
f(x)=a(x-p)(x-q)
x=p+q/2
y=f(p+q/2)
f(x) = -0.761(x-5.52)(x-22.6)
x=5.52+22.6/2=14.1
f(14.1)=55.5
The moisture content that maximizes the popping volume is about 14.1 %, and the maximum volume is about 55.5cm^3/g.
b As in Part A, the moisture content that maximizes the popping volume corresponds to the x-coordinate of the vertex, and the maximum volume corresponds to the y-coordinate. Let's determine the vertex of y= g(x).
Function
Vertex
g(x)=a(x-p)(x-q)
x=p+q/2
y=f(p+q/2)
g(x) = -0.652(x-5.35)(x-21.8)
x=5.35+21.8/2=13.6
f(13.6)=44.1
Consequently, the moisture content that maximizes the popping volume is about 13.6 %, and the maximum volume is about 44.1cm^3/g.
c In this part we must graph both function using the graphing calculator.
Let's enter the equations into a graphing calculator by pushing Y= and typing the right-hand side of the equations in the two first rows.
By pushing GRAPH, the calculator will draw the equations.
For the functions to be visible on the screen, re-size the standard window by pushing the WINDOW button. Change the settings to a more appropriate size and then push GRAPH.
Since the variables x and y represent units of measure they have to be greater than zero. Using this and the information obtained in the first two parts, we can state the following conclusions.
The domain for the hot-air popping ( f(x)) is 5.52 ≤ x ≤ 22.6, and its range is 0≤ y ≤ 55.5. It means that the moisture content for the kernels can be between 5.52 % and 22.6 % and the popping volume can be between 0 to 55.5cm^3/g.
The domain for the hot-oil popping ( g(x)) is 5.35 ≤ x ≤ 21.8, and its range is 0≤ y ≤ 44.1. It means that the moisture content for the kernels can be between 5.35 % and 21.8 % and the popping volume can be between 0 to 44.1cm^3/g.
