a Define the variables for the situation. The club's income can be found by multiplying the hourly rate by the number of families.
B
b Begin by graphing the function. To do that find its intercepts and vertex.
C
c What does the vertex represent in this content?
D
d What does the vertex represent in this content?
A
a y=- x^2+6x+475
B
bDomain: 0≤ x ≤ 25
Range: 0≤ y ≤ 484
C
c $11
D
d $484
Practice makes perfect
a We will begin by defining the variables for the situation.
& x : Number of times the hourly rate increases
& y : Club's income
The club's income can be found by multiplying the hourly rate by the number of families. Let's organize the given information on a table to write the equation that models the situation.
Verbal Expression
Algebraic Expression
Increasing the hourly rate x times ($)
0.50 x
New hourly rate ($)
9.50+0.50 x
Decreasing the number of families x times
2 x
New number of families
50-2 x
Club's income is $y.
y=(9.50+0.50 x)(50-2 x)
Now we can write the equation in the form of y=ax^2+bx+c to find its characteristics.
b To determine the domain and range of the function, we should first graph the function. A quadratic function can be graphed by finding its intercepts and vertex. Let's first find its y-intercept by substituting x= 0.
We found two x-intercepts, (-19,0) and (25,0). Finally, we will find the coordinates of the vertex. The x-coordinate can be found by using the formula for the axis of symmetry, x=- b2a. In this case, a=-1 and b=6.
The vertex of the equation is the point (3,484). Now we can graph the equation by plotting the intercepts and the vertex. Notice that the coefficient of the quadratic term is negative, a<0. Therefore, the graph will open down.
In our situation the number of times the hourly rate increases and the club's income cannot be negative. Therefore, the graph must be bound by the axes.
Looking at the graph, we can say that the values between 0 and 25 form the domain and the values between 0 and 484 form the range.
Domain:& 0≤ x ≤ 25
Range:& 0≤ y ≤ 484
c In Part A, we have defined the new hourly rate.
New hourly rate: 9.50+0.50 x
Because the graph of the quadratic function opens downward, it has a maximum value at the vertex. In this content, the vertex states that when the hourly rate increases 3 times, the income is maximum. Therefore, we can find the hourly rate that maximize the income, by substituting x=3 in the expression.
