The domain and range of a function tells us the possible x- and y-values of the function, respectively.
A
a
The green hose
B
b
y=-4/25(x-5)^2+8
C
c
Domain: 0≤ x ≤ 10
Range: 4≤ y ≤ 8
Practice makes perfect
a
We are given the information about Maura, who waters her outdoor plants. With the information we can identify a few data points that lie on the path of the water shooting from the green hose.
The water reaches a height of 8 feet.
The water lands on the plants 10 feet from where Maura is standing.
The nozzle of the hose and the flowers are 4 feet above the ground.If we let x=0 represent where Maura stands, we get two points at (0, 4) and ( 10, 4). Since the water follows a parabolic path, the greatest height must be reached halfway between these points. This gives us a third point at (5, 8). Now we can plot the parabola.
For the water shooting from the red hose we have been given its equation. Notice that it is written in vertex form. This allows us to identify its vertex ( h, k) immediately.
y = a(x- h)^2 + k
⇕
y=-(x- 3)^2+ 7
The maximum height of the water from the red hose is 7 feet. Therefore, the water from the green hose throws water higher, as it reaches a maximum height of 8 feet.
b
To write an equation that models the path of the water from the green hose, we will consider the vertex form of a quadratic function.
y= a(x- h)^2+ k
In this form, ( h, k) is the vertex and a is the stretch factor of the parabola. From Part A, we know that the vertex is ( 5, 8). With this information, we can start writing the equation.
y= a(x- h)^2+ k
⇕
y= a(x- 5)^2+ 8
Finally, to determine the stretch factor, we have to substitute a point that is not the vertex in the equation and solve for a.
Now we can finalize the equation in vertex form.
y= a(x- 5)^2+ 8
⇕
y= -4/25(x- 5)^2+ 8
c
The domain and range of a function tells us the possible x- and y-values of the function. Let's have a look at the function describing the water from the green hose.
Domain
The function goes from x=0 to x=10 including the endpoints. Therefore, its domain is all values greater than or equal to 0 but less than or equal to 10. We can write this as the following compound inequality.
0≤ x ≤ 10
Range
The function goes from y=4 to y=8 including the endpoints. Therefore, its range is all values greater than or equal to 4 but less than or equal to 8. We can write this as the following compound inequality.
4≤ y ≤ 8
