a We are given the model path of the flight of a hummingbird.
f(x)=-1/5x(x-4)^2(x-7), 0≤ x≤ 7
The hummingbird feeds at ground level, so when f(x)=0.
To solve this equation, we can use that a product is 0 when one of the terms is 0.
There are three terms that depend on x in the expression of f(x).
Term
Equation
Solution
x
x=0
x=0
(x-4)^2
(x-4)^2=0
x=4
x-7
x-7=0
x=7
We can illustrate the path and the solutions.
There are three solutions to the equation f(x)=0.
x=0, x=4, x=7
The hummingbird feeds at distances 0, 4, and 7 meters.
b We can use graph transformations to model the path of the flight of the second hummingbird.
Information
Graph Transformation
Model
Path of the first hummingbird.
y=f(x), 0≤ x≤ 7
The second hummingbird feeds 2 meters farther away.
Translation to the right by 2.
y=f(x- 2), 2≤ x≤ 7+ 2
The second hummingbird flies twice as high.
Vertical stretch by a factor of 2.
y= 2f(x-2), 2≤ x≤ 9
To find the equation for the model of the flight of the second hummingbird, we replace x with x-2 in f(x) and multiply the result by 2.
f(x)=-1/5x(x-4)^2(x-7)
⇓
2f( x-2)= 2(-1/5( x-2)( x-2-4)^2( x-2-7))
Let's simplify this expression.
We can illustrate the path of the flight of the second hummingbird (the red curve) compared to the path of the flight of the first hummingbird (the blue curve).
