Question

Suppose the path of a baseball follows the path graphed by the quadratic function ƒ(d) = –0.6d2 + 5.4d + 0.8 where d is the horizontal distance the ball traveled in yards, and ƒ(d) is the height, in yards, of the ball at d horizontal yards. Identify the domain and range that matches this situation.

Answer 1

Domain is all real numbers

Range is

`y\le 12.95`

Explanation:

The given function is

`f(d)=-0.6d^2+5.4d+0.8`

This is a maximum quadratic function therefore the domain is all real numbers.

Let us complete the square to find the vertex.

`f(d)=-0.6(d^2-9d)+0.8`

`f(d)=-0.6(d^2+9d)+-0.6((9)/(2))^2- -0.6((9)/(2))^2+ 0.8`

`f(d)=-0.6(d-(9)/(2))^2+(243)/(20)+ 0.8`

`f(d)=-0.6(d-(9)/(2))^2+(259)/(20)`

Therefore the range is

`y\le (259)/(20)`

`y\le 12.95`

See graph

Answer 2

Domain:

`[0,9.146]`

Range:

`[0,12.95]`

Explanation:

we are given

a baseball follows the path graphed by the quadratic function

`f(d)=-0.6d^2+5.4d+0.8`

where

d is the horizontal distance the ball traveled in yards

ƒ(d) is the height, in yards, of the ball at d horizontal yards

Domain:

we know that domain is all possible values of x for which any function is defined

So, for finding domain , we will take smallest x-value to largest x-value

so, we can set f(d)=0 and find zeros

`-0.6d^2+5.4d+0.8=0`

we can use quadratic formula

`d=(-b\pm √(b^2-4ac))/(2a)`

`d=(-54\pm √(54^2-4\left(-6\right)8))/(2\left(-6\right))`

`d=-(√(777)-27)/(6),\:d=(27+√(777))/(6)`

`d=-0.14579,d=9.146`

we know that d is a horizontal distance

and distance can never be negative

so, domain will be

`[0,9.146]`

Range:

Since, this is quadratic equation

so, it is also equation of parabola

so, firstly we will find vertex of parabola

Suppose, we have

`ax^2+bx+c=0`

Vertex is

`x=(-b)/(2a)`

so, we can compare

a=-0.6

b=5.4

c=0.8

now, we can find vertex

`x=(-5.4)/(2* -0.6)`

`d=4.5`

now, we can find y-value

`f(4.5)=-0.6(4.5)^2+5.4(4.5)+0.8`

`f(4.5)=12.95`

we can see

that leading coefficient is -0.6

which is negative

so, parabola opens downward

so, range will be

`[0,12.95]`