Answer:
Domain is all real numbers
Range is
[tex]y\le 12.95[/tex]
Step-by-step explanation:
The given function is
[tex]f(d)=-0.6d^2+5.4d+0.8[/tex]
This is a maximum quadratic function therefore the domain is all real numbers.
Let us complete the square to find the vertex.
[tex]f(d)=-0.6(d^2-9d)+0.8[/tex]
[tex]f(d)=-0.6(d^2+9d)+-0.6(\frac{9}{2})^2- -0.6(\frac{9}{2})^2+ 0.8[/tex]
[tex]f(d)=-0.6(d-\frac{9}{2})^2+\frac{243}{20}+ 0.8[/tex]
[tex]f(d)=-0.6(d-\frac{9}{2})^2+\frac{259}{20}[/tex]
Therefore the range is
[tex]y\le \frac{259}{20}[/tex]
[tex]y\le 12.95[/tex]
See graph
Answer:
Domain:
[tex][0,9.146][/tex]
Range:
[tex][0,12.95][/tex]
Step-by-step explanation:
we are given
a baseball follows the path graphed by the quadratic function
[tex]f(d)=-0.6d^2+5.4d+0.8[/tex]
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
[tex]-0.6d^2+5.4d+0.8=0[/tex]
we can use quadratic formula
[tex]d=\frac{-b\pm \sqrt{b^2-4ac}}{2a}[/tex]
[tex]d=\frac{-54\pm \sqrt{54^2-4\left(-6\right)8}}{2\left(-6\right)}[/tex]
[tex]d=-\frac{\sqrt{777}-27}{6},\:d=\frac{27+\sqrt{777}}{6}[/tex]
[tex]d=-0.14579,d=9.146[/tex]
we know that d is a horizontal distance
and distance can never be negative
so, domain will be
[tex][0,9.146][/tex]
Range:
Since, this is quadratic equation
so, it is also equation of parabola
so, firstly we will find vertex of parabola
Suppose, we have
[tex]ax^2+bx+c=0[/tex]
Vertex is
[tex]x=\frac{-b}{2a}[/tex]
so, we can compare
a=-0.6
b=5.4
c=0.8
now, we can find vertex
[tex]x=\frac{-5.4}{2\times -0.6}[/tex]
[tex]d=4.5[/tex]
now, we can find y-value
[tex]f(4.5)=-0.6(4.5)^2+5.4(4.5)+0.8[/tex]
[tex]f(4.5)=12.95[/tex]
we can see
that leading coefficient is -0.6
which is negative
so, parabola opens downward
so, range will be
[tex][0,12.95][/tex]
