The function h(t) = -4.9t² + 19.6t is used to model the height of an object projected in the air where h(t) is the height (in meters) and t is the time (in seconds). What is the domain and range?yw:)

Hello!The answer is:Domain:[tex]0\leq t\leq 4[/tex]Range:[tex]0\leq h\leq 19.6[/tex]Why?To solve this problem, we need to remember that the height is given as afunction of the time, and the time can only take positive values, also, the range of the function will be the distance from the top of the ground or zero, to the highest point of the function (parabola).So, finding the domain and the range of the given function, we have:[tex]h(t)=-4.9t^{2}+19.6t[/tex]Where,[tex]a=-4.9\\b=19.6\\c=0[/tex]Using the quadratic equation in order to find where the function (height) tends to 0, we have:[tex]\frac{-b+-\sqrt{b^{2} -4ac} }{2a}[/tex]Substituting we have:[tex]\frac{-19.6+-\sqrt{19.6^{2} -4(-4.9*0)} }{2*(-4.9)}=\frac{-19.6+-\sqrt{19.6^{2}-0} }{-9.8}\\\\\frac{-19.6+-\sqrt{19.6^{2} -0} }{-9.8}=\frac{-19.6+-(19.6)}{-9.8}\\\\t_{1}=\frac{-19.6+(19.6)}{-9.8}=\frac{0}{-9.8}=0\\\\t_{1}=\frac{-19.6-(19.6)}{-9.8}=\frac{-39.2}{-9.8}=4[/tex]So, from the domain we have that it goes from 0, to 4, or:[tex]0\leq t\leq 4[/tex]Then, finding the range, we have:To know the range of the function, we need to calculate the highest point y-coordinate, the highest or lowest point of a parabola is the vertex, for this case, the we are looking for the highest point since we have a parabola opening downwards.We can calculate the vertex using the following formula:[tex]x_{vertex}=\frac{-b}{2a}\\\\x_{vertex}=\frac{-19.6}{2*(-4.9)}=\frac{-19.6}{-9.8)}=2\\[/tex][tex]x=t=2[/tex]Now, substituting the x-coordinate(t) of the vertex into the function, to find the y-coordinate value, we have:[tex]h(t)=-4.9t^{2}+19.6t[/tex][tex]y=-4.9(2)^{2}+19.6(2)=-4.9*(4)+19.6*(2)=19.6[/tex]Therefore, the y-coordinate is equal to the highest point of the parabola which is 19.6 (feet).Hence, the range of the parabola goes from 0 to 19.6, or:[tex]0\leq h\leq 19.6[/tex]Note: Look for the graphic  (attached) for better understanding.Have a nice day!