Answer:
Part A) The graph in the attached figure (see the explanation)
Part B) 16 feet
Part C) see the explanation
Step-by-step explanation:
Part A) Graph the function
Let
h(t) ----> the height in feet of the ball above the ground
t -----> the time in seconds
we have
![h(t)=-16t^{2}+98](https://tex.z-dn.net/?f=h%28t%29%3D-16t%5E%7B2%7D%2B98)
This is a vertical parabola open downward (the leading coefficient is negative)
The vertex is a maximum
To graph the parabola, find the vertex, the intercepts, and the axis of symmetry
<em>Find the vertex</em>
The function is written in vertex form
so
The vertex is the point (0,98)
Find the y-intercept
The y-intercept is the value of the function when the value of t is equal to zero
For t=0
![h(t)=-16(0)^{2}+98](https://tex.z-dn.net/?f=h%28t%29%3D-16%280%29%5E%7B2%7D%2B98)
![h(0)=98](https://tex.z-dn.net/?f=h%280%29%3D98)
The y-intercept is the point (0,98)
Find the t-intercepts
The t-intercepts are the values of t when the value of the function is equal to zero
For h(t)=0
![-16t^{2}+98=0](https://tex.z-dn.net/?f=-16t%5E%7B2%7D%2B98%3D0)
![t^{2}=\frac{98}{16}](https://tex.z-dn.net/?f=t%5E%7B2%7D%3D%5Cfrac%7B98%7D%7B16%7D)
square root both sides
![t=\pm\frac{\sqrt{98}}{4}](https://tex.z-dn.net/?f=t%3D%5Cpm%5Cfrac%7B%5Csqrt%7B98%7D%7D%7B4%7D)
![t=\pm7\frac{\sqrt{2}}{4}](https://tex.z-dn.net/?f=t%3D%5Cpm7%5Cfrac%7B%5Csqrt%7B2%7D%7D%7B4%7D)
therefore
The t-intercepts are
![(-7\frac{\sqrt{2}}{4},0), (7\frac{\sqrt{2}}{4},0)](https://tex.z-dn.net/?f=%28-7%5Cfrac%7B%5Csqrt%7B2%7D%7D%7B4%7D%2C0%29%2C%20%287%5Cfrac%7B%5Csqrt%7B2%7D%7D%7B4%7D%2C0%29)
![(-2.475,0), (2.475,0)](https://tex.z-dn.net/?f=%28-2.475%2C0%29%2C%20%282.475%2C0%29)
Find the axis of symmetry
The equation of the axis of symmetry in a vertical parabola is equal to the x-coordinate of the vertex
so
----> the y-axis
To graph the parabola, plot the given points and connect them
we have
The vertex is the point ![(0,98)](https://tex.z-dn.net/?f=%280%2C98%29)
The y-intercept is the point ![(0,98)](https://tex.z-dn.net/?f=%280%2C98%29)
The t-intercepts are ![(-2.475,0), (2.475,0)](https://tex.z-dn.net/?f=%28-2.475%2C0%29%2C%20%282.475%2C0%29)
The axis of symmetry is the y-axis
The graph in the attached figure
Part B) How far is the artifact fallen from the time t=0 to time t=1
we know that
For t=0
![h(t)=-16(0)^{2}+98](https://tex.z-dn.net/?f=h%28t%29%3D-16%280%29%5E%7B2%7D%2B98)
![h(0)=98\ ft](https://tex.z-dn.net/?f=h%280%29%3D98%5C%20ft)
For t=1
![h(t)=-16(1)^{2}+98](https://tex.z-dn.net/?f=h%28t%29%3D-16%281%29%5E%7B2%7D%2B98)
![h(1)=82\ ft](https://tex.z-dn.net/?f=h%281%29%3D82%5C%20ft)
Find the difference
![98\ ft-82\ ft=16\ ft](https://tex.z-dn.net/?f=98%5C%20ft-82%5C%20ft%3D16%5C%20ft)
Part C) Does the artifact fall the same distance from time t=1 to time t=2 as it does from the time t=0 to time t=1?
we know that
For t=1
![h(t)=-16(1)^{2}+98](https://tex.z-dn.net/?f=h%28t%29%3D-16%281%29%5E%7B2%7D%2B98)
![h(1)=82\ ft](https://tex.z-dn.net/?f=h%281%29%3D82%5C%20ft)
For t=2
![h(t)=-16(2)^{2}+98](https://tex.z-dn.net/?f=h%28t%29%3D-16%282%29%5E%7B2%7D%2B98)
![h(2)=34\ ft](https://tex.z-dn.net/?f=h%282%29%3D34%5C%20ft)
Find the difference
![82\ ft-34\ ft=48\ ft](https://tex.z-dn.net/?f=82%5C%20ft-34%5C%20ft%3D48%5C%20ft)
so
The artifact fall 48 feet from time t=1 to time t=2 and fall 16 feet from time t=0 to time t=1
therefore
The distance traveled from t=1 to t=2 is greater than the distance traveled from t=0 to t=1