Question

Question1: A ball is shot from a cannon into the air with an upward velocity of 10 ft/sec. The function h(t) =-2t^2 + 10t + 12= represents the height (h) of the ball after t seconds. 1. Determine what is the axis of symmetry of the function? 2. Identify the roots of the quadratic function? 3. What is the y-intercept of the quadratic function? .4.) Determine the vertex of the function?

Answer 1

1)
`x=(5)/(2)`

2)
`t_1=-1` and
`t_2=6`

3) (0,12)

4)
`\left((5)/(2),(49)/(2)\right)`

Explanation:

The function
`h(t) =-2t^2 + 10t + 12` represents the height (h) of the ball after t seconds.

Find the vertex of the parabola:

`t_v=-(b)/(2a)=-(10)/(2\cdot (-2))=(5)/(2)\\ \\h_v=h\left((5)/(2)\right)=-2\cdot\left((5)/(2)\right)^2+10\cdot (5)/(2)+12=-2\cdot (25)/(4)+25+12=-(25)/(2)+37=(49)/(2)`

Hence, the vertex of the function is at point
`\left((5)/(2),(49)/(2)\right).`

The axis of symmetry of the function is vertical line which passes through the vertex, so its equation is

`x=(5)/(2)`

To find y-intercept, equat t to 0 and find h:

`h=-2\cdot 0^2+10\cdot 0+12=12`

Hence, y-intercept is at point (0,12)

To find the roots of the quadratic finction, equate h to 0 and solve the equation for t:

`h=0\Rightarrow -2t^2+10t+12=0\\ \\t^2-5t-6=0\ [\text{Divided by -2}]\\ \\D=(-5)^2-4\cdot 1\cdot (-6)=25+24=49\\ \\t_(1,2)=(-(-5)\pm √(49))/(2\cdot 1)=(5\pm 7)/(2)=6,\ -1`

Therefore, two roots are
`t_1=-1` and
`t_2=6`