Question

F(x) =(x+3)2−1 G(x) =−2x2+8x+3 H (t) =−16t2+28t Part A: What are the x- and y-intercepts of the graph of the function f(x)? Identify the coordinates of any maxima or minima of the function. Explain your answers. Part B: Write the function g(x) in the form y=a(x−h)2+k . Write the equation for the line of symmetry for the graph of g(x). Show all your work and explain your answer. Part C: The function h(t) gives the approximate height, in feet, of a dolphin's jump out of water, where t represents the time, in seconds. What is the total length of time, in seconds, it took the dolphin to complete one jump? Show your work.

Answer 1

Part 1)
we have that
F(x) =(x+3)²−1

using a graph tool
see the attached figure N 1

we know that
the x-intercepts of the graph is when y=0
the points are (-4,0) and (-2,0)

the y-intercepts of the graph is when x=0
the point is (0,8)

the graph is a vertical parabola that opens up
so
the vertex of the parabola is a minimum
the vertex is the point (-3,-1)

Part 2) Write the function g(x) in the form y=a(x−h)²+k . Write the equation for the line of symmetry for the graph of g(x)

we have that
G(x) =−2x²+8x+3
Group terms that contain the same variable
G(x) =(−2x²+8x)+3
Factor the leading coefficient
G(x) =-2*(x²-4x)+3
Complete the square. Remember to balance the equation
G(x) =-2*(x²-4x+4-4)+3
G(x) =-2*(x²-4x+4)+3 +8
G(x) =-2*(x²-4x+4)+11
Rewrite as perfect squaresG(x) =-2*(x-2)²+11

the answer Part 2) isG(x) =-2*(x-2)²+11
see the attached figure N 2
the equation for the line of symmetry for the graph of g(x) is the vertical line that passes through the vertex

the vertex is the point (2,11)
so
the equation for the line of symmetry is x=2

Part 3) we have that
H (t) =−16t²+28t

see the attached figure N 3

the total length of time, in seconds, it took the dolphin to complete one jump is the difference for t=0 sec and t=1.75 sec
(1.75-0)=1.75 sec

the answer Part 3) is 1.75 sec