A)The x-intercepts are at x=-2 and x=-4; the y-intercept is at y=8. B) Vertex form is g(x) = -2(x-2)²+11, and the axis of symmetry is at x=2. C) It takes 1.75 seconds for the dolphin to complete one jump.
Explanation: A) To find the x-intercepts, we replace y (in this case, f(x)) with 0 and solve: 0 = (x+3)²-1
First, add 1 to both sides: 0+1 = (x+3)²-1+1 1 = (x+3)²
Take the square root of both sides: √1 = √(x+3)² +/- 1 = x+3
Subtract 3 from both sides: -3 +/- 1 = x
This means x=-3-1 and x=-3+1, or x=-4 and x=-2.
B) To write g(x) in vertex form, we must complete the square. First we will factor the GCF of the first two terms out: g(x) = -2(x²-4x)+3
To find the value that we add (and subtract, to maintain equality) to this function, we take half of the value of b (the one in parentheses) and square it: (-4/2)²=(-2)²=4
This is what we will add in parentheses. Since this will be multiplied by -2, this means to cancel it, we will add 8 as well (2*-4 = -8; -8+8=0): g(x)=-2(x²-4x+4)+3+8 g(x)=-2(x²-4x+4)+11
Factoring the trinomial in parentheses, we have g(x) = -2(x-2)²+11.
The axis of symmetry is given by x = -b/2a: x = -8/2(-2) = -8/-4 = 2
C) We will set this equal to 0 to solve it: 0 = -16t² + 28t
Now we will factor out the GCF, 4t: 0 = 4t(-4t+7)
Using the zero product property, we know that either 4t=0 or -4t+7=0: 4t=0
Divide both sides by 4: 4t/4 = 0/4 t = 0
This is when the dolphin first jumps.
-4t+7=0
Subtract 7 from both sides: -4t+7-7=0-7 -4t = -7
Divide both sides by -4: -4t/-4 = -7/-4 t = 7/4 = 1.75
