Algebra 1 benchmark

Question

Eduardo and Cynthia launched four different rockets into the air. Their flight paths are modeled by the four quadratic functions whose attributes are shown below.

1. Rocket 1: Has a vertex point of (4, 600)
2. Rocket 2: Is represented by the function f(x) =- (x - 30)^2 + 200.
3. Rocket 3: Is represented by the equation f(x) =-8 (x - 6) (x - 16).
4. Rocket 4: Is represented by the table given below:

x | 8 | 10 | 12 |

y | 400 | 650 | 400 |

Assume that the x-axis represents the ground and

f (x) represents the height in feet. Which rocket went the highest?

Answer 1

1) The vertex at (4, 600) indicates that the maximum height is 600.

2) This function is given in vertex form f(x) = a( x - h)² + k where the vertex is at (h, k). So the vertex in example (2) is at (30,200). Maximum height is 200.

3) If you do all the multiplications, you'll get f(x) = -8x² +176x - 768. The x value at the vertex is at -b/2a, where b is 176 and a is -8. -b/2a is -176/-16 = 11. Substituting x = 11 into the function to find the related y value, you get -8(5)(-5) = 200. Same maximum as (2).

4) The table clearly shows a symmetrical shape with 650 as the maximum y value. Looks like that's the winner!

Good idea to graph (2) and (3) on Desmos Graphing Calculator to demonstrate the above results.

Answer 2

Hi Jess T

Rocket 4

No equation is given for Rocket 1 but the vertex is given typically since rockets are launched from the ground up and then come down the shape of the trajectory of a projectile in motion at constant velocity horizontally but experiencing a changing velocity vertically is typically a parabola that opens downward with height, h, as a function of time, t, h(t) = -at² + bt + c

f(x) = ax²+ bx + c is a parabola in the equation above x = t or represents the time component

Rocket 1 has vertex (4, 600)

Rocket 2 has vertex (30,200) this equation is given in vertex form f(x)=-(x-30)²+200

f(x)=a(x-h)²+k where a = -1 h=30 and k =200; h, k are the coordinates of the vertex

Rocket 3 has vertex (11, 200) this one is given in factored form f(x)= -8(x-6)(x-16)

Rocket 3 multiplies out to -8(x²-16x-6x+96)=-8(x²-22x+96) = -8x²+176x - 768

You can get the x coordinate of the vertex as x= -b/2a then plug this value into your quadratic to obtain the height

Rocket 4 has vertex (10, 650)

You will need to take the values in the table and graph them or use them to determine coefficients in

f(x) = ax² + bx + c

You can graph all your functions on the same grid at Desmos.com to see and confirm each vertex.