Question

1. a ball is thrown upward with an initial velocity of 30 meters per second. the position of the ball over time is recorded in the table below.

time in seconds, x	distance from the ground in meters, f(x)
1	25
2	40
3	45
4	40
5	25

which quadratic function models the data?

a. ( f(x) = -5x^2 + 30x )
b. ( f(x) = 30x^2 )
c. ( f(x) = 30x^2 - 5x )
d. ( f(x) = -5x^2 - 30x )

Explanation:

Step1: Test x=0 in each option

• For A: \( f(0) = -5(0)^2 + 30(0) = 0 \) (matches table)
• For B: \( f(0) = 30(0)^2 = 0 \) (matches table)
• For C: \( f(0) = 30(0)^2 - 5(0) = 0 \) (matches table)
• For D: \( f(0) = -5(0)^2 - 30(0) = 0 \) (matches table)

Step2: Test x=1 in remaining options

• For A: \( f(1) = -5(1)^2 + 30(1) = 25 \) (matches table)
• For B: \( f(1) = 30(1)^2 = 30 \) (does not match)
• For C: \( f(1) = 30(1)^2 - 5(1) = 25 \) (matches table)
• For D: \( f(1) = -5(1)^2 - 30(1) = -35 \) (does not match)

Step3: Test x=2 in A and C

• For A: \( f(2) = -5(2)^2 + 30(2) = 40 \) (matches table)
• For C: \( f(2) = 30(2)^2 - 5(2) = 110 \) (does not match)

Answer:

A. \( f(x) = -5x^2 + 30x \)