Question

1. robert drops a ball from his balcony. the height of the ball is modeled by the function ( f(x) = -2x^2 + x + 11 ), where ( f(x) ) represents the height of the ball and ( x ) represents the number of seconds. which of the following best represents the number of seconds that will pass before the ball reaches the ground?

a. 1.4
b. 1.9
c. 2.1
d. 2.6

1. which type of function is represented by the table of values below?

Explanation:

Step1: Set the function to zero (ground level)

To find when the ball reaches the ground, we set \( f(x) = 0 \), so the equation is \( -2x^{2}+x + 11=0 \). Multiply both sides by -1 to get \( 2x^{2}-x - 11 = 0 \).

Step2: Use the quadratic formula

The quadratic formula for a quadratic equation \( ax^{2}+bx + c = 0 \) is \( x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a} \). Here, \( a = 2 \), \( b=- 1 \), and \( c=-11 \).

First, calculate the discriminant \( D=b^{2}-4ac=(-1)^{2}-4\times2\times(-11)=1 + 88 = 89 \).

Then, \( x=\frac{-(-1)\pm\sqrt{89}}{2\times2}=\frac{1\pm\sqrt{89}}{4} \).

We know that \( \sqrt{89}\approx9.43 \), so we have two solutions:

\( x_1=\frac{1 + 9.43}{4}=\frac{10.43}{4}\approx2.6 \)

\( x_2=\frac{1 - 9.43}{4}=\frac{-8.43}{4}\approx - 2.1 \)

Since time cannot be negative, we take the positive solution \( x\approx2.6 \).

Answer:

d. 2.6