Question

a ball is thrown upward and outward from a height of 5 feet. the table shows four measurements indicating the balls height at various horizontal distances from where it was thrown. a graphing calculator displays a quadratic function that models the balls height, y, in feet, in terms of its horizontal distance, x, in feet. answer parts (a)-(c) below.

x, balls horizontal distance (feet)	y, balls height (feet)
1	7.7
3	5
4	3.1

quadreg
y = ax²+bx + c
a=-0.8
b = 2.4
c = 5.4
in the quadratic regression screen shown in the problem statement, why is the value of the coefficient a negative?
a. since the height increases then decreases, the quadratic graph must open upward, not downward, which means a must be negative.
b. since the height increases then decreases, the quadratic graph must open downward, not upward, which means a must be negative.
c. since the height of the ball starts at the height of the person throwing the ball, the quadratic graph must be shifted up, which makes a negative.
b. use the graphing calculator screen (shown in the box above) to express the model in function notation.
f(x)=-0.8x² + 2.4x+5.4
c. use the model from part (b) to determine the x - coordinate of the quadratic functions vertex.
the x - coordinate of the vertex is
(type an integer or a decimal rounded to one decimal place as needed.)

Explanation:

Step1: Recall vertex - x formula

The formula for the x - coordinate of the vertex of a quadratic function \(y = ax^{2}+bx + c\) is \(x=-\frac{b}{2a}\).

Step2: Identify a and b values

From the function \(f(x)=- 0.8x^{2}+2.4x + 5.4\), we have \(a=-0.8\) and \(b = 2.4\).

Step3: Calculate x - coordinate of vertex

Substitute \(a=-0.8\) and \(b = 2.4\) into the formula \(x=-\frac{b}{2a}\), so \(x=-\frac{2.4}{2\times(-0.8)}=\frac{2.4}{1.6}=1.5\).

Answer:

1.5