Question

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) = -0.9x² + 2.7x + 6.7
c. use the model from part (b) to determine the x - coordinate of the quadratic functions vertex.
the x - coordinate of the vertex is 1.5. (type an integer or a decimal rounded to one decimal place as needed.)
complete the statement below.
the maximum height of the ball occurs □ feet from where it was thrown and the maximum height is □ feet. (type integers or decimals rounded to one decimal place as needed.)

Explanation:

Step1: Recall the formula for the x - coordinate of the vertex of a quadratic function

For a quadratic function \(y = ax^{2}+bx + c\), the x - coordinate of the vertex is given by \(x=-\frac{b}{2a}\). In the function \(f(x)=-0.9x^{2}+2.7x + 6.7\), \(a=-0.9\) and \(b = 2.7\).

Step2: Calculate the x - coordinate of the vertex

Substitute \(a=-0.9\) and \(b = 2.7\) into the formula \(x =-\frac{b}{2a}\).

$$
LATEXBLOCK0
$$

Step3: Calculate the y - coordinate of the vertex

Substitute \(x = 1.5\) into the function \(f(x)=-0.9x^{2}+2.7x + 6.7\).

$$
LATEXBLOCK1
$$

Answer:

The maximum height of the ball occurs \(1.5\) feet from where it was thrown and the maximum height is \(8.7\) feet.