William Curry

Answered

2022-01-16

A ball is thrown upward and outward from a height of 6 feet. The table shows four measurements indicating the ball's height at various horizontal distances from where it was thrown. A graphing calculator displays a quadratic function that models the ball's height, y, in feet, in terms of its horizontal distance, x, in feet. Answer

$$\begin{array}{|cc|}\hline x,\text{Ball's Horizontal Distance (feet)}& y,\text{Ball's Height (feet)}\\ 0& 6\\ 1& 8.1\\ 3& 6\\ 4& 2.9\\ \hline\end{array}$$

a. Explain why a quadratic function was used to model the data.

In the quadratic regression screen shown in the problem statement, why is the value of the coefficient a negative?

b. Use the graphing calculator screen (shown in the box above) to express the model in function notation.

$f\left(x\right)=?$

c. Use the model from part (b) to determine the k-coordinate of the quadratic function's vertex.

The x-coordinate of the vertex is ?

a. Explain why a quadratic function was used to model the data.

In the quadratic regression screen shown in the problem statement, why is the value of the coefficient a negative?

b. Use the graphing calculator screen (shown in the box above) to express the model in function notation.

c. Use the model from part (b) to determine the k-coordinate of the quadratic function's vertex.

The x-coordinate of the vertex is ?

Answer & Explanation

Lindsey Gamble

Expert

2022-01-17Added 38 answers

a) According to the table above,, that height of the ball first increases and then decreases, which looks like a quadratic function.

The coefficient a in the regression screen is negative because the height of the ball increases and then decreases, hence, the quadratic must open downward.

b) Using the quadratic regression model,

$y=a{x}^{2}+bx+c$

$a=-0.9,b=2.6,c=6.1$

Hence, $y=f\left(x\right)=-0.9{x}^{2}+2.6x+6.1$

c) We have that, the vertex of the parabola $y=a{x}^{2}+bx+c$ provided by

$x=-\frac{b}{2a}$

$\Rightarrow x=-\frac{2.6}{2(-0.9)}=1.44$

The x coordinate of the vertex is 1.4

The vertex is where the height is at its highest.

Thus, ${y}_{max}=f\left(1.44\right)=-0.9{\left(1.44\right)}^{2}+2.6\left(1.44\right)+6.1=7.978$

$\approx 8$ feet

The maximum height of the ball occurs 1.4 feet from where it was thrown and the maximum height is 8 feet.

Esta Hurtado

Expert

2022-01-18Added 39 answers

a.) After the point at which horizontal distance is 1ft, the height decreases, and decreases more rapidly with time. It appears to be a quadratic function. So, the response is A.

The graph must open downwards, as the height first increase then decrease. So, the value of the coefficient of $x}^{2$ must be negative. So, the response is A.

b.) $f\left(x\right)=-0.7{x}^{2}+2.1x+6.1$

c.) At vertex, the slope of the graph will be 0. Therefore, the function's differentiation will be zero. So,

$f\left(x\right)=-0.7{x}^{2}+2.1x+6.1$

${f}^{\prime}\left(x\right)=-1.4x+2.1=0$

$x=\frac{2.1}{1.4}=\frac{3}{2}=1.5$

alenahelenash

Expert

2022-01-24Added 366 answers

b) $y=-0.68{x}^{2}+2.05x+6.01$ $y=-0.68({x}^{2}\frac{-205}{68})$ c) ${y}^{\prime}=-2(0.68)x+2.05=0$ $x=\frac{2.05}{2\times 0.68}=1.5074$ x-coordinate of vertex $=1.5074$ d) $y(x)=-0.68(1.5074{)}^{2}+2.05(1.5074)+6.01$ $y=7.5550$ Maximum height $=7.5550$

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