A ball is thrown upward and outward from a height of 6 feet. The table...

William Curry

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
x,Ball's Horizontal Distance (feet)y,Ball's Height (feet)0618.13642.9
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(x)=?
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

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=ax2+bx+c 
a=0.9,b=2.6,c=6.1 
Hence, y=f(x)=0.9x2+2.6x+6.1 
c) We have that, the vertex of the parabola y=ax2+bx+c provided by
x=b2a 
x=2.62(0.9)=1.44 
The x coordinate of the vertex is 1.4 
The vertex is where the height is at its highest.
Thus, ymax=f(1.44)=0.9(1.44)2+2.6(1.44)+6.1=7.978 
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

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 x2 must be negative. So, the response is A.
b.) f(x)=0.7x2+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(x)=0.7x2+2.1x+6.1
f(x)=1.4x+2.1=0
x=2.11.4=32=1.5

alenahelenash

alenahelenash

Expert

2022-01-24Added 366 answers

b) y=0.68x2+2.05x+6.01y=0.68(x220568)c) y=2(0.68)x+2.05=0x=2.052×0.68=1.5074x-coordinate of vertex =1.5074d) y(x)=0.68(1.5074)2+2.05(1.5074)+6.01y=7.5550Maximum height =7.5550

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