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Question

A concave curve is plotted in quadrant 1 of a coordinate plane. Eleven unlabeled points are plotted on the curve.

Carla is watching a professional baseball game on television and sees a batter hit a home run over the fence. A graphic overlay comes up during the replay that shows how high and far the ball traveled in feet. Carla recognizes the shape to be a quadratic and wants to write an equation to represent the function. Which of the following functions accurately models the graph?

(1 point)
Responses

The height reached by the ball, h(x)
, is given by the equation h(x)=43/1,620(x−90)^2+215
, where x represents the horizontal distance.
The height reached by the ball, h left parenthesis x right parenthesis , is given by the equation h left parenthesis x right parenthesis equals Start Fraction 43 over 1,620 End Fraction left parenthesis x minus 90 right parenthesis squared plus 215 , where x represents the horizontal distance.

The height reached by the ball, h(x)
, is given by the equation h(x)=−18/9,245(x−215)^2+90
, where x represents the horizontal distance.
The height reached by the ball, h left parenthesis x right parenthesis , is given by the equation h left parenthesis x right parenthesis equals negative Start Fraction 18 over 9,245 End Fraction left parenthesis x minus 215 right parenthesis squared plus 90 , where x represents the horizontal distance.

The height reached by the ball, h(x)
, is given by the equation h(x)=−43/1,620(x−90)^2+215
, where x represents the horizontal distance.
The height reached by the ball, h left parenthesis x right parenthesis , is given by the equation h left parenthesis x right parenthesis equals negative Start Fraction 43 over 1,620 End Fraction left parenthesis x minus 90 right parenthesis squared plus 215 , where x represents the horizontal distance.

The height reached by the ball, h(x)
, is given by the equation h(x)=18/9,245(x−215)^2+90
, where x represents the horizontal distance.
The height reached by the ball, h left parenthesis x right parenthesis , is given by the equation h left parenthesis x right parenthesis equals Start Fraction 18 over 9,245 End Fraction left parenthesis x minus 215 right parenthesis squared plus 90 , where x represents the horizontal distance.
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Answer 1

To determine which function accurately models the height of the ball in relation to its horizontal distance, consider the characteristics of quadratic functions, especially those that open downwards, since a ball reaches a maximum height before descending after being hit.

1. The maximum point of the parabola corresponds to the peak height of the ball.
2. The leading coefficient should be negative to indicate that the parabola opens downwards.

Given these observations, the options should be evaluated by their structures. Functions with a negative leading coefficient are plausible candidates. Looking at the options:

• The first option features a positive coefficient: \( \frac{43}{1,620} \) which indicates it opens upwards.

• The second option also opens upwards due to the negative sign in front of the fraction being ignored in the context.

• The third option opens downwards because it has a negative leading coefficient \( -\frac{18}{9,245} \).

• The fourth option similarly has a negative leading coefficient and therefore opens downwards.

• The fifth option has a negative leading coefficient \( -\frac{43}{1,620} \), indicating it opens downwards.

• The last option with a positive coefficient indicates it opens upwards.

Both the third and fifth options present equations of parabolas that open downward, reflecting a realistic trajectory of a baseball.

However, we have to also assess their vertex positions (indicated by the values subtracted from \( x \)) and the maximum heights.

If we assume the vertex (maximum height) should reasonably fall within expected values based on how high and far one might expect a baseball to travel after being hit, one can check where these heights land within the context typically seen in baseball.

Without a graph to visualize which vertex represents a probable angle and height of a baseball's trajectory, one may conjecture that common values for homers could often be in the area of around 215 feet high.

Based on all factors (the parabola opening downwards, the coefficients suggesting maximum heights), the function that accurately models the parabola for this scenario is likely:

\( h(x) = -\frac{43}{1,620}(x - 90)^2 + 215 \), as it opens downwards and reflects the maximum height capably.

Hence, the answer is:

The height reached by the ball, \( h(x) \) is given by the equation \( h(x)= -\frac{43}{1,620}(x - 90)^2 + 215 \), where \( x \) represents the horizontal distance.