Question
Lorraine hits a golf ball into the air. It lands 60 feet away from her. At its highest point, the ball is 30 feet in the air. Which parabolic equation describes the height of the ball as it moves away from Lorraine.
(1 point)
Responses
f(x)=−130(x−30)2+30
f of x is equal to negative 1 over 30 times open paren x minus 30 close paren squared plus 30
f(x)=130(x−30)2+60
f of x is equal to 1 over 30 times open paren x minus 30 close paren squared plus 60
f(x)=130(x−60)2+30
f of x is equal to 1 over 30 times open paren x minus 60 close paren squared plus 30
f(x)=−130(x+60)2−30
(1 point)
Responses
f(x)=−130(x−30)2+30
f of x is equal to negative 1 over 30 times open paren x minus 30 close paren squared plus 30
f(x)=130(x−30)2+60
f of x is equal to 1 over 30 times open paren x minus 30 close paren squared plus 60
f(x)=130(x−60)2+30
f of x is equal to 1 over 30 times open paren x minus 60 close paren squared plus 30
f(x)=−130(x+60)2−30
Answers
Answered by
GPT-4o mini
To find the correct parabolic equation that describes the height of the golf ball as it moves away from Lorraine, we can use the vertex form of a parabolic equation, which is given by:
\[ f(x) = a(x - h)^2 + k \]
where \((h, k)\) is the vertex of the parabola. In this scenario:
- The highest point of the ball, which is its vertex, occurs at \(x = 30\) feet (halfway between Lorraine and where the ball lands).
- The maximum height (k) is 30 feet.
So, the vertex is at (30, 30).
Since the ball is going up and down in a parabolic trajectory and it lands at 60 feet away from Lorraine, this means the parabola opens downward. Therefore, \(a\) should be negative.
Calculating \(a\): Since the ball hits the ground (height = 0) at \(x = 60\), we can use this point to find \(a\).
1. Plugging in the point (60, 0) into the equation:
\[ 0 = a(60 - 30)^2 + 30 \]
\[ 0 = a(30)^2 + 30 \]
\[ 0 = 900a + 30 \]
\[ 900a = -30 \]
\[ a = -\frac{30}{900} = -\frac{1}{30} \]
Now that we have \(a\), the complete equation is:
\[ f(x) = -\frac{1}{30}(x - 30)^2 + 30 \]
Thus, the correct choice from the options you provided is:
**f(x)=−\frac{1}{30}(x−30)^2+30**
\[ f(x) = a(x - h)^2 + k \]
where \((h, k)\) is the vertex of the parabola. In this scenario:
- The highest point of the ball, which is its vertex, occurs at \(x = 30\) feet (halfway between Lorraine and where the ball lands).
- The maximum height (k) is 30 feet.
So, the vertex is at (30, 30).
Since the ball is going up and down in a parabolic trajectory and it lands at 60 feet away from Lorraine, this means the parabola opens downward. Therefore, \(a\) should be negative.
Calculating \(a\): Since the ball hits the ground (height = 0) at \(x = 60\), we can use this point to find \(a\).
1. Plugging in the point (60, 0) into the equation:
\[ 0 = a(60 - 30)^2 + 30 \]
\[ 0 = a(30)^2 + 30 \]
\[ 0 = 900a + 30 \]
\[ 900a = -30 \]
\[ a = -\frac{30}{900} = -\frac{1}{30} \]
Now that we have \(a\), the complete equation is:
\[ f(x) = -\frac{1}{30}(x - 30)^2 + 30 \]
Thus, the correct choice from the options you provided is:
**f(x)=−\frac{1}{30}(x−30)^2+30**