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)=−1/30(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)=1/30(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)=1/30(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)=−1/30(x+60)^2−30
(1 point)
Responses
f(x)=−1/30(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)=1/30(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)=1/30(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)=−1/30(x+60)^2−30
Answers
Answered by
GPT-4o mini
To find the correct parabolic equation describing the height of the golf ball as it moves away from Lorraine, we need to know the vertex of the parabola and the general form of the equation.
Given the following points:
- The highest point (vertex) of the parabola is \( (30, 30) \), since it reaches a height of 30 feet at a horizontal distance of 30 feet.
- The ball lands 60 feet away from Lorraine, which means the ball touches the ground (height = 0) at the other side, specifically at the point \( (60, 0) \).
Using the vertex form of the parabolic equation:
\[ f(x) = a(x - h)^2 + k \]
where \( (h, k) \) is the vertex.
In this case, \( (h, k) = (30, 30) \):
\[ f(x) = a(x - 30)^2 + 30 \]
Now, we need to find the value of \( a \). We know that \( f(60) = 0 \) because the ball lands at that point:
\[ 0 = a(60 - 30)^2 + 30 \]
\[ 0 = a(30)^2 + 30 \]
\[ 0 = 900a + 30 \]
\[ 900a = -30 \]
\[ a = -\frac{1}{30} \]
Thus, the equation becomes:
\[ f(x) = -\frac{1}{30}(x - 30)^2 + 30 \]
Therefore, the correct parabolic equation that describes the height of the ball as it moves away from Lorraine is:
**f(x) = -1/30(x - 30)^2 + 30**
This corresponds to the first option listed in your responses.
Given the following points:
- The highest point (vertex) of the parabola is \( (30, 30) \), since it reaches a height of 30 feet at a horizontal distance of 30 feet.
- The ball lands 60 feet away from Lorraine, which means the ball touches the ground (height = 0) at the other side, specifically at the point \( (60, 0) \).
Using the vertex form of the parabolic equation:
\[ f(x) = a(x - h)^2 + k \]
where \( (h, k) \) is the vertex.
In this case, \( (h, k) = (30, 30) \):
\[ f(x) = a(x - 30)^2 + 30 \]
Now, we need to find the value of \( a \). We know that \( f(60) = 0 \) because the ball lands at that point:
\[ 0 = a(60 - 30)^2 + 30 \]
\[ 0 = a(30)^2 + 30 \]
\[ 0 = 900a + 30 \]
\[ 900a = -30 \]
\[ a = -\frac{1}{30} \]
Thus, the equation becomes:
\[ f(x) = -\frac{1}{30}(x - 30)^2 + 30 \]
Therefore, the correct parabolic equation that describes the height of the ball as it moves away from Lorraine is:
**f(x) = -1/30(x - 30)^2 + 30**
This corresponds to the first option listed in your responses.