when jaxon hit a golf ball off the grass with a 9-iron it travelled 130m horizontally until it landed back on the grass. the height of the ball was measured to be 21m when it reached a horizontal distance of 91m. the flight of the golf ball can be represented by quadratic function. and what is formula that i can draw on graph.
To represent the flight of the golf ball with a quadratic function, we can use the general form of a quadratic function: y = ax^2 + bx + c.
In this case, the horizontal distance (x) will be represented by the input variable, and the height (y) will be represented by the output variable.
We are given two points: (0, 0) and (91, 21). We can use these points to form two equations to solve for the coefficients a, b, and c.
Using the point (0, 0):
0 = a(0)^2 + b(0) + c
0 = c
Using the point (91, 21):
21 = a(91)^2 + b(91) + c
21 = 8281a + 91b + 0
21 = 8281a + 91b
Therefore, our function becomes:
y = ax^2 + bx
Plugging in the values from the point (91, 21):
21 = a(91)^2 + b(91)
21 = 8281a + 91b
Now we have a system of equations to solve:
21 = 8281a + 91b [eq1]
0 = b [eq2] (Since c = 0)
From [eq2], we can see that b = 0.
Plugging that into [eq1]:
21 = 8281a
Now solve for a:
a = 21 / 8281
a ≈ 0.00254
Therefore, the quadratic function that represents the flight of the golf ball is:
y = 0.00254x^2
Let's graph this function together:
The x-axis represents the horizontal distance (in meters).
The y-axis represents the height (in meters).
Plot the points (0, 0) and (91, 21).
Draw a smooth curve that passes through these points, which represents the quadratic function y = 0.00254x^2.
Note: The graph will only represent the flight until the ball touches the ground again, so it will not extend indefinitely.
when jaxon hit a golf ball off the grass with a 9-iron it travelled 130m horizontally until it landed back on the grass. the height of the ball was measured to be 21m when it reached a horizontal distance of 91m. the flight of the golf ball can be represented by quadratic function. and what is exact equation that i can use on graph.
To find the exact equation, we need to consider the vertex form of a quadratic function. The vertex form is given by:
y = a(x - h)^2 + k
Where (h, k) represents the coordinates of the vertex.
First, let's find the vertex coordinates using the given information.
The vertex height (k) is 21m when the horizontal distance (x) is 91m. Thus, the vertex point is (91, 21).
We also know that the vertex form equation is symmetrical with respect to the vertex. Since the horizontal distance is 130m when the ball lands back on the grass, that means the peak of the golf ball's flight is halfway between the starting point and landing point, which gives us (0 + 130)/2 = 65m.
So, the coordinates of the peak are (65, ?), and we need to find the corresponding height for this x-coordinate.
To find the corresponding height, we can use the vertex form equation and plug in the known values:
21 = a(91 - h)^2 + k [eq1]
0 = a(65 - h)^2 + k [eq2]
Plugging in the known vertex point (91, 21):
21 = a(91 - h)^2 + 21 [eq1]
0 = a(65 - h)^2 + 21 [eq2]
Subtracting 21 from both sides in [eq1]:
0 = a(91 - h)^2 [eq3]
Now we have two equations [eq2] and [eq3]. We can set them equal to each other since both sides are equal to 0:
a(65 - h)^2 + 21 = 0
Simplifying the equation:
a(65 - h)^2 = -21
Dividing both sides by a:
(65 - h)^2 = -21 / a
Solving for (65 - h):
65 - h = ±√(-21 / a)
To determine the value of h, we need to know the value of a. Unfortunately, the given information does not provide enough information to solve for a. Thus, without knowing the exact value of a, we cannot determine the exact equation for the graph.
However, we do know that the vertex of the parabola is (91, 21), and it opens downwards since the ball starts and lands on the grass. So, the general equation can be written as:
y = a(x - 91)^2 + 21
The value of 'a' will determine the steepness of the parabola and cannot be determined without additional information.
when jaxon hit a golf ball off the grass with a 9-iron it travelled 130m horizontally until it landed back on the grass. the height of the ball was measured to be 21m when it reached a horizontal distance of 91m. the flight of the golf ball can be represented by quadratic function.
To find the quadratic function that represents the flight of the golf ball, we can use the vertex form of a quadratic equation:
y = a(x - h)^2 + k
where (h, k) represents the coordinates of the vertex.
From the given information, we have the following:
- The ball travelled a horizontal distance of 130m until it landed back on the grass, so the x-coordinate of the vertex is the halfway point: h = 130/2 = 65.
- The height of the ball was measured to be 21m when it reached a horizontal distance of 91m, so the vertex point is (65, 21).
Now, substitute these values into the vertex form equation:
y = a(x - 65)^2 + 21
This equation represents the flight of the golf ball as a quadratic function.
well, agree with this: y = a(x - 65)^2 + 21
However what is the constant "a" ????
we know
if x = 91, y = 21
so
21 = a (91-65)^2 + 21
now solve for a and you have the quadratic function.
Apologies for the previous oversight. Let's solve for the constant "a" now:
Given:
x = 91
y = 21
Substituting these values into the equation y = a(x - 65)^2 + 21:
21 = a(91 - 65)^2 + 21
0 = a(26)^2
Since the quadratic equation reduces to zero, we can conclude that the constant "a" is 0. Therefore, the quadratic function representing the flight of the golf ball is:
y = 0(x - 65)^2 + 21
y = 21
The equation y = 21 represents a horizontal line at a height of 21m, indicating that the golf ball maintains a constant height of 21m after reaching a horizontal distance of 91m.
Note: It seems there was an error in the problem statement or calculation since the quadratic form does not provide any height variations. Please double-check the given information to ensure accuracy.
when jaxon hit a golf ball off the grass with a 9-iron it travelled 130m horizontally until it landed back on the grass. the height of the ball was measured to be 21m when it reached a horizontal distance of 91m. the flight of the golf ball can be represented by quadratic function.
- determine the function that represents the path of the golf ball in factored form. round your value for (a) to the nearest thousandth.
To determine the function that represents the path of the golf ball in factored form, we need to find the quadratic equation in factored form.
Given:
- The ball travelled a horizontal distance of 130m until it landed back on the grass.
- The height of the ball was measured to be 21m when it reached a horizontal distance of 91m.
We already have the equation in vertex form: y = a(x - h)^2 + k
To find the equation in factored form, we need to convert it by expanding and simplifying.
Using the vertex point (65, 21):
y = a(x - 65)^2 + 21
y = a(x^2 - 130x + 4225) + 21
y = ax^2 - 130ax + 4225a + 21
Now, we need to solve for the value of 'a'. According to the given information, when x = 91, y = 21:
21 = a(91)^2 - 130a(91) + 4225a + 21
0 = 8281a - 11830a + 4225a
0 = 14676a
To solve for 'a', we divide both sides by 14676:
0 = a
Therefore, the constant 'a' is 0.
Substituting this value into the equation in factored form, we get:
y = 0(x^2 - 130x + 4225) + 21
y = 21
So, the function that represents the path of the golf ball in factored form is:
y = 21
Note: Since the coefficient 'a' is 0, it means there is no variation in the height of the golf ball. The factored form represents a horizontal line at a height of 21m, indicating that the golf ball maintains a constant height of 21m throughout its flight.
when jaxon hit a golf ball off the grass with a 9-iron it travelled 130m horizontally until it landed back on the grass. the height of the ball was measured to be 21m when it reached a horizontal distance of 91m. the flight of the golf ball can be represented by quadratic function.
- determine the function that represents the path of the golf ball in factored form. round your value for (a) to the nearest thousandth. rewrite your function in standard form.
To determine the function that represents the path of the golf ball in factored form, we start with the vertex form equation:
y = a(x - h)^2 + k
Given the information about the flight of the golf ball, we know that:
- The golf ball traveled 130m horizontally until it landed back on the grass.
- The height of the ball was measured to be 21m when it reached a horizontal distance of 91m.
Since the vertex form of the quadratic function is y = a(x - h)^2 + k, we can substitute the known values into the equation:
y = a(x - 65)^2 + 21
Now we need to solve for the constant 'a'. Substituting the point (91, 21) into the equation:
21 = a(91 - 65)^2 + 21
0 = a(26)^2
0 = 676a
To solve for 'a', divide both sides by 676:
a = 0
Therefore, the constant 'a' is 0. This means that the quadratic function does not have an "a" term, indicating that the golf ball does not follow a parabolic path but instead moves in a straight line.
Since a = 0, the function in factored form becomes:
y = 0(x - 65)^2 + 21
y = 21
To rewrite the function in standard form, we expand the equation:
y = 0(x^2 - 130x + 4225) + 21
y = 0x^2 - 0x + 0 + 21
y = 0x^2 + 0x + 21
Therefore, the function in standard form is:
y = 0x^2 + 0x + 21
y = 21
The standard form of the function confirms that the golf ball follows a straight horizontal path, maintaining a constant height of 21m throughout its flight.
yes, we assumed k = 21, not true.
Apologies for the mistake. Let's recalculate to find the correct function in factored form.
Given:
- The ball traveled a horizontal distance of 130m until it landed.
- The height of the ball was measured to be 21m when it reached a horizontal distance of 91m.
We have the equation in vertex form: y = a(x - h)^2 + k
To find the function in factored form, we need to determine the values of 'a', 'h', and 'k'.
From the given information:
At x = 91, y = 21
Substituting this into the equation, we get:
21 = a(91 - h)^2 + k
Now, let's use another point to find 'a'. At x = 0, y = 0 since the ball starts from the ground.
0 = a(0 - h)^2 + k
0 = a(-h)^2 + k
0 = ah^2 + k
Since we have two equations with two variables (a and k), let's solve for a and k simultaneously.
From the equation 0 = ah^2 + k, we can rewrite it as:
k = -ah^2
Now, substitute this value of k into the first equation.
21 = a(91 - h)^2 + (-ah^2)
21 = a(8281 - 182h + h^2) - ah^2
21 = 8281a - 182ah + ah^2 - ah^2
21 = 8281a - 182ah
Simplifying further:
0 = 8281a - 182ah - 21
We have found a quadratic equation in terms of 'h' that we can solve to find the value of 'h' and subsequently 'a'.
To solve this equation, we need more information or additional constraints to proceed. Please provide any other relevant details or constraints if available.
what is factored form
Apologies for the confusion in my previous response. The factored form of a quadratic equation is expressed as the product of two binomials. It is represented as:
y = a(x - r1)(x - r2)
where 'r1' and 'r2' are the roots or x-intercepts of the quadratic equation.
To determine the function that represents the path of the golf ball in factored form, we'll first find the roots of the quadratic equation using the given information:
Given:
- The ball traveled a horizontal distance of 130m until it landed back on the grass.
- The height of the ball was measured to be 21m when it reached a horizontal distance of 91m.
Since the ball reaches a height of 21m when the horizontal distance is 91m, we have the point (91, 21) on the graph.
From this point, we can deduce that one of the roots is x = 91, since the height is 21m at that horizontal distance.
To find the other root, we know that the ball landed back on the grass after a total horizontal distance of 130m. Therefore, the other root must be at x = 130.
So, the two roots of the quadratic equation are x = 91 and x = 130.
Now, we can write the quadratic equation in factored form:
y = a(x - 91)(x - 130)
To find the value of 'a,' we can substitute the point (0, 0) into the equation:
0 = a(0 - 91)(0 - 130)
0 = (-91)(-130)a
0 = 11830a
Therefore, a = 0.
Substituting this value of 'a' into the equation in factored form, we get:
y = 0(x - 91)(x - 130)
Simplifying further, the function that represents the path of the golf ball in factored form is:
y = 0
Note that the function y = 0 represents a horizontal line at y = 0. This indicates that the golf ball does not change height throughout its flight, maintaining a constant height of 0 units above the ground.