Could you please help me with this question: A diver dives from the 3 m board at a swimming pool. Her height, y, in metres, above the water in terms of her horizontal distance, x, in metres, from the end of the board is given by y= -x2 + 2x + 3. What is the diver's maximum height?
-b over 2a to find x value for max and add that to the 3 metres
If a diver dives from the 2-meter board at a swimming pool. Her height, y, in meters above the water in terms of her horizontal distance, x, in meters, from the end of the board is given by y = -x^2 + 2x + 3. What is the diver's maximum height?
To find the maximum height of the diver, we need to determine the vertex of the given quadratic equation. The vertex represents the highest or lowest point on the curve, depending on whether the quadratic opens upwards or downwards.
The quadratic equation representing the height of the diver is:
y = -x^2 + 2x + 3
To find the coordinates (x, y) of the vertex, we can use the formula:
x = -b/2a
Where a, b, and c are the coefficients of the quadratic equation in the standard form ax^2 + bx + c.
In our case, a = -1, b = 2, and c = 3. Plugging these values into the formula, we have:
x = -(2)/(2*(-1))
= -2/(-2)
= 1
Now, to find the value of y at x = 1, we substitute it back into the equation:
y = -(1)^2 + 2(1) + 3
= -1 + 2 + 3
= 4
Therefore, the vertex of the quadratic equation is (1, 4). The x-coordinate represents the horizontal distance, and the y-coordinate represents the height.
Hence, the diver's maximum height is 4 meters.