Find the minimum and maximum value of the function y=−(x−4)^2+9. Enter infinity or -infinity if the function never stops increasing or decreasing.
This is a parabola, and since they gave you the equation in vertex form, you can find the vertex, which will be at (-h, k). The vertex here, then, is at (4, 9).
Since there's a negative sign out in front, we know this parabola opens down, meaning 9 is the highest y value the function will ever reach.
And because this is a parabola, it will just keep going down forever, so the lowest y value it will ever reach is negative infinity.
Nice answer !
To find the minimum and maximum values of the function y = -(x-4)^2 + 9, we can start by analyzing the equation.
The function y = -(x-4)^2 + 9 is in the form of a quadratic equation. In this form, the coefficient of x^2 is negative, which means the graph of the function will be a downward-opening parabola.
To find the minimum and maximum values, we need to locate the vertex of the parabola. The vertex is the point where the function reaches its highest or lowest point, depending on the direction of the parabola.
In general, the vertex of a quadratic function in the form y = a(x-h)^2 + k is given by the coordinates (h, k). In our case, the vertex can be obtained by observing that h = 4 and k = 9.
Therefore, the vertex of the parabola is (4, 9).
Since the parabola is downward-opening, the y-coordinate of the vertex represents the maximum value of the function. Hence, the maximum value of the function is y = 9.
However, there is no minimum value since the function is a downward-opening parabola and never stops decreasing. In other words, as x approaches infinity or negative infinity, the value of y continues to decrease without bound.
So, the maximum value of the function is 9, and there is no minimum value (which can be expressed as negative infinity).