Identify the vertex, the axis of symmetry, the maximum or minimum value, and the domain and range of function
f(x) = -(x-9)^2
Identify the vertex.
the coordinates of the vertex are
(the an ordered pair)
its (9, -20)
If the coordinates of the vertex are (9, -20), then we have:
- Vertex: (9, -20)
- Axis of symmetry: x = 9
- Maximum or minimum value: Since the leading coefficient is negative, the parabola opens downwards and the vertex represents the maximum point of the function. Therefore, the maximum value is f(9) = -20.
- Domain: All real numbers
- Range: The range is bounded by the vertex, since the maximum value is at the vertex. Therefore, the range is (-∞, -20].
To identify the vertex of the function f(x) = -(x-9)^2, we can observe that the function is in the form f(x) = a(x-h)^2 + k, where (h, k) represents the vertex.
In this case, we have f(x) = -(x-9)^2. Comparing this with the general form, we can see that h = 9. Therefore, the x-coordinate of the vertex is 9.
To find the y-coordinate of the vertex, we substitute x = 9 into the equation:
f(9) = -(9-9)^2
f(9) = -(0)^2
f(9) = 0
So, the y-coordinate of the vertex is 0.
Hence, the vertex of the function is (9, 0).
To identify the vertex of the function f(x) = -(x-9)^2, we can start by rewriting the function in vertex form, which is given by f(x) = a(x-h)^2 + k, where (h,k) represents the coordinates of the vertex.
In our case, the function is already in the form f(x) = -(x-9)^2, so we can directly identify the vertex.
From the equation, we see that h = 9, which corresponds to the x-coordinate of the vertex.
However, since the coefficient in front of the squared term is negative, we know that the parabola opens downwards, indicating that the vertex is a maximum point rather than a minimum.
So, the coordinates of the vertex are (9, f(9)). To find the y-coordinate, we can substitute x = 9 into the function:
f(9) = -(9-9)^2 = -(0)^2 = 0
Therefore, the vertex of the function is at (9, 0).
Next, let's identify the axis of symmetry. The axis of symmetry is a vertical line that passes through the vertex and divides the parabola into two symmetrical halves. For a parabola in the form f(x) = a(x-h)^2 + k, the equation of the axis of symmetry is given by x = h.
In our case, the x-coordinate of the vertex is h = 9. Therefore, the equation of the axis of symmetry is x = 9.
Moving on, let's determine the maximum or minimum value of the function based on its vertex. Since the parabola opens downwards and the vertex is located at the highest point, the function has a maximum value.
Therefore, the maximum value of the function f(x) = -(x-9)^2 is 0, which occurs at x = 9.
Finally, let's find the domain and range of the function.
The domain of a function is the set of all possible x-values for which the function is defined. In this case, the function is a quadratic polynomial, and there are no restrictions on the x-values. Therefore, the domain is all real numbers, or (-∞, ∞).
The range of a function is the set of all possible y-values that the function takes on. Since the function is in the form f(x) = -(x-9)^2, we can observe that the y-values will be negative or zero, as the function is always multiplied by -1. So, the range is (-∞, 0].
To summarize:
- The vertex of the function is (9, 0).
- The axis of symmetry is x = 9.
- The maximum value of the function is 0, which occurs at x = 9.
- The domain is (-∞, ∞).
- The range is (-∞, 0].
The vertex is at (9, 0).
Identify the axis of symmetry.
The axis of symmetry is the vertical line that passes through the vertex. Therefore, the axis of symmetry is the vertical line x = 9.
Identify the maximum or minimum value.
Since the leading coefficient is negative, the parabola opens downwards and the vertex represents the maximum point of the function. Therefore, the maximum value is f(9) = 0.
Identify the domain and range of the function.
The domain is all real numbers, since there are no restrictions on the values of x that can be inputted into the function. The range, however, is bounded by the vertex, since the maximum value is at the vertex. Therefore, the range is (-∞, 0].