Final answer:
- -In this graph, the vertex is (3, 3).
- - The y-intercept is (0, -6).
- - The reflection of the vertex is (3, -3).
- - The graph is a downward-facing parabola.
- - The function has a maximum value at the vertex.
- - The range is (-∞, 3].
Step-by-step explanation:
To graph the function f(x) = -x² + 6x - 6, we can follow these steps:
Step 1: Find the vertex.
The vertex of a quadratic function can be found using the formula x = -b / (2a), where a and b are coefficients of the quadratic term and linear term, respectively. In this case, a = -1 and b = 6.
x = -b / (2a)
x = -6 / (2(-1))
x = -6 / -2
x = 3
To find the corresponding y-coordinate, substitute the x-value back into the function:
f(3) = -(3)² + 6(3) - 6
f(3) = -9 + 18 - 6
f(3) = 3
Therefore, the vertex of the function is (3, 3).
Step 2: Find the y-intercept.
The y-intercept is the point where the function intersects the y-axis. To find it, substitute x = 0 into the function:
f(0) = -(0)² + 6(0) - 6
f(0) = 0 + 0 - 6
f(0) = -6
Therefore, the y-intercept is (0, -6).
Step 3: Plot the points and sketch the graph.
Using the vertex (3, 3) and the y-intercept (0, -6), we can plot these points on the coordinate plane. Additionally, we can plot the reflection of the vertex, which is (3, -3), since the coefficient of the quadratic term is negative.
Next, we can draw a smooth curve that passes through these points, forming a downward-facing parabola. The graph should be symmetrical with respect to the vertical line passing through the vertex.
Step 4: Determine the maximum/minimum and the range.
Since the coefficient of the quadratic term is negative, the parabola opens downward, indicating a maximum. Therefore, the function has a maximum value at the vertex.
The range of the function can be determined by looking at the y-values of the graph. In this case, the graph is a downward-facing parabola with a maximum value of 3. Thus, the range is (-∞, 3].