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Graph the following function. Remember to plot and label the vertex, y-intercept, and its reflection. Below your sketch, define whether there is a Maximum/Minimum and the range.

f(x) = -x^2 + 6x - 6

User Czetsuya
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1 Answer

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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].

Graph the following function. Remember to plot and label the vertex, y-intercept, and-example-1
User Saccharine
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