Question

Use graphing technology to find the range of the function f(x)=-x^2+6x-3

Answer 1

To find the range of f(x) = -x^2 + 6x - 3, calculate the vertex of the quadratic function. As the parabola opens downwards, the range of the function are all y-values below or equal to the y-coordinate of the vertex. Hence, the range is y ≤ -3.

Step-by-step explanation:

To find the range of a function using graphing technology, you need to start by finding the vertex of the function because the vertex gives us the maximum or minimum point of a parabola, which is crucial in determining the range. In this case, the function given is f(x) = -x^2 + 6x - 3, which is a quadratic function in the form of f(x)= ax^2 + bx + c. The graph of this function is a parabola which opens downwards because the coefficient of x² is negative.

The vertex (h, k) of a quadratic function f(x) = a(x-h)² + k can be calculated using the formula h = -b/2a. In our function, a = -1 and b = 6. Substituting these in the formula, we get h = -6/(-2) = 3. To find k (the y-coordinate of the vertex), we substitute x = 3 into our function: f(3) = -(3)² + 6*3 - 3 = -3. Thus, the vertex of the function is at (3, -3).

Since the parabola opens downwards, the range of the function will be all the y-values that are lower than or equal to the y-coordinate of the vertex. So, the range of the function f(x) = -x^2 + 6x - 3 will be y ≤ -3.

Learn more about Range of a Function

Answer 2

The range of the function is [-∞,6]

Step-by-step explanation:

Range of a Function

It's the set of values the function takes when x moves into the function's domain.

It can be graphically determined by placing imaginary horizontal lines from minus infinite to plus infinite. Where the line meets the function, that value of y belongs to the range.

The given function is:

`f(x)=-x^2+6x-3`

Using a graphing tool, we draw the graph shown in the image below.

It can be clearly seen that if we imagine horizontal lines from above y=6 to infinity, they won't ever touch the graph of the function. From y=6 and below, every horizontal line will touch the function one time or twice.

We can conclude the range of the function is [-∞,6]