Use a graphing utility to graph the function and approximate (to two decimal places) any relative minima or maxima. (If an answer does not exist, enter DNE.)
f(x) = −x2 + 9x − 8
http://www.wolframalpha.com/input/?i=-x%5E2%2B9x-8
To graph the function f(x) = −x^2 + 9x − 8 and approximate any relative minima or maxima, you can use a graphing utility such as Desmos or GeoGebra. Here's how you can do it using Desmos:
1. Open a web browser and go to www.desmos.com.
2. Click on the "Graphing Calculator" option to open the graphing utility.
3. In the equation input box, enter the function f(x) = −x^2 + 9x − 8.
4. Once you enter the function, the graph of the function will automatically appear on the screen.
Now let's approximate any relative minima or maxima:
To find the relative minima or maxima, we need to locate the highest or lowest point on the graph of the function. In this case, since the coefficient of x^2 is negative (-1), the graph of the function will be concave down, indicating a maximum point.
To approximate the maximum point, you can follow these steps:
1. Look for the highest point on the graph.
2. Zoom in or out as needed to get a clear view of the graph.
3. If necessary, you can adjust the x and y-axis scales to get a better approximation.
4. Once you have a clear view of the graph, find the highest point and read the x-coordinate and y-coordinate.
In the case of the function f(x) = −x^2 + 9x − 8, the maximum point occurs at the vertex of the parabola. The x-coordinate of the vertex can be found using the formula x = -b/2a, where a is the coefficient of x^2 and b is the coefficient of x.
In this case, a = -1 and b = 9. So, the x-coordinate of the vertex is x = -9/(2*(-1)) = -9/(-2) = 4.5.
To find the y-coordinate of the vertex, substitute the x-coordinate back into the equation f(x) = −x^2 + 9x − 8. So, f(4.5) = -(4.5)^2 + 9*(4.5) - 8 = 3.25.
Therefore, the maximum point on the graph is approximately (4.5, 3.25).