Question

Consider the function f left parenthesis x right parenthesis equals 2 left parenthesis x minus 5 right parenthesis squared minus 3.
​(a) Give the coordinates of the vertex of the graph of the function.
​(b) Graph the function on a window that includes the vertex.
Question content area bottom
Part 1
​(a) The vertex of the function is
  
enter your response here. ​(Type an ordered​ pair.)
Part 2
​(b) Choose the correct graph below.
A.



x y graph
B.



x y graph
C.



x y graph
D.



x y graph
​[minus​10,10] by ​[minus​10,10], Xsclequals​1, Ysclequals1

Answer 1

To analyze the function \( f(x) = 2(x - 5)^2 - 3 \), we'll follow the prompts for both parts (a) and (b):

### Part 1: Finding the Vertex

The function is given in vertex form \( f(x) = a(x - h)^2 + k \), where \( (h, k) \) is the vertex of the parabola.

For the function \( f(x) = 2(x - 5)^2 - 3 \):

- \( h = 5 \)
- \( k = -3 \)

Thus, the vertex of the function is \( (5, -3) \).

**Answer for Part 1: The vertex of the function is \( (5, -3) \).**

### Part 2: Graphing the Function

To graph the function on a window that includes the vertex, we will note the following:

- The vertex is at \( (5, -3) \).
- The parabola opens upwards since the coefficient of the squared term (which is 2) is positive.
- As you plot the vertex, you may also want to check some other points by plugging in values for \( x \) in the function \( f(x) \) to see how it behaves around the vertex.

Here are a few points you might evaluate:
- \( f(4) = 2(4 - 5)^2 - 3 = 2(1) - 3 = -1 \)
- \( f(6) = 2(6 - 5)^2 - 3 = 2(1) - 3 = -1 \)
- \( f(3) = 2(3 - 5)^2 - 3 = 2(4) - 3 = 8 - 3 = 5 \)
- \( f(7) = 2(7 - 5)^2 - 3 = 2(4) - 3 = 8 - 3 = 5 \)

This shows that the parabola reaches \( (4, -1) \) and \( (6, -1) \) on either side of the vertex, and rises again to \( (3, 5) \) and \( (7, 5) \).

### Choosing the Correct Graph

You want to find the graph that includes the vertex \( (5, -3) \) and has the general shape of a parabola opening upwards.

Since I don't have visual access to graphs A, B, C, or D, please check each graph to find one that matches these criteria:
- The vertex at \( (5, -3) \)
- The parabola opens upwards
- Other points around the vertex follow the calculated values.

Once you do that, you'll be able to select the correct graph representing the function \( f(x) = 2(x - 5)^2 - 3 \).