Answer:
vertex: (-5, 3)
Step-by-step explanation:
Besides providing the graph of the given quadratic equation, I will also demonstrate the algebraic method of solving for the vertex of a parabola.
Method 1: Algebraic Solution
Given the quadratic equation, y = -x² - 10x - 22, where:
a = -1, b = -10, and c = -22.
Find the x-coordinate of the vertex:
In order to find the vertex of the given equation, use the following formula to find the x-coordinate of the vertex: [tex]x = \frac{-b}{2a}[/tex].
[tex]x = \frac{-b}{2a}[/tex]
[tex]x = \frac{-(-10)}{2(-1)}[/tex]
[tex]x = \frac{10}{-2}[/tex]
x = -5
The x-coordinate of the vertex is: x = 5.
Find the y-coordinate of the vertex:
Next, substitute the value of the x-coordinate into the given quadratic equation to find its corresponding y-coordinate:
y = -x² - 10x - 22
y = -(-5)² - 10(-5) - 22
y = -25 + 50 - 22
y = 3
Therefore, the vertex of the given parabola is: (-5, 3), where it is the maximum point.
One of the attachments is the screenshot of the graphed quadratic equation (using Desmos), where it shows the vertex as the maximum point.
Method 2: Graphing Calculator
If you have a graphing calculator (such as the Texas Instruments graphing calculator):
- Press Y=
- Then, simply enter the given quadratic equation, -x² - 10x - 22.
- Next, press 2ND ⇒ Quit.
- Press the GRAPH button.
- Press 2ND ⇒ TRACE
- Select option 4: maximum
- It will ask for the "Left Bound?": move the cursor as close to the maximum point as possible
- Press ENTER.
- Next, it will ask for the "RightBound?" Repeat the same process as in step #7.
- Press ENTER.
- It will ask for "Guess?" ⇒ press ENTER, and it will provide the same coordinates for the vertex, (-5, 3).
Please let me know if you have any questions or concerns regarding this post.
