Answer:
The line of symmetry is x = -3
Explanation:
Given a quadratic function, we know that the graph is a parabola. The general form of a parabola is:
![y=ax^2+bx+c](https://tex.z-dn.net/?f=y%3Dax%5E2%2Bbx%2Bc)
The line of symmetry coincides with the x-axis of the vertex. To find the x-coordinate of the vertex, we can use the formula:
![x_v=-\frac{b}{2a}](https://tex.z-dn.net/?f=x_v%3D-%5Cfrac%7Bb%7D%7B2a%7D)
In this problem, we have:
![y=-x^2-6x-13](https://tex.z-dn.net/?f=y%3D-x%5E2-6x-13)
Then:
a = -1
b = -6
We write now:
Part 3:For this part, we need to find the x-intercepts. This is, when y = 0:
![-x^2-6x-13=0](https://tex.z-dn.net/?f=-x%5E2-6x-13%3D0)
To solve this, we can use the quadratic formula:
![x_{1,2}=\frac{-(-6)\pm\sqrt{(-6)^2-4\cdot(-1)\cdot(-13)}}{2(-1)}](https://tex.z-dn.net/?f=x_%7B1%2C2%7D%3D%5Cfrac%7B-%28-6%29%5Cpm%5Csqrt%7B%28-6%29%5E2-4%5Ccdot%28-1%29%5Ccdot%28-13%29%7D%7D%7B2%28-1%29%7D)
And solve:
![x_{1,2}=\frac{6\pm\sqrt{36-52}}{-2}](https://tex.z-dn.net/?f=x_%7B1%2C2%7D%3D%5Cfrac%7B6%5Cpm%5Csqrt%7B36-52%7D%7D%7B-2%7D)
![x_{1,2}=\frac{-6\pm\sqrt{-16}}{2}](https://tex.z-dn.net/?f=x_%7B1%2C2%7D%3D%5Cfrac%7B-6%5Cpm%5Csqrt%7B-16%7D%7D%7B2%7D)
Since there is no solution to the square root of a negative number, the function does not have any x-intercept. The correct option is ZERO x-intercepts.
Part 4:To find the y intercept, we need to find the value of y when x = 0:
![y=-0^2-6\cdot0-13=-13](https://tex.z-dn.net/?f=y%3D-0%5E2-6%5Ccdot0-13%3D-13)
The y-intercept is at (0, -13)
Part 5:
Now we need to find two points in the parabola. Let-s evaluate the function when x = 1 and x = -1:
![x=1\Rightarrow y=-1^2-6\cdot1-13=-1-6-13=-20](https://tex.z-dn.net/?f=x%3D1%5CRightarrow%20y%3D-1%5E2-6%5Ccdot1-13%3D-1-6-13%3D-20)
![x=-1\Rightarrow y=-(-1)^2-6\cdot(-1)-13=-1+6-13=-8](https://tex.z-dn.net/?f=x%3D-1%5CRightarrow%20y%3D-%28-1%29%5E2-6%5Ccdot%28-1%29-13%3D-1%2B6-13%3D-8)
The two points are:
(1, -20)
(-1, -8)
Part 6:
Now, we can use 3 points to find the graph of the parabola.
We can locate (1, -20) and (-1, -8)
The third could be the vertex. We need to find the y-coordinate of the vertex. We know that the x-coordinate of the vertex is x = -3
Then, y-coordinate of the vertex is:
![y=-(-3)^2-6(-3)-13=-9+18-13=-4](https://tex.z-dn.net/?f=y%3D-%28-3%29%5E2-6%28-3%29-13%3D-9%2B18-13%3D-4)
The third point we can use is (-3, -4)
Now we can locate them in the cartesian plane:
And that's enough to get the full graph: