Answer:
1) C
2) C
Step-by-step explanation:
Question 1)
We want a parabola that has the vertex (-8, -7) and also passes through the point (-7, -4).
So, we can use the vertex form. Remember that the vertex form is:
![y=a(x-h)^2+k](https://tex.z-dn.net/?f=y%3Da%28x-h%29%5E2%2Bk)
Where a is our leading co-efficient and (h, k) is our vertex.
We know that our vertex is (-8, -7). So, substitute this into the equation:
![y=a(x-(-8))^2+(-7)](https://tex.z-dn.net/?f=y%3Da%28x-%28-8%29%29%5E2%2B%28-7%29)
Simplify:
![y=a(x+8)^2-7](https://tex.z-dn.net/?f=y%3Da%28x%2B8%29%5E2-7)
Now, we need to determine the value of our a. To do so, we can use the point the problem had given us. We know that the graph passes through (-7, -4).
So, when x is -7, y is -4. Substitute -7 for x and -4 for y:
![(-4)=a((-7)+8)^2-7](https://tex.z-dn.net/?f=%28-4%29%3Da%28%28-7%29%2B8%29%5E2-7)
Solve for a. Add within the parentheses:
![-4=a(1)^2-7](https://tex.z-dn.net/?f=-4%3Da%281%29%5E2-7)
Square:
![-4=a-7](https://tex.z-dn.net/?f=-4%3Da-7)
Add 7 to both sides. Therefore, the value of a is:
![a=3](https://tex.z-dn.net/?f=a%3D3)
So, our entire equation in vertex form is:
![y=3(x+8)^2-7](https://tex.z-dn.net/?f=y%3D3%28x%2B8%29%5E2-7)
Our answer is C.
Question 2)
We are given a graph and are asked to find the equation of the graph.
Again, let's use the vertex form. From the graph, we can see that the vertex is at (-2, 2). Let's substitute this into our vertex equation:
![y=a(x-(-2))^2+2](https://tex.z-dn.net/?f=y%3Da%28x-%28-2%29%29%5E2%2B2)
Simplify:
![y=a(x+2)^2+2](https://tex.z-dn.net/?f=y%3Da%28x%2B2%29%5E2%2B2)
Again, we need to find the value of a.
Notice that the graph crosses the point (-1, 5).
So, let's substitute -1 for x and 5 for y. This yields:
![5=a((-1)+2)^2+2](https://tex.z-dn.net/?f=5%3Da%28%28-1%29%2B2%29%5E2%2B2)
Solve for a. Add within the parentheses.
![5=a(1)^2+2](https://tex.z-dn.net/?f=5%3Da%281%29%5E2%2B2)
Square:
![5=a+2](https://tex.z-dn.net/?f=5%3Da%2B2)
Subtract 2 from both sides. So, the value of a is:
![a=3](https://tex.z-dn.net/?f=a%3D3)
Therefore, our entire equation is:
![y=3(x+2)^2-2](https://tex.z-dn.net/?f=y%3D3%28x%2B2%29%5E2-2)
Our answer is C.
And we're done!