Answer:
Options (1) and (5)
Step-by-step explanation:
Expression that defines the function is,
![f(x)=\frac{1}{2}x+\frac{3}{2}](https://tex.z-dn.net/?f=f%28x%29%3D%5Cfrac%7B1%7D%7B2%7Dx%2B%5Cfrac%7B3%7D%7B2%7D)
Option 1
![f(-\frac{1}{2})=\frac{1}{2}(-\frac{1}{2})+\frac{3}{2}](https://tex.z-dn.net/?f=f%28-%5Cfrac%7B1%7D%7B2%7D%29%3D%5Cfrac%7B1%7D%7B2%7D%28-%5Cfrac%7B1%7D%7B2%7D%29%2B%5Cfrac%7B3%7D%7B2%7D)
![=-\frac{1}{4}+\frac{3}{2}](https://tex.z-dn.net/?f=%3D-%5Cfrac%7B1%7D%7B4%7D%2B%5Cfrac%7B3%7D%7B2%7D)
![=-\frac{1}{4}+1+\frac{1}{2}](https://tex.z-dn.net/?f=%3D-%5Cfrac%7B1%7D%7B4%7D%2B1%2B%5Cfrac%7B1%7D%7B2%7D)
![=1+\frac{1}{4}](https://tex.z-dn.net/?f=%3D1%2B%5Cfrac%7B1%7D%7B4%7D)
![=1\frac{1}{4}](https://tex.z-dn.net/?f=%3D1%5Cfrac%7B1%7D%7B4%7D)
So,
is false.
Option 2
f(0) = ![\frac{1}{2}(0)+\frac{3}{2}](https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7B2%7D%280%29%2B%5Cfrac%7B3%7D%7B2%7D)
= ![\frac{3}{2}](https://tex.z-dn.net/?f=%5Cfrac%7B3%7D%7B2%7D)
True.
Option 3
f(1) = ![\frac{1}{2}(1)+\frac{3}{2}](https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7B2%7D%281%29%2B%5Cfrac%7B3%7D%7B2%7D)
= ![\frac{3+1}{2}](https://tex.z-dn.net/?f=%5Cfrac%7B3%2B1%7D%7B2%7D)
= 2
Therefore, f(1) = -1 is false.
Option 4
![f(2)=\frac{1}{2}(2)+\frac{3}{2}](https://tex.z-dn.net/?f=f%282%29%3D%5Cfrac%7B1%7D%7B2%7D%282%29%2B%5Cfrac%7B3%7D%7B2%7D)
![=1+1+\frac{1}{2}](https://tex.z-dn.net/?f=%3D1%2B1%2B%5Cfrac%7B1%7D%7B2%7D)
![=2\frac{1}{2}](https://tex.z-dn.net/?f=%3D2%5Cfrac%7B1%7D%7B2%7D)
Therefore, f(2) = 1 is false.
Option 5
f(4)
![=2+\frac{3}{2}](https://tex.z-dn.net/?f=%3D2%2B%5Cfrac%7B3%7D%7B2%7D)
![=\frac{4+3}{2}](https://tex.z-dn.net/?f=%3D%5Cfrac%7B4%2B3%7D%7B2%7D)
![=\frac{7}{2}](https://tex.z-dn.net/?f=%3D%5Cfrac%7B7%7D%7B2%7D)
True.
Options (1) and (5) are the correct options.
- Vertex form:
, with (h,k) as the vertex.
For this, we will be using vertex form. Firstly, plug the vertex into the vertex form equation:
![y=a(x-(-3))^2+1\\y=a(x+3)^2+1](https://tex.z-dn.net/?f=%20y%3Da%28x-%28-3%29%29%5E2%2B1%5C%5Cy%3Da%28x%2B3%29%5E2%2B1%20)
Next, we need to solve for a. Plug in (-2,4) into the x and y coordinates to solve for a as such:
![4=a(-2+3)^2+1\\4=a(1)^2+1\\4=a+1\\3=a](https://tex.z-dn.net/?f=%204%3Da%28-2%2B3%29%5E2%2B1%5C%5C4%3Da%281%29%5E2%2B1%5C%5C4%3Da%2B1%5C%5C3%3Da%20)
Putting our equation together,<u> it's
</u>
<u>*Additional section*</u>
- Standard form:
![y=ax^2+bx+c](https://tex.z-dn.net/?f=%20y%3Dax%5E2%2Bbx%2Bc%20)
Converting to standard form as such:
![y=3(x+3)^2+1\\y=3(x^2+6x+9)+1\\y=3x^2+18x+27+1\\y=3x^2+18x+28](https://tex.z-dn.net/?f=%20y%3D3%28x%2B3%29%5E2%2B1%5C%5Cy%3D3%28x%5E2%2B6x%2B9%29%2B1%5C%5Cy%3D3x%5E2%2B18x%2B27%2B1%5C%5Cy%3D3x%5E2%2B18x%2B28%20)
Assuming the car's speed
does not change, the car will travel
miles.
Hope this helps :)