<u>We will proceed to find the equations in the vertex form in each case</u>
<u>Part a)</u> ![y= 4x^{2}+24x+38](https://tex.z-dn.net/?f=y%3D%204x%5E%7B2%7D%2B24x%2B38)
Group terms that contain the same variable, and move the constant to the opposite side of the equation
![y-38= 4x^{2}+24x](https://tex.z-dn.net/?f=y-38%3D%204x%5E%7B2%7D%2B24x)
Factor the leading coefficient
![y-38= 4(x^{2}+6x)](https://tex.z-dn.net/?f=y-38%3D%204%28x%5E%7B2%7D%2B6x%29)
Complete the square. Remember to balance the equation by adding the same constants to each side
![y-38+36= 4(x^{2}+6x+9)](https://tex.z-dn.net/?f=y-38%2B36%3D%204%28x%5E%7B2%7D%2B6x%2B9%29)
![y-2= 4(x+3)^{2}](https://tex.z-dn.net/?f=y-2%3D%204%28x%2B3%29%5E%7B2%7D)
![y= 4(x+3)^{2}+2](https://tex.z-dn.net/?f=y%3D%204%28x%2B3%29%5E%7B2%7D%2B2)
<u>the vertex is the point
</u>
<u>Part b)</u> ![y=4x^{2}-24x+38](https://tex.z-dn.net/?f=y%3D4x%5E%7B2%7D-24x%2B38)
Group terms that contain the same variable, and move the constant to the opposite side of the equation
![y-38= 4x^{2}-24x](https://tex.z-dn.net/?f=y-38%3D%204x%5E%7B2%7D-24x)
Factor the leading coefficient
![y-38= 4(x^{2}-6x)](https://tex.z-dn.net/?f=y-38%3D%204%28x%5E%7B2%7D-6x%29)
Complete the square. Remember to balance the equation by adding the same constants to each side
![y-38+36= 4(x^{2}-6x+9)](https://tex.z-dn.net/?f=y-38%2B36%3D%204%28x%5E%7B2%7D-6x%2B9%29)
![y-2= 4(x-3)^{2}](https://tex.z-dn.net/?f=y-2%3D%204%28x-3%29%5E%7B2%7D)
![y= 4(x-3)^{2}+2](https://tex.z-dn.net/?f=y%3D%204%28x-3%29%5E%7B2%7D%2B2)
<u>the vertex is the point
</u>
<u>Part c) </u> ![y=4x^{2}+12x+2](https://tex.z-dn.net/?f=y%3D4x%5E%7B2%7D%2B12x%2B2)
Group terms that contain the same variable, and move the constant to the opposite side of the equation
![y-2= 4x^{2}+12x](https://tex.z-dn.net/?f=y-2%3D%204x%5E%7B2%7D%2B12x)
Factor the leading coefficient
![y-2= 4(x^{2}+3x)](https://tex.z-dn.net/?f=y-2%3D%204%28x%5E%7B2%7D%2B3x%29)
Complete the square. Remember to balance the equation by adding the same constants to each side
![y-2+9= 4(x^{2}+3x+2.25)](https://tex.z-dn.net/?f=y-2%2B9%3D%204%28x%5E%7B2%7D%2B3x%2B2.25%29)
![y+7= 4(x+1.5)^{2}](https://tex.z-dn.net/?f=y%2B7%3D%204%28x%2B1.5%29%5E%7B2%7D)
![y= 4(x+1.5)^{2}-7](https://tex.z-dn.net/?f=y%3D%204%28x%2B1.5%29%5E%7B2%7D-7)
<u>the vertex is the point
</u>
<u>Part d)</u> ![y=4x^{2}+16x+13](https://tex.z-dn.net/?f=y%3D4x%5E%7B2%7D%2B16x%2B13)
Group terms that contain the same variable, and move the constant to the opposite side of the equation
![y-13= 4x^{2}+16x](https://tex.z-dn.net/?f=y-13%3D%204x%5E%7B2%7D%2B16x)
Factor the leading coefficient
![y-13= 4(x^{2}+4x)](https://tex.z-dn.net/?f=y-13%3D%204%28x%5E%7B2%7D%2B4x%29)
Complete the square. Remember to balance the equation by adding the same constants to each side
![y-13+16= 4(x^{2}+4x+4)](https://tex.z-dn.net/?f=y-13%2B16%3D%204%28x%5E%7B2%7D%2B4x%2B4%29)
![y+3= 4(x+2)^{2}](https://tex.z-dn.net/?f=y%2B3%3D%204%28x%2B2%29%5E%7B2%7D)
![y= 4(x+2)^{2}-3](https://tex.z-dn.net/?f=y%3D%204%28x%2B2%29%5E%7B2%7D-3)
<u>the vertex is the point
</u>
therefore
<u>the answer is </u>
![y= 4x^{2}+24x+38](https://tex.z-dn.net/?f=y%3D%204x%5E%7B2%7D%2B24x%2B38)