Signment
Complete the square to write the given function as and find the vertex.

Write the equation in vertex form.

Ask by Zimmerman Nunez. in the United States

Mar 20,2025

Question

Signment Complete the square to write the given function as $$ f(x)=a(x-h)^{2}+k $$ and find the vertex. $$ f(x)=x^{2}+4 x-8 $$ Write the equation in vertex form. $$ f(x)=\square $$

UpStudy AI · Accepted Answer

1. **Identify the quadratic and constant parts:**  
   The function is  
   $$
   f(x)=x^2+4x-8.
   $$  
   We focus on rewriting $$x^2+4x$$ by completing the square.

2. **Complete the square for the quadratic part:**  
   Take the coefficient of $$x$$, which is 4, halve it, and square the result:  
   $$
   \left(\frac{4}{2}\right)^2 = 2^2 = 4.
   $$  
   Add and subtract this value inside the function:  
   $$
   f(x)=x^2+4x+4-4-8.
   $$

3. **Rewrite as a perfect square:**  
   The expression $$x^2+4x+4$$ is a perfect square:  
   $$
   x^2+4x+4=(x+2)^2.
   $$  
   Substitute it back in:  
   $$
   f(x)=(x+2)^2-4-8.
   $$

4. **Simplify the constant terms:**  
   Combine $$-4$$ and $$-8$$:  
   $$
   -4-8=-12.
   $$  
   Thus,  
   $$
   f(x)=(x+2)^2-12.
   $$

5. **Determine the vertex:**  
   The vertex form of a quadratic is  
   $$
   f(x)=a(x-h)^2+k,
   $$  
   where $$(h,k)$$ is the vertex. Here,  
   $$
   f(x)=(x+2)^2-12=(x-(-2))^2-12.
   $$  
   So, the vertex is at $$(-2,-12)$$.

6. **Final Answer:**  
   $$
   f(x)=(x+2)^2-12.
   $$
$$ f(x) = (x + 2)^2 - 12 $$

Signment
Complete the square to write the given function as and find the vertex.

Write the equation in vertex form.

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Signment Complete the square to write the given function as and find the vertex. Write the equation in vertex form.