Complete the square of the given quadratic expression. Then, graph the function using the technique of shifting.
\[
f(x)=x^{2}+8 x+13
\]
Answer
\(\boxed{\text{Final Answer: The completed square form of the given quadratic expression is } (x+4)^2 - 3 \text{ and the graph of the function is a parabola that opens upwards with the vertex at the point } (-4, -3).}\)
Step 1 ：Given the quadratic expression \(f(x)=x^{2}+8 x+13\), we are to complete the square and graph the function.
Step 2 ：First, we rewrite the quadratic expression in the form of \((x-h)^2 + k\). The value of \(h\) is given by \(-\frac{b}{2a}\) and the value of \(k\) is \(f(h)\) where \(f(x)\) is the given quadratic expression. In this case, \(a=1\), \(b=8\) and \(c=13\).
Step 3 ：Calculating the values of \(h\) and \(k\), we get \(h = -4.0\) and \(k = -3.0\).
Step 4 ：We can now rewrite the quadratic expression as \((x+4)^2 - 3\).
Step 5 ：Next, we graph the function using the technique of shifting. The vertex of the graph will be at the point \((-4, -3)\) and the graph will open upwards since \(a=1\).
Step 6 ：\(\boxed{\text{Final Answer: The completed square form of the given quadratic expression is } (x+4)^2 - 3 \text{ and the graph of the function is a parabola that opens upwards with the vertex at the point } (-4, -3).}\)
