Question
find the extreme value of the following quadratic function by completing the square
f(x)=x^2-4x+35/9
a. 2
b. -2
c. 1/9
d. -1/9
f(x)=x^2-4x+35/9
a. 2
b. -2
c. 1/9
d. -1/9
Answers
GPT 3.5
To find the extreme value of the quadratic function \(f(x) = x^2 - 4x + \frac{35}{9}\) by completing the square, we will follow these steps:
1. Rewrite the function in the form \(f(x) = A(x - h)^2 + k\).
2. Find the vertex \(V(h,k)\) which corresponds to the extreme value.
Given function: \(f(x) = x^2 - 4x + \frac{35}{9}\)
Step 1:
\(f(x) = x^2 - 4x + \frac{35}{9}\)
\(f(x) = (x^2 - 4x) + \frac{35}{9}\)
\(f(x) = (x^2 - 4x + 4) + \frac{35}{9} - 4\)
\(f(x) = (x - 2)^2 + \frac{35}{9} - \frac{36}{9}\)
\(f(x) = (x - 2)^2 - \frac{1}{9}\)
Step 2:
Given the function is now in the form \(f(x) = A(x - h)^2 + k\) where \(A = 1\), \(h = 2\), and \(k = -\frac{1}{9}\).
The extreme value occurs at the vertex \((h,k) = (2, -\frac{1}{9})\).
Therefore, the extreme value of the quadratic function is the value of \(f(x)\) at the vertex, which is -1/9.
Correct answer: d. -1/9
1. Rewrite the function in the form \(f(x) = A(x - h)^2 + k\).
2. Find the vertex \(V(h,k)\) which corresponds to the extreme value.
Given function: \(f(x) = x^2 - 4x + \frac{35}{9}\)
Step 1:
\(f(x) = x^2 - 4x + \frac{35}{9}\)
\(f(x) = (x^2 - 4x) + \frac{35}{9}\)
\(f(x) = (x^2 - 4x + 4) + \frac{35}{9} - 4\)
\(f(x) = (x - 2)^2 + \frac{35}{9} - \frac{36}{9}\)
\(f(x) = (x - 2)^2 - \frac{1}{9}\)
Step 2:
Given the function is now in the form \(f(x) = A(x - h)^2 + k\) where \(A = 1\), \(h = 2\), and \(k = -\frac{1}{9}\).
The extreme value occurs at the vertex \((h,k) = (2, -\frac{1}{9})\).
Therefore, the extreme value of the quadratic function is the value of \(f(x)\) at the vertex, which is -1/9.
Correct answer: d. -1/9