A quadratic in vertex form has a squared part plus a constant, so its turning point is visible without expanding. When the square has a positive coefficient, its smallest possible contribution is , making the constant the minimum output. Keep the input that makes the square separate from the output the question asks for.
Hints
- Hint 1
A square is never negative, whether the number inside it is positive, negative, or zero. What is the smallest value the squared part of this function can have?
- Hint 2
The square reaches zero when the expression inside it is zero. At that input, what does the constant outside the square do to the function’s output?
Step-by-step
Read the minimum from vertex form
Step 1Find the square’s smallest contribution
The function is already in vertex form, which shows its lowest output. A square is never negative:
So the function’s lowest output occurs when that square reaches .
- Step 2
Find the lowest output
At , the inside of the square is . Graph in Desmos and click its lowest point, the vertex. Desmos shows , so the output there is .
The minimum value is the output, not the input where it occurs. The minimum value of is . Grid in -5.