A quadratic in vertex form shows its turning point through a square plus a constant. For a minimum, graph the function in Desmos and click its vertex, the lowest point when the number multiplying the square is positive. Read the output coordinate, not the input where the minimum occurs. A square cannot be negative, which explains why that point is lowest.
Hints
- Hint 1
A parabola is the U-shaped graph of a squared expression. A positive number multiplying the square makes it open upward, so its turning point is the lowest point. Where would you click to find that point?
- Hint 2
A point on the graph of is written . The minimum value is the smallest output, not the input that produces it. Once Desmos shows the lowest point, which coordinate should you report?
Step-by-step
Approach 1: Graph and read the lowest point
- Step 1
Identify the point you need
The number multiplying the square is positive, so the graph is an upward-opening parabola, a U-shaped curve. Its vertex, or turning point, is its lowest point.
- Step 2
Read the minimum output
Type in Desmos. Click the graph, then click its lowest marked point. Desmos shows . The minimum value is the vertex’s output, not its input. So the minimum value of is . Choice B.
Approach 2: Use the square to prove the minimum
- Step 1
Bound the squared term
A square is never negative. Multiplying it by the positive number keeps it nonnegative: .
- Step 2
Find when the square is smallest
The square reaches its smallest value, , when its inside is . Set the inside to zero: . Add : .
- Step 3
Find the output at that input
Substitute : . Simplify inside the square: . Evaluate: . The squared term cannot go below , so no output can fall below this reached value. The minimum value of is . Choice B.