Question
If \( a>0 \) and \( f(x)=a(x-h)^{2}+k \), then \( k \) is the ___ value of the function.
Ask by Lewis Estrada. in the United States
Feb 25,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
\( k \) is the minimum value of the function.
Solution
In the vertex form of a quadratic function, f(x) = a(x - h)² + k, the vertex is the point (h, k). When a > 0, the graph of the quadratic function opens upward, meaning that the vertex is the lowest point on the graph. Hence, k is the minimum value of the function.
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Simplify this solution Extra Insights
If \( a>0 \) and \( f(x)=a(x-h)^{2}+k \), then \( k \) is the minimum value of the function. This is because the graph of a quadratic function opens upwards when the coefficient \( a \) is positive, which means that the vertex (\( h, k \)) represents the lowest point on the graph. The vertex form of a quadratic function is not just useful for identifying the minimum (or maximum) value; it also allows you to quickly sketch the graph. By knowing the vertex \( (h, k) \), you can easily plot the turning point and determine the direction the parabola opens, providing a beautiful picture of how the function behaves!
