Question
the maximum or minimum value of a parabola
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ANSWER
Answers
GPT-4o mini
To determine the maximum or minimum value of a parabola, we first need to understand its general form. A parabola can be represented by the equation:
\[ y = ax^2 + bx + c \]
where:
- \( a \), \( b \), and \( c \) are constants,
- \( a \) determines the direction of the parabola (if \( a > 0 \), the parabola opens upward and has a minimum value; if \( a < 0 \), it opens downward and has a maximum value).
### Finding the Vertex
The vertex of the parabola, given by the coordinates \( (h, k) \), is where the maximum or minimum value occurs. The \( x \)-coordinate of the vertex can be found using the formula:
\[ h = -\frac{b}{2a} \]
After calculating \( h \), you can substitute this value back into the original equation to find \( k \), which is the maximum or minimum value:
\[ k = a(h)^2 + b(h) + c \]
### Summary
- If \( a > 0 \), the parabola has a **minimum value** at the vertex.
- If \( a < 0 \), the parabola has a **maximum value** at the vertex.
### Example
Consider the parabola defined by the equation \( y = 2x^2 - 4x + 1 \).
1. Here, \( a = 2 > 0 \), so it opens upwards and has a minimum value.
2. Calculate \( h \):
\[
h = -\frac{-4}{2 \times 2} = \frac{4}{4} = 1
\]
3. Now calculate \( k \):
\[
k = 2(1)^2 - 4(1) + 1 = 2 - 4 + 1 = -1
\]
Thus, the minimum value of the parabola is \( -1 \) at \( x = 1 \).
\[ y = ax^2 + bx + c \]
where:
- \( a \), \( b \), and \( c \) are constants,
- \( a \) determines the direction of the parabola (if \( a > 0 \), the parabola opens upward and has a minimum value; if \( a < 0 \), it opens downward and has a maximum value).
### Finding the Vertex
The vertex of the parabola, given by the coordinates \( (h, k) \), is where the maximum or minimum value occurs. The \( x \)-coordinate of the vertex can be found using the formula:
\[ h = -\frac{b}{2a} \]
After calculating \( h \), you can substitute this value back into the original equation to find \( k \), which is the maximum or minimum value:
\[ k = a(h)^2 + b(h) + c \]
### Summary
- If \( a > 0 \), the parabola has a **minimum value** at the vertex.
- If \( a < 0 \), the parabola has a **maximum value** at the vertex.
### Example
Consider the parabola defined by the equation \( y = 2x^2 - 4x + 1 \).
1. Here, \( a = 2 > 0 \), so it opens upwards and has a minimum value.
2. Calculate \( h \):
\[
h = -\frac{-4}{2 \times 2} = \frac{4}{4} = 1
\]
3. Now calculate \( k \):
\[
k = 2(1)^2 - 4(1) + 1 = 2 - 4 + 1 = -1
\]
Thus, the minimum value of the parabola is \( -1 \) at \( x = 1 \).