To determine the intervals of increasing and decreasing for a quadratic function, we can analyze the function in vertex form or standard form. The general form of a quadratic function is:
\[ f(x) = ax^2 + bx + c \]
where \( a \) determines the direction of the parabola (opening upwards if \( a > 0 \) and downwards if \( a < 0 \)).
Step 1: Find the Vertex
The vertex of the quadratic function can be determined using the vertex formula:
\[ x = -\frac{b}{2a} \]
This x-value of the vertex is where the function changes from increasing to decreasing (or vice versa).
Step 2: Determine Intervals
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If the parabola opens upwards (\( a > 0 \)), the function will be:
- Decreasing on the interval \((- \infty, -\frac{b}{2a})\)
- Increasing on the interval \((- \frac{b}{2a}, \infty)\)
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If the parabola opens downwards (\( a < 0 \)), the function will be:
- Increasing on the interval \((- \infty, -\frac{b}{2a})\)
- Decreasing on the interval \((- \frac{b}{2a}, \infty)\)
Example
Assuming you have a specific quadratic function, you can plug in the values of \( a \) and \( b \) to find the intervals.
Conclusion
To provide specific intervals, please provide the coefficients \( a \) and \( b \) for the quadratic function you are analyzing. Then, intervals of increasing/decreasing can be specified accurately.