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To determine the behavior of a function, the concept of the derivative plays a central role. In simple terms, the derivative of a function gives us the rate at which the function changes. It essentially tells us how steep a function is or how flat it is at any given point.
When we say "taking the derivative," we refer to finding the derivative function. For the function given in the problem, which is \[ h(t) = t^2 + 2t - 3 \], the derivative \( h'(t) \) was found to be \( 2t + 2 \). This was done using the power rule, which states that the derivative of \( t^n \) is \( nt^{n-1} \), and the constant rule, where the derivative of a constant is 0.
Hence, if you can understand these rules, you open the door to understanding much of calculus.

Critical points help determine key features of functions, like peaks, valleys, or any point where the behavior might change. These points occur where the derivative equals zero or is undefined. In our example, since \( h'(t) = 2t + 2 \) is defined everywhere, we're only interested in where it's zero.
We set the equation \( 2t + 2 = 0 \) and solve for \( t \), leading to \( t = -1 \). This simple equation means that at \( t = -1 \), the slope of the tangent line to \( h(t) \) is zero. It's the tipping point where the curve might switch from increasing to decreasing or vice versa.
This simplicity helps when analyzing functions, ensuring we have a good overview of their behavior.

An increasing function is essentially moving upwards as you progress along the x-axis. Mathematically, for a function to be increasing over an interval, its derivative must be positive on that interval.
After finding critical points, we test intervals around these points. For the function \( h(t) = t^2 + 2t - 3 \), we found that when \( t \) is greater than \(-1\), \( h'(t) = 2t + 2 \) is positive. This positive value indicates the function is increasing in the interval of \((-1, \infty)\).
Understanding intervals where the function increases helps in plotting and analyzing the function's overall growth.

A decreasing function behaves oppositely to an increasing one. It appears to slope downward as you move right along the x-axis. For this to happen, the derivative of the function should be negative on the specific interval.
In our example, by examining the derivative \( h'(t) = 2t + 2 \), we understood that for \( t < -1 \), \( h'(t) < 0 \). This means the function \( h(t) = t^2 + 2t - 3 \) decreases wherever \( t \) is less than \(-1\).
This analysis across different intervals highlights how the function drops off as it moves along, adding a more complete understanding of its overall behavior.