Maximum and minimum values are points that represent the highest and lowest y-values of a function within a domain. These points are useful for solving optimisation problems.
Consider the quadratic function $f(x) = -x^2 + 4x + 5$. Using a graphing calculator, we can determine that the maximum point occurs at (2, 9). This means the highest y-value the function reaches is 9, occurring when x = 2.
When using technology to find max/min values, always check if they are global (absolute) or local (relative) extrema. Some functions may have multiple local extrema but only one global maximum or minimum.
Intercepts are the points where a graph crosses the $x$ or $y$-axis.
For the function $g(x) = x^2 - 4x + 3$, using a graphing calculator reveals:
Alternatively, it should be obvious that
$$g(0)=3$$
and
$$g(x) = (x-3)(x-1)$$
So the $x$ and $y$ intercepts can also be found analytically.
Intercepts are particularly useful in analyzing the behavior of functions and solving equations graphically.
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