Key Features of functions
Summary:
x-intercept: When the value of the function touches the x-axis, so \(y = 0\)
y-intercept: When the value of the function touches the y-axis, so \(x=0\)
Absolute minimum: When the value of the function is at its lowest point over the whole domain
Absolute maximum: When the value of the function is at its highest point over the whole domain
Relative minimum: When the function goes from decreasing to increasing, causing the point to be the lowest in an interval around it.
Relative maximum: When the function goes from increasing to decreasing, causing the point to be the highest in an interval around it.
For domain and range see Domain and Range
In the first example, we will identify absolute and relative minimums and maximums from a graph. This is quite simple since it is entirely visual. Note that Absolute maximums do not exist when a function is increasing infinately in either direction, and absolute minimums do not exist when a function is decreasing infinately in either direction.
Example 1:
Find the absolute minimum and relative minimums of the following graph. Also find the relative maximum.
We can see that before the point \((5, -1)\) the function is decreasing and after it the function is increasing, therefore it is a local minimum. However, we can see that there are lower points in other parts of the graph, so this is not the absolute minimum.
We can also see that before the point \((9.5, 4)\) the function is increasing and after it the function is decreasing, therefore it is a local maximum. However, we can see that the function increases infinately as \(x\) increases towards infinity, so there is no absolute maximum.
Finally, we see that \((19, -6)\) is the lowest point on the graph so it is both the absolute minimum and a relative minimum.
Therefore, as our final answer, we get:
Relative minimums: \((5, -1), (19, -6)\)
Absolute minimum: \((19, -6)\)
Relative maximum: \((9.5, 4)\)
In the next example, we will find the \(y\) intercept and \(x\) intercepts of a function without graphing it. To find the \(x\) intercepts, we will set \(y\) equal to zero and solve for \(x\). To find the y intercept, we will set \(x\) equal to zero and solve for \(y\)
Example 2:
Find the \(y\) intercept and \(x\) intercepts of \(y = x^2-x-6\)
First, lets find the \(x\) intercept. To do this, we will set \(y\) equal to 0, and get: $$x^2-x-6 = 0$$ We can then factor the expression to get: $$(x-3)(x+2) = 0$$ Finally, we get that \(y = 0\) when \(x = 3\) or\(-2\), so the \(x\) intercepts are \((3, 0)\) and \((-2, 0)\). When \(x= 0\), we get \(y=-6\), so the \(y\) intercept is \((0, -6)\)
Conlcusion:
You can now find absolute and relative minimums and maximums of a function without a graph, and the \(x\) and \(y\) intercepts of a function without the graph
Next: Direct Variation