Functions and Their Graphs
Introduction
Consider the function ᡘ䙦ᡶ䙧 = ᡶ⡰ − 6ᡶ + 5. The following information allows us to graph it.
1. By inspection, we see that this is a quadratic function (or 2nd degree polynomial function). Its graph is a parabola that opens upwards.
2. We also know that the ᡷ -intercept is ᡷ = 5. The gives us the point on the graph (0, 5).
3. This function may be factored as ᡘ䙦ᡶ䙧 = 䙦ᡶ − 1䙧䙦ᡶ − 5䙧, which gives us zeros
at ᡶ = 1 and at ᡶ = 5. These give us points on the graph (1, 0) and (5, 0).
4. Furthermore, the axis of symmetry is ᡶ = ⡹〩⡰〨 = ⡹䙦⡹⡴䙧⡰䙦⡩䙧 = 3. This is also the ᡶ – value of the vertex;
the ᡷ – value of the vertex is ᡘ䙦3䙧 = 3⡰ − 6䙦3䙧 + 5 = −4. Thus, the vertex is located at (3,−4).
We can also find the vertex by completing the square. Here ᡘ䙦ᡶ䙧 = ᡶ⡰ − 6ᡶ + 5 = 䙦ᡶ − 3䙧⡰ − 4, and we see that the vertex is at (3,−4). [Recall that ᡘ䙦ᡶ䙧 = ᡓ䙦ᡶ − ℎ䙧⡰ + ᡣ has vertex located at (ℎ, ᡣ).]
Putting this information together allows us to graph the function.
From the graph, we can answer these questions:
a) Where is ᡘ䙦ᡶ䙧 decreasing? b) Where is ᡘ䙦ᡶ䙧 increasing?
From 䙦−∞,3䙧. From 䙦3,∞䙧.
Let’s now consider the new function ᡙ䙦ᡶ䙧 = ᡘ䙦ᡶ䙧 + 4.
The effect of adding 4 to the function ᡘ䙦ᡶ䙧 is to shift the graph up four units. Simplifying, we see that ᡙ䙦ᡶ䙧 = 䙦ᡶ⡰ − 6ᡶ + 5䙧 + 4
= ᡶ⡰ − 6ᡶ + 9
= 䙦ᡶ − 3䙧⡰
This function has a double root, or a root of multiplicity two, which
occurs at ᡶ = 3.
The function is still decreasing on the interval 䙦−∞,3䙧 and increasing
on the interval 䙦3,∞䙧.
The local minimum occurs at ᡶ = 3.
Complete and turn in the following four problems.
