Linear Function
Graphing
is equivalent to graphing the equation 
,
which is a straight line.
Recall that
is the slope of the line and 
is
the y-intercept of the line.If the slope is positive, the line is rising going from left to right, if the slope
is negative the line is falling from left to right, if the slope is zero the line is
horizontal.
See Example 2, page 227.
Quadratic Function
The graph of a quadratic function is called a parabola.
If
the parabola opens up. 
If
the parabola opens down.
The vertex V(h,K) is the highest point or the lowest point on the parabola..
If the parabola opens up, the vertex is the lowest point of the parabola.
If the parabola opensdown, the vertex is the highest point of the parabola.
The vertical line going though the vertex,
,
is called the axis of symmetry of the parabola. It divides the parabola into two symmetric halves, like a straight line
through the center of a circle.
To find the coordinates of the vertex, the quadratic function must be written in the
standard form
.When the function is not given in standard, we must complete the squares to rewrite
it in standard form in order to read off the coordinates of the vertex, V(h,k).
Example
Put
in standard form.First, factor out
as follows,
Complete the squares for the expression within the parentheses,
.Multiplying using the distributive law, everything inside the parentheses
is multiplied by
.In particular,
which is not in the function, so we must cancel it in order to not change the function.
Outside of the parentheses we subtract it away.
is in standard form.Read off the coordinates of the vertex, V(-3, -19).
V(-3, -19)
If the parabola opens up, the vertex is the lowest point on the graph.
Then the y-coordinate of the vertex gives the minimum value of the function.
Here the minimum value is -19.
If the parabola down, the vertex is the highest point on the graph.
Then the y-coordinate of the vertex gives the maximum value of the function.
Example
The parabola opens down, the vertex is V(5/2, 11/2),
and the maximum value of the function is 11/2.
See the example on pages 228 - 229, and Example 3, page 231.
These graphs are examples of graphs of continuous functions.
A graph is continuous if it can be completely drawn without lifting
the pencil off the paper.
To draw the graph
the pencil must be lifted in order to go from the left branch to the right branch.
To draw the complete graph, we must jump from the third quadrant into the
first quadrant. The graph is not continuous. The graph is discontinuous.
A function that is defined by using different formulas for different parts
of its domain is called a piecewise-defined function.
Example
