September 25, 2015
quadratic function
f(x) = ax + bx2 + c where a does not = 0; it's graph is in
the shape of a curve called a parabola
; ax is called the
quadratic term; bx is called the linear term; andc is the
constant
axis of symmetry
2-4
Analyzing Graphs of Quadratic Functions
the imaginary line a parabola is symmetric about
vertex
where the axis of symmetry intersects the parabola; the point at
which the parabola changes direction
maximum
the y-value of the vertex of a downward opening parabola
minimum
the y-value of the vertex of an upward opening parabola
Example 1
Find the vertex, axis of symmetry and the maximum or
2
minimum value of y = (x - 3) + 2. Then draw
its graph.
vertex form
y = a(x - h) + k
2
vertex: (h, k)
axis of symmetry: x = h
To find the vertex of a parabola:
1) plug into the formula,
in order to find the x-coordinate
2) plug THAT number into the original function to find the y-coordinate
Example 2
Find the vertex, axis of symmetry and the maximum or
minimum value of the following. Then draw its graph.
2 + 4
a) y = x + 2x
2
September 25, 2015
Example 2 - continued
Find the vertex, axis of symmetry and the maximum or
minimum value of the following. Then draw its graph.
b) y = -2x - 4x 2+ 2
2 + 23
a) f(x) = x + 10x
Example 4
Let's use the graphs to help us find the intervals on which each of these
functions is increasing or decreasing. Locate the maximum or minimum
value(s).
2 + 23
a) f(x) = x + 10x
Example 3 - on your own
Find the vertex, axis of symmetry and the maximum or minimum value
of the following. Then draw the graph. Write in vertex form.
b) f(x) = -x - 10x2 - 23
Example 6
A model rocket is launched with an initial velocity of 100 ft/sec from the
top of a hill that is 20 ft. high. Its height
t
seconds after it has been
launched is given by the function s(t) = -16t + 100t + 20. Determine 2the
time when the rocket reaches its maximum height and find that height.
b) f(x) = -x - 10x2 - 23
Example 5
A stone mason has enough stones to enclose a rectangular patio with
60 ft. of stone wall. If the house forms one side of the rectangle, what is
the maximum area that the mason can enclose? What should the
dimensions of the patio be in order to yield this area?