Download Ch. 2 Polynomial and Rational Functions 2.1 Quadratic Functions

Ch. 2 Polynomial and Rational Functions
2.1 Quadratic Functions
Polynomial functions are classified by degree:
Constant function f(x) = c degree 0
Linear function f(x) = ax + b, a≠ 0 degree 1
Quadratic function f(x) = ax2 + bx + c, a≠ 0 degree 2
* the graph of a quadratic function is a parabola
Parabolas
­ all parabolas are symmetric with respect to the axis of symmetry
(vertical line)
­ the vertex is the point where the axis intersects the parabola
­ the minimum and maximum points on a parabola occur at
the vertex
Standard Form of a Quadratic Function
f(x) = a(x­h)2 + k, a≠ 0
­ axis of symmetry is the vertical line x = h
­ vertex is the point (h, k)
­ if a > 0, the graph opens upward
­ if a < 0, the graph opens downward
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Sketching Graphs of Quadratic Functions
recall: y = f(x± c), or y = f(x)± c, or y = f(­x), or y= ­ f(x) are all RIGID transformations of the graph of y = f(x) b/c they do not change the basic shape of the graph
y = af(x), a≠ 1 is NONRIGID
* if the |a| is small, the parabola opens more widely than
if |a| is large
ex.
Sketch f(x) = 2x2 +8x +7 and find the vertex
and the axis of symmetry.
Rewrite the equation in standard form by completing
the square.
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Finding the x­intercepts (zeros) of a qudaratic function
­ if the function can't be factored, then use the quadratic formula
Writing the Equation of a Parabola
ex.
Write the standard form of the equation of the parabola whose vertex is (1, 2) and that passes thru (0, 0).
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* some quadratic functions aren't easily written in
standard form, so it is a good idea to know how
to find the vertex another way:
­ the x­coordinate of the vertex is:
ex.
Given: y = 3x2 + 4x ­1
a) Find the vertex
b) Write the equation in standard form
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Ex.
A baseball is hit at a point 3 feet above the ground at a velocity of 100
feet per second and at an angle of 45o with respect to the ground. The
path of the baseball is given by f(x) = ­0.0032x2 + x + 3,
where f(x) is the height of the ball and x is the horizontal distance from
home plate.
a) Find the maximum height of the baseball.
b) How far does the baseball travel horizontally?
* see graph below after solving
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HW
pg. 142
#2­8(E), 30, 38, 74,
80, 88, 90, 94
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