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Unformatted text preview: Math 110 FS 2007
CH 3.4—3.5 Quadratic Functions. Lectures #11 and #12 ‘3' General and Standard forms of the equation of Quadratic
Functions. Deﬁnition.
f(x) 2 cm2 + bx + c, where a, b, c are real numbers, a i 0 is called quadratic function (or second degree polynomial
function) in general form. f (x) = a(x — h)2 + k is a standard form of the equation of a quadratic function, (11,16) is the vertex. . Graphs of Quadratic Functions. Problem #1. Indicate how the graph of f(x) = (x — 3)2 — 2 is
related to the graph of basic function f (x) = x2. Sketch
the graph of y = (x — 3)2 — 2. State the Range of f Math 110 FS 2007
CH 3.4—3.5 Quadratic Functions. Lectures #11 and #12 Problem #2.
Indicate how the graph of f (x) = — (x — 3)2 — 2 is 2 related to the graph of the basic function f (x) = x . Sketch the graph of y = — (x — 3)2 — 2.
State the Range of f. Question. When a quadratic function has a maximum
value? Minimum value? Math 110 FS 2007
CH 3.4—3.5 Quadratic Functions. Lectures #11 and #12 . Properties of a Quadratic Function.
f(x) = atx2 + bx + c, a ¢ 0  general form f(x) = a(x — hf + k — standard form, (h, k) is vertex.
1. Domain: ( —oo, oo) 2. Vertex is the highest point on the parabola if a < 0
(quadratic function has a max value),
and it is the lowest point on the parabola if a < O
(quadratic function has a min value), 3. Range: (— oo, yvmex] if a < O; [yvmw 00) if a > 0. 4. Quadratic function increases on one of the vertex and
decreases on the other. 5. Axis of symmetry: x = h (parallel to x—axis). 6. Intercepts. Math 110 FS 2007 CH 3.4—3.5 Quadratic Functions. Lectures #11 and #12
Problem #3.
3 2
For f(x) = 2(x —Zj — 8 ﬁnd the following:
a) Vertex. b) Equation of the axis of symmetry.
0) y intercept.
d) x — intercepts. e) Maximum/minimum.
t) Range. g) Using the info above sketch the graph of f (x) Math 110 FS 2007
CH 3.4—3.5 Quadratic Functions. Lectures #11 and #12 . Conversion general form of a quadratic function into
standard form (completing square procedure). Problem #4.
Convert the function f (x) = ——2x2 + 5x — 5 into standard form f(x)=a(x—h)2+k.  Coordinates of the vertex when a quadratic function is
given by an equation in general form. For f(x)=ax2+bx+c, a¢0, the vertex is [—3, fﬂ—LD
2a 2a Problem #5.
Find the vertex for y =— 3x2 + 6x — 5. State the Range for y = f (x). Math 110 FS 2007
CH 3.43.5 Quadratic Functions. Lectures #11 and #12 . Applications. Optimization. Verbal problems involving optimization require to
model functions based on. given or implied
conditions. If the resulting function is quadratic, we can use
the x—coordinate of the vertex to determine the
maximum or minimum value of the function. Note: When xcoordinate of the vertex is used to
determine maximum or minimum value of a
quadratic function, you need to present a reason,
why given function has a maximum (minimum)
value. As the reason can be used info about the
coefﬁcient a in the equation of the quadratic
function. If the coefficient a > 0, parabola opens upward,
the function has a minimum value. If the coefﬁcient a < O, parabola opens
downward, the function has a maximum value. Math 110 FS 2007
CH 3.43.5 Quadratic Functions. Lectures #11 and #12 I Enclosing the Most Area with a Fence.
Problem #6. A farmer with 2000 meters of fencing wants to
enclose a rectangular plot that borders on a straight
highway. If the farmer does not fence the side along
highway, what is the largest area that can be
enclosed? Problem #7. A rancher has 600 yards of fencing to put around a
rectangular field and then subdivide the field into two
identical plots by placing a fence parallel to field's
shorter sides. Find the dimensions that maximize
enclosed area. What is the maximum area? Math 110 FS 2007
CH 3.43.5 Quadratic Functions. Lectures #11 and #12 Problem #8. Shannise Cole makes and sell candy. She has found
that the cost per box for making x boxes of candy is
given by C(x) = x2 —40x+405. a) How much does it cost per box to make 15 boxes? b) What point on the graph corresponds to the
number of boxes that will make the cost per box
as small as possible? c) How many boxes should she make in order to keep
the cost per box at a minimum?
What is the minimum cost per box? Problem #9. A bullet is ﬁred upward from the ground level. Its height above the ground (in feet) at time t seconds
is given by H = 4612 +100t Find the maximum height of the bullet and the time
at which it hits the ground. Math 110 FS 2007
CH 3.4—3.5 Quadratic Functions. Lectures #11 and #12 Problem #10 (#12 p. 177 textbook). Find the equilibrium point and equilibrium price for
the commodity whose supply and demand functions
are given. Supply function: p = q 2 + q +10 Demand: p = — qu + 3060 Problem #11. The number of women employed fulltime in civilian
jobs has increased dramatically in the past century.
The following table shows the number of employed
women (in millions) in selected years. IYear 1910 1930 1940 1950 1960 1970 1980 10662007) Working 7.4 10.8 12.8 18.6 23.6 31.5 45.5 56.8 68.6
Women a) Display this info graphically.
b) Find a quadratic regression model for the data. c) Use the regression model to estimate the number
of women in the workforce in 2009. ...
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 Spring '08
 WORMER

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