1
2
Any function of the formAny function of the form
y = f (x) = ax 2 + bx + cy = f (x) = ax 2 + bx + c
where a 0 is called a Quadratic Functionwhere a 0 is called a Quadratic Function
3
Example:Example:
yy = 3 = 3x x 22 - 2 - 2xx + + 11yy = 3 = 3x x 22 - 2 - 2xx + + 11
aa = 3,= 3, bb = -2,= -2, cc = 1= 1
Note that if Note that if aa = 0 = 0 we simply have the we simply have the linear function linear function
yy = = bxbx + + cc
4
Consider the simplest quadratic Consider the simplest quadratic equationequation
yy = = x x 22Here Here aa = 1, = 1, bb = 0, = 0, cc = 0= 0Plotting some ordered pairs (Plotting some ordered pairs (xx, , yy) we have:) we have:
yy = = f f ((x x ) = ) = x x 22
xx f f ((x x ) () (xx, , y y ))
-3 9 (-3, -3 9 (-3, 9)9)-2 4 (-2, -2 4 (-2, 4)4)-1 1 (-1, -1 1 (-1, 1)1)0 0 (0, 0)0 0 (0, 0) 1 1 (1, 1 1 (1, 1)1) 2 4 (2, 2 4 (2, 4)4) 3 9 (3, 3 9 (3, 9)9)
5
((xx, , yy))
(-3, 9)(-3, 9)(-2, 4)(-2, 4)(-1, (-1, 1)1)(0, 0)(0, 0)(1, 1)(1, 1)(2, 4)(2, 4)
(3, 9)(3, 9)
xx
yy
-3 -2 -1 1 2 3-3 -2 -1 1 2 3
44
33
22
11
(-2, 4)(-2, 4)
Vertex (0, 0)Vertex (0, 0)
(2, 4)(2, 4)
y = xy = x22
A parabola with the A parabola with the yy-axis as the axis of -axis as the axis of symmetry.symmetry.
6
Graphs of Graphs of yy = = ax ax 22 will have similar form will have similar form and the value of the coefficient ‘and the value of the coefficient ‘a a ’ ’ determines the graph’s shape.determines the graph’s shape.
xx
yy
-3 -2 -1 1 2 3-3 -2 -1 1 2 3
44
33
22
11
yy = = x x 22yy = 2 = 2x x 22
yy = = 11//2 2 x x 22
aa > 0 > 0
opening upopening up
7
xx
yy
yy = -2 = -2x x 22
aa < 0 < 0
opening downopening down
In general the quadratic term In general the quadratic term ax ax 22 in the in the
quadratic function quadratic function f f ((x x ) = ) = ax ax 22 + +bxbx + + cc
determines the way the graph opens. determines the way the graph opens.
8
Consider Consider f f ((x x ) = ) = ax ax 22 + +bxbx + + cc In a general sense the linear term In a general sense the linear term bxbx acts to acts to shift the plot of shift the plot of f f ((x x ) from side to side and the ) from side to side and the constant term constant term cc (= (=cx cx 00) acts to shift the plot up ) acts to shift the plot up or down.or down.
xx
yy
cc
aa > 0 > 0cc
aa < 0 < 0
yy-intercept-intercept
xx-intercept-intercept
Notice thatNotice that cc is is the the y y --intercept intercept where where xx = 0 = 0 and and f f (0) = (0) = cc
Note also that the Note also that the x x -intercepts (if they exist) -intercepts (if they exist) are obtained by solving: are obtained by solving: yy = = ax ax 22 + +bx bx + + cc = 0 = 0
9
It turns out that the details of a It turns out that the details of a quadratic function can be found by quadratic function can be found by considering its coefficients considering its coefficients aa, , bb and and cc as as follows:follows:(1) Opening up ((1) Opening up (aa > 0), down ( > 0), down (aa < 0) < 0)
(2) (2) y y –intercept: –intercept: cc
(3) (3) x x -intercepts from solution of -intercepts from solution of
yy = = ax ax 22 + + bxbx + + cc = 0 = 0
(4) v(4) vertertex = ex =
You solve by factoring You solve by factoring or the quadratic or the quadratic formulaformula
2ab-
f,2ab-
10
Example:Example: yy = = f f ((x x ) = ) = x x 22 - - xx - 2 - 2
here here aa = 1, = 1, bb = -1 and = -1 and cc = = -2-2(1) opens upwards since (1) opens upwards since aa > 0 > 0
(2) (2) y y –intercept: -2–intercept: -2
(3) (3) x x -intercepts from -intercepts from x x 22 - - xx - 2 = 0 - 2 = 0
or (or (x x -2)(-2)(x x +1) = 0 +1) = 0
xx = 2 = 2 or or xx = -1 = -1
(4) vertex:(4) vertex:
21
f,21
21
f,21
41
2- ,21
41
2- ,21
2ab-
h2ab-
h
11
xx
yy
-2 -1 0 1 2-2 -1 0 1 2
-1-1
-2-2
-3-3
(-1, 0)(-1, 0) (2, 0)(2, 0)
41
2-,21
41
2-,21
yy = = x x 22 - - xx - - 22
12
Example:Example: yy = = j j ((x x ) = ) = x x 22 - 9 - 9
here here aa = 1, = 1, bb = 0 and = 0 and cc = = -9-9(1) opens upwards since (1) opens upwards since aa > 0 > 0
(2) (2) y y –intercept: -9–intercept: -9
(3) (3) xx -intercepts from -intercepts from x x 22 - 9 = 0 - 9 = 0
or or x x 22 = 9 = 9 xx = = 33
(4) vertex at (0, -9) (4) vertex at (0, -9)
2ab-
j,2ab-
13
xx
yy
-3 0 3-3 0 3
-9-9
(-3, 0)(-3, 0) (3, 0)(3, 0)
yy = = x x 22 - 9 - 9
(0, -9)(0, -9)
14
Example:Example: yy = = g g ((x x ) = ) = x x 22 - 6 - 6xx + 9 + 9
here here aa = 1, = 1, bb = -6 and = -6 and cc = = 99(1) opens upwards since (1) opens upwards since aa > 0 > 0
(2) (2) yy –intercept: 9 –intercept: 9
(3) (3) xx -intercepts from -intercepts from x x 22 - 6 - 6xx + 9 = 0 + 9 = 0
or (or (xx - 3)( - 3)(xx - 3) = 0 - 3) = 0 xx = 3 = 3 onlyonly
(4) vertex: (4) vertex:
3g3, 3g3, 2ab-
h2ab-
h 03, 03,
15
xx
yy
33
99
(3, 0)(3, 0)
(0, 9)(0, 9)
yy = = x x 22 - 6 - 6xx + + 99
16
Example:Example: yy = = f f ((x x ) = -3) = -3x x 22 + 6 + 6xx - 4 - 4
here here aa = -3, = -3, bb = 6 and = 6 and cc = = -4-4(1) opens downwards since (1) opens downwards since aa < 0 < 0
(2) (2) yy –intercept: -4 –intercept: -4
(3) (3) xx -intercepts from -3 -intercepts from -3x x 22 + 6 + 6xx - 4 = 0 - 4 = 0
(there are (there are nono x x -intercepts here)-intercepts here)
(4) vertex at (1, -(4) vertex at (1, -1) 1)
2ab-
f,2ab-
Vertex is below x-axis, and parabola opens Vertex is below x-axis, and parabola opens down!down!
17
xx
yy
1 21 2
-1-1
-4-4
(1, -1)(1, -1)
(0, -4)(0, -4)
yy = -3 = -3x x 22 + 6 + 6xx - 4 - 4
18
The Quadratic FormulaThe Quadratic Formula
It is not always easy to find It is not always easy to find x x -intercepts -intercepts by factoring by factoring ax ax 22 + + bxbx + + cc when solving when solving
ax ax 22 + + bxbx + + cc = = 00
Quadratic equations of this form can be Quadratic equations of this form can be solved for solved for xx using the formula: using the formula:
4acbb-x
2
2a
19
Example:Example: Solve Solve x x 22 − 6 − 6xx + 9 = 0 + 9 = 0here here aa = 1, = 1, bb = -6 and = -6 and cc = = 99
only 326
206
x
only 326
206
x
as found previouslyas found previously
2a4acbb-
x2
2(1)4(1)(9)(-6)(-6)-
x2
Note: the expression Note: the expression inside the radical is inside the radical is called the called the “discriminant”“discriminant”
Note: discriminant = Note: discriminant = 00 one solution one solution
20
Example:Example: Solve Solve x x 22 - - xx - 2 = 0 - 2 = 0
here here aa = 1, = 1, bb = -1 and = -1 and cc = = -2-2
)1(2
)2)(1(4)1()1(x
2
)1(2
)2)(1(4)1()1(x
2
2x or 1x231
x or 231
x
231
291
x
2x or 1x231
x or 231
x
231
291
x
Note: discriminant > Note: discriminant > 00 two solutions two solutions
21
Example:Example: Find Find x x -intercepts of -intercepts of yy = = x x 22 - - 99Solve Solve x x 22 - 9 = 0 - 9 = 0Solve Solve x x 22 - 9 = 0 - 9 = 0
aa = 1, = 1, bb = 0, = 0, cc = -9 = -9
2
)9(40x
2
)9(40x
26
236
x
26
236
x
xx = 3 or = 3 or xx = -3 = -3
Note: discriminant > Note: discriminant > 00 two solutions two solutions
22
Example:Example: Find the Find the x x -intercepts of -intercepts of
yy = = f f ((xx) = -3) = -3x x 22 + 6 + 6xx - 4 - 4
a a = -3, = -3, b b = 6 and = 6 and cc = = -4-4Solve Solve -3-3x x 22 + 6 + 6xx - 4 - 4 = 0= 0
a2ac4bb
x2
a2
ac4bbx
2
)3(2
)4)(3(4366x
)3(2
)4)(3(4366x
?6
126x
?6
126x
undefined is 12 undefined is 12
there are no there are no x x -intercepts as we -intercepts as we discovered in an earlier plot of discovered in an earlier plot of yy = -3 = -3x x 22 + + 66xx - 4 - 4
Note: discriminant < Note: discriminant < 00 no Real solutions no Real solutions
23
The end.The end.
Top Related