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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

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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

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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)

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((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

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(-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.

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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

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yy = = x x 22yy = 2 = 2x x 22

yy = = 11//2 2 x x 22

aa > 0 > 0

opening upopening up

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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.

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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

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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-

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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:

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f,21

21

f,21

41

2- ,21

41

2- ,21

2ab-

h2ab-

h

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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)

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2-,21

41

2-,21

yy = = x x 22 - - xx - - 22

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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-

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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)

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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,

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xx

yy

33

99

(3, 0)(3, 0)

(0, 9)(0, 9)

yy = = x x 22 - 6 - 6xx + + 99

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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!

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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

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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

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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

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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

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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

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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

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The end.The end.