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MATH 31 LESSONS PreCalculus 8. Sketching Functions
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MATH 31 LESSONS PreCalculus 8. Sketching Functions.

Jan 20, 2016

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Page 1: MATH 31 LESSONS PreCalculus 8. Sketching Functions.

MATH 31 LESSONS

PreCalculus

8. Sketching Functions

Page 2: MATH 31 LESSONS PreCalculus 8. Sketching Functions.

A. Relations and Functions

Whenever one variable (y) is affected by another variable (x),

we say they form a relation.

A function is a special relation in which every x-value has

at most one y-value.

That is, if an x-value could have more than one y-value,

it is not a function.

All functions are relations, but not all relations are functions.

Page 3: MATH 31 LESSONS PreCalculus 8. Sketching Functions.

Vertical Line Test

If a relation is a function, then any vertical line can cross the graph at most once.

If the vertical line crosses the graph more than once,

then it is not a function.

Page 4: MATH 31 LESSONS PreCalculus 8. Sketching Functions.

e.g. Which of the following relations are functions?

Page 5: MATH 31 LESSONS PreCalculus 8. Sketching Functions.

This is a function.

The vertical lines never cross twice (or more).

Page 6: MATH 31 LESSONS PreCalculus 8. Sketching Functions.

This is not a function.

A vertical line can cross twice.

Page 7: MATH 31 LESSONS PreCalculus 8. Sketching Functions.

This is not a function.

A vertical line can cross twice.

Page 8: MATH 31 LESSONS PreCalculus 8. Sketching Functions.

This is a function.

The vertical lines never cross twice (or more).

Page 9: MATH 31 LESSONS PreCalculus 8. Sketching Functions.

B. Parabolas

A parabola is a function of the form

cbxaxy 2

Page 10: MATH 31 LESSONS PreCalculus 8. Sketching Functions.

Basic form

The basic (simplest) form

of a parabola is

2xy

Page 11: MATH 31 LESSONS PreCalculus 8. Sketching Functions.

Vertex: (0, 0)

Opens upward

2xy y

x

Page 12: MATH 31 LESSONS PreCalculus 8. Sketching Functions.

Standard form

The standard form (also called the completed-square form)

of a parabola is

or

khxay 2

2hxaky

Page 13: MATH 31 LESSONS PreCalculus 8. Sketching Functions.

Vertex: (h, k)

khxay 2y

x

k

h

V(h, k)

Page 14: MATH 31 LESSONS PreCalculus 8. Sketching Functions.

Vertex: (h, k)

Axis of symmetry: x = h

khxay 2y

x

x = h

V(h, k)

The axis of symmetry cuts the graph “in half”

Page 15: MATH 31 LESSONS PreCalculus 8. Sketching Functions.

Vertex: (h, k)

Axis of symmetry: x = h

Max / Min Value: y = k

khxay 2y

x

y = kV(h, k)

If the graph opens down, k is a maximum value.

If the graph opens up, k is a minimum value.

Page 16: MATH 31 LESSONS PreCalculus 8. Sketching Functions.

If a < 0, then the parabola

opens downward

e.g. y = - x2

y = -3 (x - 2)2 + 7

khxay 2y

x

Page 17: MATH 31 LESSONS PreCalculus 8. Sketching Functions.

If a > 0, then the parabola

opens upward

e.g. y = x2 + 11

y = 3 (x - 1)2 - 4

khxay 2y

x

Page 18: MATH 31 LESSONS PreCalculus 8. Sketching Functions.

khxay 2y

x

y = k

x = h

V(h, k)

Page 19: MATH 31 LESSONS PreCalculus 8. Sketching Functions.

Ex. 1 Find the vertex of

by completing the square.

Try this example on your own first.Then, check out the solution.

20243 2 xxy

Page 20: MATH 31 LESSONS PreCalculus 8. Sketching Functions.

20243 2 xxy

xxy 24320 2

Isolate the x-variables

Page 21: MATH 31 LESSONS PreCalculus 8. Sketching Functions.

20243 2 xxy

xxy 24320 2

xxy 8320 2

Factor out the coefficient of the squared term

Page 22: MATH 31 LESSONS PreCalculus 8. Sketching Functions.

20243 2 xxy

xxy 24320 2

xxy 8320 2

Determine the constant that is needed to make a perfect square, by squaring half of the middle (linear) term.

162

82

Page 23: MATH 31 LESSONS PreCalculus 8. Sketching Functions.

20243 2 xxy

xxy 24320 2

xxy 8320 2

168316320 2 xxy

Add the coefficient to both sides.

Don’t forget that the 16 is multiplied by 3 on the right side. You must do that on the left as well.

Page 24: MATH 31 LESSONS PreCalculus 8. Sketching Functions.

20243 2 xxy

xxy 24320 2

xxy 8320 2

168316320 2 xxy

24328 xyComplete the square

Page 25: MATH 31 LESSONS PreCalculus 8. Sketching Functions.

24328 xy

2843 2 xy

Put the parabola in standard form

Page 26: MATH 31 LESSONS PreCalculus 8. Sketching Functions.

Vertex: (-4, -28)

24328 xy

2843 2 xy

Page 27: MATH 31 LESSONS PreCalculus 8. Sketching Functions.

Ex. 2 Fully sketch

Identify key features of the graph, including vertex,

max/min, axis of symmetry, and intercepts.

Try this example on your own first.Then, check out the solution.

21102 xxy

Page 28: MATH 31 LESSONS PreCalculus 8. Sketching Functions.

Complete the square to put in standard form

21102 xxy

Page 29: MATH 31 LESSONS PreCalculus 8. Sketching Functions.

21102 xxy

xxy 1021 2

Isolate the x-variables

Page 30: MATH 31 LESSONS PreCalculus 8. Sketching Functions.

21102 xxy

xxy 1021 2

xxy 1021 2

Factor out the coefficient of the squared variable

Page 31: MATH 31 LESSONS PreCalculus 8. Sketching Functions.

21102 xxy

xxy 1021 2

xxy 1021 2

Find the constant required to make a perfect square:

252

102

Page 32: MATH 31 LESSONS PreCalculus 8. Sketching Functions.

21102 xxy

xxy 1021 2

xxy 1021 2

25102521 2 xxy

Add the constant to both sides.

Don’t forget that the 25 is actually negative, due to the coefficient in front of the brackets.

Page 33: MATH 31 LESSONS PreCalculus 8. Sketching Functions.

21102 xxy

xxy 1021 2

xxy 1021 2

25102521 2 xxy

254 xy

Complete the square

Page 34: MATH 31 LESSONS PreCalculus 8. Sketching Functions.

Vertex: (5, 4)

Opens downward (a < 0)

Axis of symmetry: x = 5

Max Value: y = 4

254 xy

45 2 xy Put in standard form.

Page 35: MATH 31 LESSONS PreCalculus 8. Sketching Functions.

Find intercepts

y-int: (x = 0)

21102 xxy

Page 36: MATH 31 LESSONS PreCalculus 8. Sketching Functions.

Find intercepts

y-int: (x = 0)

(0, -21)

21102 xxy

210100 2

21

Page 37: MATH 31 LESSONS PreCalculus 8. Sketching Functions.

x-int: (y = 0)

21102 xxy

21100 2 xx

Page 38: MATH 31 LESSONS PreCalculus 8. Sketching Functions.

x-int: (y = 0)

21102 xxy

21100 2 xx

021102 xx

Page 39: MATH 31 LESSONS PreCalculus 8. Sketching Functions.

x-int: (y = 0)

21102 xxy

21100 2 xx

021102 xx

073 xx

Page 40: MATH 31 LESSONS PreCalculus 8. Sketching Functions.

x-int: (y = 0)

(3, 0) (7, 0)

21102 xxy

21100 2 xx

021102 xx

073 xx

7,3x

Page 41: MATH 31 LESSONS PreCalculus 8. Sketching Functions.

Sketch

y

x

y = 4

x = 5

V(5, 4)

3 7