math hands
TRIGONOMETRY
graphing trig functions
math hands
CH 03 SEC 03 NOTES
THE IDEA We now contemplate the consequences on the graph of an equation when every ”x” in the equation is replaced by a ”−x”. As the title of this section suggests, the graph gets reflected. In this case it gets reflected about the y-axis, whereas if we replace every ”y” in an equation by a ”−y” then the graph gets reflected about the x axis. The most direct way to illustrated this is by observing a very simple equation, such as x = 4 Basic Equation: x = 4
-5
-4
-3
-2
Tweaked Equation: −x = 4 can also be written x = −4
5
5
4
4
3
3
2
2
1
1
-1
1
2
3
4
5
-5
-4
-3
-2
-1
1
-1
-1
-2
-2
-3
-3
-4
-4
-5
-5
2
3
4
5
The logic is penetrating and powerful, the result is clear, and the implications are far reaching. Indeed, not just for the above equation, but for all equations with x’s and y’s, replacing all such x’s with ”−x” always results in the reflecting or flipping of the graph across the y axis. Similarly, replacing y’s by −y results in a reflection across the x axis. EXAMPLES Parabolas Graph the following equation by tweaking the corresponding basic equation/graph. 2 −y = x
2 We start by observing which famous, basic equation − y = x is related to. In deed, this equation is just a tweak away from the basic parabola equation y = x2 , thus we start by graphing this basic equation. We then observe 2 the necessary modification on this basic equation to obtain the desired equation, − y = x . The result goes something like this (note the blue frame-box is used as a reference to ’grab’ the basic graph and stretch or shrink it): 2 Basic Equation: y = x2 Tweaked Equation: − y = x
pg. 4
c
2007-2011 MathHands.com v.1012
TRIGONOMETRY
5
5
4
4
3
3
2 1
-5
-4
-3
-2
graphing trig functions
math hands
CH 03 SEC 03 NOTES
-1
1
2
3
4
-1 -2
Horiz. Factor: none Vert. Factor: replace y 7→ − y on graph multiply all y’s by − 1 -5 5 Horiz. Shift: none Vert. Shift: none
2 1
-4
-3
-2
-1
1
2
3
4
5
-1 -2
-3
-3
-4
-4
-5
-5
EXAMPLE Absolute Value Function Graph the following equation by tweaking the corresponding basic equation/graph. 1 − y = x 2 We start by observing which famous, basic equation − y = 12 x is related to. In deed, this equation is just a tweak away from the basic equation, y = |y|. Thus, we start by graphing this basic equation. Basic Equation: y = |x| Tweaked Equation: − y = 12 x 5
5
4
4
3
3
2 1
-5
-4
-3
-2
-1
1 -1 -2
pg. 5
2
3
4
Horiz. Factor: replace x 7→ 21 x on graph multiply all x’s by 2 Vert. Factor: replace y 7→ − y on graph multiply -5 5 all y’s by − 1 Horiz. Shift: none Vert. Shift: none
2 1
-4
-3
-2
-1
1
2
3
4
5
-1 -2
-3
-3
-4
-4
-5
-5
c
2007-2011 MathHands.com v.1012
TRIGONOMETRY
graphing trig functions
math hands
CH 03 SEC 03 NOTES
EXAMPLE the Other Parabola Graph the following equation by tweaking the corresponding basic equation/graph. 1 − x= 2
1 y 2
2
2 We start by observing which famous, basic equation − 12 x = 21 y is related to. In deed, this equation is just a tweak away from the basic equation, x = y 2 . Thus, we start by graphing this basic equation. 2 Basic Equation: x = y 2 Tweaked Equation: − 12 x = 21 y 5
5
4
4 3
3 2 1
-5
-4
-3
-2
-1
1 -1 -2
2
3
4
Horiz. Factor: replace x 7→ − 21 x on graph multiply all x’s by − 2 Vert. Factor: replace y 7→ 12 y 5 on graph multiply -5 all y’s by 2 Horiz. Shift: none Vert. Shift: none
2 1
-4
-3
-2
-1
1
2
3
4
5
-1 -2
-3
-3
-4
-4
-5
-5
EXAMPLE The Sine Graph
pg. 6
c
2007-2011 MathHands.com v.1012
TRIGONOMETRY CH 03 SEC 03 HW WORKSHEET
1. Graph
y = sin x
graphing trig functions
math hands note changes on eq. and on graph
graph
y = sin(−x)
replace 3 2 1 0 ◦ 0 -1 -2 -3
90◦
180◦ 270◦ 360◦
new eq.
3 2 1 0 ◦ 0 -1 -2 -3
90◦
180◦ 270◦ 360◦
on graph 2. Graph
y = sin x
note changes on eq. and on graph
graph
−2y = sin(x)
replace 3 2 1 0 ◦ 0 -1 -2 -3
90◦
180◦ 270◦ 360◦
new eq.
3 2 1 0 ◦ 0 -1 -2 -3
90◦
180◦ 270◦ 360◦
on graph 3. Graph
y = cos x
note changes on eq. and on graph
graph
y = cos(−x)
replace 3 2 1 0 ◦ 0 -1 -2 -3
90◦
180◦ 270◦ 360◦
new eq.
3 2 1 0 ◦ 0 -1 -2 -3
90◦
180◦ 270◦ 360◦
on graph 4. Graph
pg. 1
y = cos x
note changes on eq. and on graph
graph
−.5y = cos(x)
c
2007-2011 MathHands.com v.1012
TRIGONOMETRY CH 03 SEC 03 HW WORKSHEET
graphing trig functions
math hands replace
3 2 1 0 ◦ 0 -1 -2 -3
90◦
180◦ 270◦ 360◦
new eq.
3 2 1 0 ◦ 0 -1 -2 -3
90◦
180◦ 270◦ 360◦
on graph 5. Graph
y = sin x
note changes on eq. and on graph
graph
−.5y = sin(x)
replace 3 2 1 0 ◦ 0 -1 -2 -3
90◦
180◦ 270◦ 360◦
new eq.
3 2 1 0 ◦ 0 -1 -2 -3
90◦
180◦ 270◦ 360◦
on graph 6. Graph
y = sin x
note changes on eq. and on graph
graph
−.5y = sin(4x)
replace 3 2 1 0 ◦ 0 -1 -2 -3
90◦
180◦ 270◦ 360◦
new eq.
3 2 1 0 ◦ 0 -1 -2 -3
90◦
180◦ 270◦ 360◦
on graph 7. Graph
pg. 2
y = sin x
note changes on eq. and on graph
graph
−.5y = sin(−4x)
c
2007-2011 MathHands.com v.1012
TRIGONOMETRY CH 03 SEC 03 HW WORKSHEET
graphing trig functions
math hands replace
3 2 1 0 ◦ 0 -1 -2 -3
90◦
180◦ 270◦ 360◦
new eq.
3 2 1 0 ◦ 0 -1 -2 -3
90◦
180◦ 270◦ 360◦
on graph
pg. 3
c
2007-2011 MathHands.com v.1012
graphing trig functions
math hands
CH 03 SEC 03 NOTES
THE IDEA We now contemplate the consequences on the graph of an equation when every ”x” in the equation is replaced by a ”−x”. As the title of this section suggests, the graph gets reflected. In this case it gets reflected about the y-axis, whereas if we replace every ”y” in an equation by a ”−y” then the graph gets reflected about the x axis. The most direct way to illustrated this is by observing a very simple equation, such as x = 4 Basic Equation: x = 4
-5
-4
-3
-2
Tweaked Equation: −x = 4 can also be written x = −4
5
5
4
4
3
3
2
2
1
1
-1
1
2
3
4
5
-5
-4
-3
-2
-1
1
-1
-1
-2
-2
-3
-3
-4
-4
-5
-5
2
3
4
5
The logic is penetrating and powerful, the result is clear, and the implications are far reaching. Indeed, not just for the above equation, but for all equations with x’s and y’s, replacing all such x’s with ”−x” always results in the reflecting or flipping of the graph across the y axis. Similarly, replacing y’s by −y results in a reflection across the x axis. EXAMPLES Parabolas Graph the following equation by tweaking the corresponding basic equation/graph. 2 −y = x
2 We start by observing which famous, basic equation − y = x is related to. In deed, this equation is just a tweak away from the basic parabola equation y = x2 , thus we start by graphing this basic equation. We then observe 2 the necessary modification on this basic equation to obtain the desired equation, − y = x . The result goes something like this (note the blue frame-box is used as a reference to ’grab’ the basic graph and stretch or shrink it): 2 Basic Equation: y = x2 Tweaked Equation: − y = x
pg. 4
c
2007-2011 MathHands.com v.1012
TRIGONOMETRY
5
5
4
4
3
3
2 1
-5
-4
-3
-2
graphing trig functions
math hands
CH 03 SEC 03 NOTES
-1
1
2
3
4
-1 -2
Horiz. Factor: none Vert. Factor: replace y 7→ − y on graph multiply all y’s by − 1 -5 5 Horiz. Shift: none Vert. Shift: none
2 1
-4
-3
-2
-1
1
2
3
4
5
-1 -2
-3
-3
-4
-4
-5
-5
EXAMPLE Absolute Value Function Graph the following equation by tweaking the corresponding basic equation/graph. 1 − y = x 2 We start by observing which famous, basic equation − y = 12 x is related to. In deed, this equation is just a tweak away from the basic equation, y = |y|. Thus, we start by graphing this basic equation. Basic Equation: y = |x| Tweaked Equation: − y = 12 x 5
5
4
4
3
3
2 1
-5
-4
-3
-2
-1
1 -1 -2
pg. 5
2
3
4
Horiz. Factor: replace x 7→ 21 x on graph multiply all x’s by 2 Vert. Factor: replace y 7→ − y on graph multiply -5 5 all y’s by − 1 Horiz. Shift: none Vert. Shift: none
2 1
-4
-3
-2
-1
1
2
3
4
5
-1 -2
-3
-3
-4
-4
-5
-5
c
2007-2011 MathHands.com v.1012
TRIGONOMETRY
graphing trig functions
math hands
CH 03 SEC 03 NOTES
EXAMPLE the Other Parabola Graph the following equation by tweaking the corresponding basic equation/graph. 1 − x= 2
1 y 2
2
2 We start by observing which famous, basic equation − 12 x = 21 y is related to. In deed, this equation is just a tweak away from the basic equation, x = y 2 . Thus, we start by graphing this basic equation. 2 Basic Equation: x = y 2 Tweaked Equation: − 12 x = 21 y 5
5
4
4 3
3 2 1
-5
-4
-3
-2
-1
1 -1 -2
2
3
4
Horiz. Factor: replace x 7→ − 21 x on graph multiply all x’s by − 2 Vert. Factor: replace y 7→ 12 y 5 on graph multiply -5 all y’s by 2 Horiz. Shift: none Vert. Shift: none
2 1
-4
-3
-2
-1
1
2
3
4
5
-1 -2
-3
-3
-4
-4
-5
-5
EXAMPLE The Sine Graph
pg. 6
c
2007-2011 MathHands.com v.1012
TRIGONOMETRY CH 03 SEC 03 HW WORKSHEET
1. Graph
y = sin x
graphing trig functions
math hands note changes on eq. and on graph
graph
y = sin(−x)
replace 3 2 1 0 ◦ 0 -1 -2 -3
90◦
180◦ 270◦ 360◦
new eq.
3 2 1 0 ◦ 0 -1 -2 -3
90◦
180◦ 270◦ 360◦
on graph 2. Graph
y = sin x
note changes on eq. and on graph
graph
−2y = sin(x)
replace 3 2 1 0 ◦ 0 -1 -2 -3
90◦
180◦ 270◦ 360◦
new eq.
3 2 1 0 ◦ 0 -1 -2 -3
90◦
180◦ 270◦ 360◦
on graph 3. Graph
y = cos x
note changes on eq. and on graph
graph
y = cos(−x)
replace 3 2 1 0 ◦ 0 -1 -2 -3
90◦
180◦ 270◦ 360◦
new eq.
3 2 1 0 ◦ 0 -1 -2 -3
90◦
180◦ 270◦ 360◦
on graph 4. Graph
pg. 1
y = cos x
note changes on eq. and on graph
graph
−.5y = cos(x)
c
2007-2011 MathHands.com v.1012
TRIGONOMETRY CH 03 SEC 03 HW WORKSHEET
graphing trig functions
math hands replace
3 2 1 0 ◦ 0 -1 -2 -3
90◦
180◦ 270◦ 360◦
new eq.
3 2 1 0 ◦ 0 -1 -2 -3
90◦
180◦ 270◦ 360◦
on graph 5. Graph
y = sin x
note changes on eq. and on graph
graph
−.5y = sin(x)
replace 3 2 1 0 ◦ 0 -1 -2 -3
90◦
180◦ 270◦ 360◦
new eq.
3 2 1 0 ◦ 0 -1 -2 -3
90◦
180◦ 270◦ 360◦
on graph 6. Graph
y = sin x
note changes on eq. and on graph
graph
−.5y = sin(4x)
replace 3 2 1 0 ◦ 0 -1 -2 -3
90◦
180◦ 270◦ 360◦
new eq.
3 2 1 0 ◦ 0 -1 -2 -3
90◦
180◦ 270◦ 360◦
on graph 7. Graph
pg. 2
y = sin x
note changes on eq. and on graph
graph
−.5y = sin(−4x)
c
2007-2011 MathHands.com v.1012
TRIGONOMETRY CH 03 SEC 03 HW WORKSHEET
graphing trig functions
math hands replace
3 2 1 0 ◦ 0 -1 -2 -3
90◦
180◦ 270◦ 360◦
new eq.
3 2 1 0 ◦ 0 -1 -2 -3
90◦
180◦ 270◦ 360◦
on graph
pg. 3
c
2007-2011 MathHands.com v.1012
