Specific function examples

1. A Discussion of Dierent Functions
Mathematics 4
June 27, 2012
Mathematics 4 () A Discussion of Dierent Functions June 27, 2012 1 / 14

2. Linear Functions
f(x) =mx+b
Linear Function
A linear function has the formf(x) =mx+bwheremis the slope
andbis the y-intercept.
The domain of a linear function isfxjx2Rg The range isfyjy2Rg
Mathematics 4 () A Discussion of Dierent Functions June 27, 2012 2 / 14

3. Linear Functions
f(x) =mx+b
Linear Function
A linear function has the formf(x) =mx+bwheremis the slope
andbis the y-intercept.
The domain of a linear function isfxjx2Rg The range isfyjy2Rg
Mathematics 4 () A Discussion of Dierent Functions June 27, 2012 2 / 14

4. Linear Functions
f(x) =mx+b
Linear Function
A linear function has the formf(x) =mx+bwheremis the slope
andbis the y-intercept.
The domain of a linear function isfxjx2Rg The range isfyjy2Rg
Mathematics 4 () A Discussion of Dierent Functions June 27, 2012 2 / 14

5. Linear Functions
f(x) =mx+b
Linear Function
f(x) =f
1
(x) =
Mathematics 4 () A Discussion of Dierent Functions June 27, 2012 3 / 14

6. Quadratic Functions
f(x) =ax
2
+bx+c
Quadratic Function
A quadratic function has the formf(x) =ax
2
+bx+cwhere
a; b; c2R; a6= 0.
The graph of a quadratic function is aparabola. The graphopens
upifa >0andopens downwhena <0.
Thevertexof a parabola is given by the vertex equation

b
2a
; f

b
2a

.
The vertex can also be determined by usingcompleting the square
and transforming the equation into thevertex formof the quadratic
equation:(yk) =a(xh)
2
.
Mathematics 4 () A Discussion of Dierent Functions June 27, 2012 4 / 14

7. Quadratic Functions
f(x) =ax
2
+bx+c
Quadratic Function
A quadratic function has the formf(x) =ax
2
+bx+cwhere
a; b; c2R; a6= 0.
The graph of a quadratic function is aparabola. The graphopens
upifa >0andopens downwhena <0.
Thevertexof a parabola is given by the vertex equation

b
2a
; f

b
2a

.
The vertex can also be determined by usingcompleting the square
and transforming the equation into thevertex formof the quadratic
equation:(yk) =a(xh)
2
.
Mathematics 4 () A Discussion of Dierent Functions June 27, 2012 4 / 14

8. Quadratic Functions
f(x) =ax
2
+bx+c
Quadratic Function
A quadratic function has the formf(x) =ax
2
+bx+cwhere
a; b; c2R; a6= 0.
The graph of a quadratic function is aparabola. The graphopens
upifa >0andopens downwhena <0.
Thevertexof a parabola is given by the vertex equation

b
2a
; f

b
2a

.
The vertex can also be determined by usingcompleting the square
and transforming the equation into thevertex formof the quadratic
equation:(yk) =a(xh)
2
.
Mathematics 4 () A Discussion of Dierent Functions June 27, 2012 4 / 14

9. Quadratic Functions
f(x) =ax
2
+bx+c
Quadratic Function
A quadratic function has the formf(x) =ax
2
+bx+cwhere
a; b; c2R; a6= 0.
The graph of a quadratic function is aparabola. The graphopens
upifa >0andopens downwhena <0.
Thevertexof a parabola is given by the vertex equation

b
2a
; f

b
2a

.
The vertex can also be determined by usingcompleting the square
and transforming the equation into thevertex formof the quadratic
equation:(yk) =a(xh)
2
.
Mathematics 4 () A Discussion of Dierent Functions June 27, 2012 4 / 14

10. Quadratic Functions
Example:
Find the vertex (use completing the square), zeros, and graph of
f(x) =2x
2
+ 8x5:
Mathematics 4 () A Discussion of Dierent Functions June 27, 2012 5 / 14

11. Quadratic Functions
f(x) =ax
2
+bx+c
Quadratic Function
Thezerosof a quadratic function can be solved by lettingf(x) = 0
and solving forx. These are also thex-interceptsof the graph.
Thedomainof a quadratic function isfxjx2Rg. Therangeisfyjykgif the graph opens up, andfyjykgwhen
the graph opens down.
Mathematics 4 () A Discussion of Dierent Functions June 27, 2012 6 / 14

12. Quadratic Functions
f(x) =ax
2
+bx+c
Quadratic Function
Thezerosof a quadratic function can be solved by lettingf(x) = 0
and solving forx. These are also thex-interceptsof the graph.
Thedomainof a quadratic function isfxjx2Rg. Therangeisfyjykgif the graph opens up, andfyjykgwhen
the graph opens down.
Mathematics 4 () A Discussion of Dierent Functions June 27, 2012 6 / 14

13. Quadratic Functions
f(x) =ax
2
+bx+c
Quadratic Function
Thezerosof a quadratic function can be solved by lettingf(x) = 0
and solving forx. These are also thex-interceptsof the graph.
Thedomainof a quadratic function isfxjx2Rg. Therangeisfyjykgif the graph opens up, andfyjykgwhen
the graph opens down.
Mathematics 4 () A Discussion of Dierent Functions June 27, 2012 6 / 14

14. Quadratic Functions
Example:
Find the vertex, zeros, domain, range and graph off(x) = 3x
2
+ 3x+ 2.
Identify the interval for which the graph is increasing and decreasing:
Mathematics 4 () A Discussion of Dierent Functions June 27, 2012 7 / 14

15. Quadratic Functions
Example:
Given the functionf(x) = 2x
2
whose graph is shown below:
1
Modify the function such that the graph will move 2 unitsup.
2
Modify the new function such that the graph will move 3 units to the
left.
Mathematics 4 () A Discussion of Dierent Functions June 27, 2012 8 / 14

16. Absolute Value Functions
f(x) =ajxhj+k
Absolute Value Function
An absolute value function has the formf(x) =ajxhj+kwhere
a2R; a6= 0.
The graph of an absolute value function forms the shape of aV. The
graphopens upifa >0andopens downwhena <0.
The slope of the legs of an absolute value function is given by botha
anda.
Thevertexof the graph of an absolute value function is given by the
(h; k). Mathematics 4 () A Discussion of Dierent Functions June 27, 2012 9 / 14

17. Absolute Value Functions
f(x) =ajxhj+k
Absolute Value Function
An absolute value function has the formf(x) =ajxhj+kwhere
a2R; a6= 0.
The graph of an absolute value function forms the shape of aV. The
graphopens upifa >0andopens downwhena <0.
The slope of the legs of an absolute value function is given by botha
anda.
Thevertexof the graph of an absolute value function is given by the
(h; k). Mathematics 4 () A Discussion of Dierent Functions June 27, 2012 9 / 14

18. Absolute Value Functions
f(x) =ajxhj+k
Absolute Value Function
An absolute value function has the formf(x) =ajxhj+kwhere
a2R; a6= 0.
The graph of an absolute value function forms the shape of aV. The
graphopens upifa >0andopens downwhena <0.
The slope of the legs of an absolute value function is given by botha
anda.
Thevertexof the graph of an absolute value function is given by the
(h; k). Mathematics 4 () A Discussion of Dierent Functions June 27, 2012 9 / 14

19. Absolute Value Functions
f(x) =ajxhj+k
Absolute Value Function
An absolute value function has the formf(x) =ajxhj+kwhere
a2R; a6= 0.
The graph of an absolute value function forms the shape of aV. The
graphopens upifa >0andopens downwhena <0.
The slope of the legs of an absolute value function is given by botha
anda.
Thevertexof the graph of an absolute value function is given by the
(h; k). Mathematics 4 () A Discussion of Dierent Functions June 27, 2012 9 / 14

20. Absolute Value Functions
Example:
Find the vertex, zeros, domain, range and graph off(x) = 2jx+ 3j 5.
Identify the interval for which the graph is increasing and decreasing:
Mathematics 4 () A Discussion of Dierent Functions June 27, 2012 10 / 14

21. Absolute Value Functions
Example:
Given the graph below of the previous
functionf(x) = 2jx+ 3j 5, nd the
equation of the function for the
following cases:
1
The graph is moved two units to
the left.
2
The graph is then moved 4 units
up.
3
The direction of the graph is then
inverted.
4
The slopes of the legs are then
reduced to0:5and0:5.
Mathematics 4 () A Discussion of Dierent Functions June 27, 2012 11 / 14

22. Absolute Value Functions
Example:
Given the graph below of the previous
functionf(x) = 2jx+ 3j 5, nd the
equation of the function for the
following cases:
1
The graph is moved two units to
the left.
2
The graph is then moved 4 units
up.
3
The direction of the graph is then
inverted.
4
The slopes of the legs are then
reduced to0:5and0:5.
Mathematics 4 () A Discussion of Dierent Functions June 27, 2012 11 / 14

23. Absolute Value Functions
Example:
Given the graph below of the previous
functionf(x) = 2jx+ 3j 5, nd the
equation of the function for the
following cases:
1
The graph is moved two units to
the left.
2
The graph is then moved 4 units
up.
3
The direction of the graph is then
inverted.
4
The slopes of the legs are then
reduced to0:5and0:5.
Mathematics 4 () A Discussion of Dierent Functions June 27, 2012 11 / 14

24. Absolute Value Functions
Example:
Given the graph below of the previous
functionf(x) = 2jx+ 3j 5, nd the
equation of the function for the
following cases:
1
The graph is moved two units to
the left.
2
The graph is then moved 4 units
up.
3
The direction of the graph is then
inverted.
4
The slopes of the legs are then
reduced to0:5and0:5.
Mathematics 4 () A Discussion of Dierent Functions June 27, 2012 11 / 14

25. The Square Root Function
Consider the functionf(x) =x
2
, whose domain isfxjx0g.
f(x) =x
2
; x0f
1
(x) =
Find the inverse of this function both algebraically and graphically.
Mathematics 4 () A Discussion of Dierent Functions June 27, 2012 12 / 14

26. The Square Root Function
Given the square root functionf(x) =
p
x, whose graph is shown below:
f(x) =
p
x
1
Determine the domain and
range.
2
Move the graph 2 units up.
3
Move the graph 3 units right.
4
Flip the graph horizontally.
5
Flip the graph vertically.
Mathematics 4 () A Discussion of Dierent Functions June 27, 2012 13 / 14

27. The Square Root Function
Given the square root functionf(x) =
p
x, whose graph is shown below:
f(x) =
p
x
1
Determine the domain and
range.
2
Move the graph 2 units up.
3
Move the graph 3 units right.
4
Flip the graph horizontally.
5
Flip the graph vertically.
Mathematics 4 () A Discussion of Dierent Functions June 27, 2012 13 / 14

28. The Square Root Function
Given the square root functionf(x) =
p
x, whose graph is shown below:
f(x) =
p
x
1
Determine the domain and
range.
2
Move the graph 2 units up.
3
Move the graph 3 units right.
4
Flip the graph horizontally.
5
Flip the graph vertically.
Mathematics 4 () A Discussion of Dierent Functions June 27, 2012 13 / 14

29. The Square Root Function
Given the square root functionf(x) =
p
x, whose graph is shown below:
f(x) =
p
x
1
Determine the domain and
range.
2
Move the graph 2 units up.
3
Move the graph 3 units right.
4
Flip the graph horizontally.
5
Flip the graph vertically.
Mathematics 4 () A Discussion of Dierent Functions June 27, 2012 13 / 14

30. The Square Root Function
Given the square root functionf(x) =
p
x, whose graph is shown below:
f(x) =
p
x
1
Determine the domain and
range.
2
Move the graph 2 units up.
3
Move the graph 3 units right.
4
Flip the graph horizontally.
5
Flip the graph vertically.
Mathematics 4 () A Discussion of Dierent Functions June 27, 2012 13 / 14

31. The Square Root Function
Given the graph of the square root function below, nd the equation of
the function.
Mathematics 4 () A Discussion of Dierent Functions June 27, 2012 14 / 14