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

2. Linear Functions
f (x) = mx + b
Linear Function
A linear function has the form f (x) = mx + b where m is the slope
and b is the y-intercept.
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3. Linear Functions
f (x) = mx + b
Linear Function
A linear function has the form f (x) = mx + b where m is the slope
and b is the y-intercept.
The domain of a linear function is {x | x ∈ R}
Mathematics 4 () A Discussion of Different Functions June 27, 2012 2 / 14

4. Linear Functions
f (x) = mx + b
Linear Function
A linear function has the form f (x) = mx + b where m is the slope
and b is the y-intercept.
The domain of a linear function is {x | x ∈ R}
The range is {y | y ∈ R}
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5. Linear Functions
f (x) = mx + b
Linear Function
f (x) = f −1 (x) =
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6. Quadratic Functions
f (x) = ax2 + bx + c
Quadratic Function
A quadratic function has the form f (x) = ax2 + bx + c where
a, b, c ∈ R, a = 0.
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7. Quadratic Functions
f (x) = ax2 + bx + c
Quadratic Function
A quadratic function has the form f (x) = ax2 + bx + c where
a, b, c ∈ R, a = 0.
The graph of a quadratic function is a parabola. The graph opens
up if a > 0 and opens down when a < 0.
Mathematics 4 () A Discussion of Different Functions June 27, 2012 4 / 14

8. Quadratic Functions
f (x) = ax2 + bx + c
Quadratic Function
A quadratic function has the form f (x) = ax2 + bx + c where
a, b, c ∈ R, a = 0.
The graph of a quadratic function is a parabola. The graph opens
up if a > 0 and opens down when a < 0.
The vertex of a parabola is given by the vertex equation
−b −b
,f .
2a 2a
Mathematics 4 () A Discussion of Different Functions June 27, 2012 4 / 14

9. Quadratic Functions
f (x) = ax2 + bx + c
Quadratic Function
A quadratic function has the form f (x) = ax2 + bx + c where
a, b, c ∈ R, a = 0.
The graph of a quadratic function is a parabola. The graph opens
up if a > 0 and opens down when a < 0.
The vertex of a parabola is given by the vertex equation
−b −b
,f .
2a 2a
The vertex can also be determined by using completing the square
and transforming the equation into the vertex form of the quadratic
equation: (y − k) = a (x − h)2 .
Mathematics 4 () A Discussion of Different Functions June 27, 2012 4 / 14

10. Quadratic Functions
Example:
Find the vertex (use completing the square), zeros, and graph of
f (x) = −2x2 + 8x − 5:
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11. Quadratic Functions
f (x) = ax2 + bx + c
Quadratic Function
The zeros of a quadratic function can be solved by letting f (x) = 0
and solving for x. These are also the x-intercepts of the graph.
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12. Quadratic Functions
f (x) = ax2 + bx + c
Quadratic Function
The zeros of a quadratic function can be solved by letting f (x) = 0
and solving for x. These are also the x-intercepts of the graph.
The domain of a quadratic function is {x | x ∈ R}.
Mathematics 4 () A Discussion of Different Functions June 27, 2012 6 / 14

13. Quadratic Functions
f (x) = ax2 + bx + c
Quadratic Function
The zeros of a quadratic function can be solved by letting f (x) = 0
and solving for x. These are also the x-intercepts of the graph.
The domain of a quadratic function is {x | x ∈ R}.
The range is {y | y ≥ k} if the graph opens up, and {y | y ≤ k} when
the graph opens down.
Mathematics 4 () A Discussion of Different Functions June 27, 2012 6 / 14

14. Quadratic Functions
Example:
Find the vertex, zeros, domain, range and graph of f (x) = 3x2 + 3x + 2.
Identify the interval for which the graph is increasing and decreasing:
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15. Quadratic Functions
Example:
Given the function f (x) = 2x2 whose graph is shown below:
1 Modify the function such that the graph will move 2 units up.
2 Modify the new function such that the graph will move 3 units to the
left.
Mathematics 4 () A Discussion of Different Functions June 27, 2012 8 / 14

16. Absolute Value Functions
f (x) = a |x − h| + k
Absolute Value Function
An absolute value function has the form f (x) = a |x − h| + k where
a ∈ R, a = 0.
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17. Absolute Value Functions
f (x) = a |x − h| + k
Absolute Value Function
An absolute value function has the form f (x) = a |x − h| + k where
a ∈ R, a = 0.
The graph of an absolute value function forms the shape of a V. The
graph opens up if a > 0 and opens down when a < 0.
Mathematics 4 () A Discussion of Different Functions June 27, 2012 9 / 14

18. Absolute Value Functions
f (x) = a |x − h| + k
Absolute Value Function
An absolute value function has the form f (x) = a |x − h| + k where
a ∈ R, a = 0.
The graph of an absolute value function forms the shape of a V. The
graph opens up if a > 0 and opens down when a < 0.
The slope of the legs of an absolute value function is given by both a
and −a.
Mathematics 4 () A Discussion of Different Functions June 27, 2012 9 / 14

19. Absolute Value Functions
f (x) = a |x − h| + k
Absolute Value Function
An absolute value function has the form f (x) = a |x − h| + k where
a ∈ R, a = 0.
The graph of an absolute value function forms the shape of a V. The
graph opens up if a > 0 and opens down when a < 0.
The slope of the legs of an absolute value function is given by both a
and −a.
The vertex of the graph of an absolute value function is given by the
(h, k).
Mathematics 4 () A Discussion of Different Functions June 27, 2012 9 / 14

20. Absolute Value Functions
Example:
Find the vertex, zeros, domain, range and graph of f (x) = 2 |x + 3| − 5.
Identify the interval for which the graph is increasing and decreasing:
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21. Absolute Value Functions
Example:
Given the graph below of the previous
function f (x) = 2 |x + 3| − 5, find the
equation of the function for the
following cases:
1 The graph is moved two units to
the left.
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22. Absolute Value Functions
Example:
Given the graph below of the previous
function f (x) = 2 |x + 3| − 5, find 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.
Mathematics 4 () A Discussion of Different Functions June 27, 2012 11 / 14

23. Absolute Value Functions
Example:
Given the graph below of the previous
function f (x) = 2 |x + 3| − 5, find 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.
Mathematics 4 () A Discussion of Different Functions June 27, 2012 11 / 14

24. Absolute Value Functions
Example:
Given the graph below of the previous
function f (x) = 2 |x + 3| − 5, find 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 to 0.5 and −0.5.
Mathematics 4 () A Discussion of Different Functions June 27, 2012 11 / 14

25. The Square Root Function
Consider the function f (x) = x2 , whose domain is {x | x ≥ 0}.
f (x) = x2 , x ≥ 0 f −1 (x) =
Find the inverse of this function both algebraically and graphically.
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26. The Square Root Function
√
Given the square root function f (x) = x, whose graph is shown below:
1 Determine the domain and
range.
√
f (x) = x
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27. The Square Root Function
√
Given the square root function f (x) = x, whose graph is shown below:
1 Determine the domain and
range.
2 Move the graph 2 units up.
√
f (x) = x
Mathematics 4 () A Discussion of Different Functions June 27, 2012 13 / 14

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

29. The Square Root Function
√
Given the square root function f (x) = x, whose graph is shown below:
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.
√
f (x) = x
Mathematics 4 () A Discussion of Different Functions June 27, 2012 13 / 14

30. The Square Root Function
√
Given the square root function f (x) = x, whose graph is shown below:
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.
√
f (x) = x 5 Flip the graph vertically.
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31. The Square Root Function
Given the graph of the square root function below, find the equation of
the function.
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