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A Discussion of Different Functions
Mathematics 4
June 27, 2012
Mathematics 4 () A Discussion of Different Functions June 27, 2012 1 / 14
Linear Functionsf(x) = mx+ b
Linear Function
A linear function has the form f(x) = mx+ b where m is the slopeand b is the y-intercept.
The domain of a linear function is {x | x ∈ R}
The range is {y | y ∈ R}
Mathematics 4 () A Discussion of Different Functions June 27, 2012 2 / 14
Linear Functionsf(x) = mx+ b
Linear Function
A linear function has the form f(x) = mx+ b where m is the slopeand b is the y-intercept.
The domain of a linear function is {x | x ∈ R}
The range is {y | y ∈ R}
Mathematics 4 () A Discussion of Different Functions June 27, 2012 2 / 14
Linear Functionsf(x) = mx+ b
Linear Function
A linear function has the form f(x) = mx+ b where m is the slopeand b is the y-intercept.
The domain of a linear function is {x | x ∈ R}
The range is {y | y ∈ R}
Mathematics 4 () A Discussion of Different Functions June 27, 2012 2 / 14
Linear Functionsf(x) = mx+ b
Linear Function
f(x) = f−1(x) =
Mathematics 4 () A Discussion of Different Functions June 27, 2012 3 / 14
Quadratic Functionsf(x) = ax2 + bx+ c
Quadratic Function
A quadratic function has the form f(x) = ax2 + bx+ c wherea, b, c ∈ R, a 6= 0.
The graph of a quadratic function is a parabola. The graph opensup if a > 0 and opens down when a < 0.
The vertex of a parabola is given by the vertex equation(−b2a
, f
(−b2a
)).
The vertex can also be determined by using completing the squareand transforming the equation into the vertex form of the quadraticequation: (y − k) = a (x− h)2.
Mathematics 4 () A Discussion of Different Functions June 27, 2012 4 / 14
Quadratic Functionsf(x) = ax2 + bx+ c
Quadratic Function
A quadratic function has the form f(x) = ax2 + bx+ c wherea, b, c ∈ R, a 6= 0.
The graph of a quadratic function is a parabola. The graph opensup if a > 0 and opens down when a < 0.
The vertex of a parabola is given by the vertex equation(−b2a
, f
(−b2a
)).
The vertex can also be determined by using completing the squareand transforming the equation into the vertex form of the quadraticequation: (y − k) = a (x− h)2.
Mathematics 4 () A Discussion of Different Functions June 27, 2012 4 / 14
Quadratic Functionsf(x) = ax2 + bx+ c
Quadratic Function
A quadratic function has the form f(x) = ax2 + bx+ c wherea, b, c ∈ R, a 6= 0.
The graph of a quadratic function is a parabola. The graph opensup if a > 0 and opens down when a < 0.
The vertex of a parabola is given by the vertex equation(−b2a
, f
(−b2a
)).
The vertex can also be determined by using completing the squareand transforming the equation into the vertex form of the quadraticequation: (y − k) = a (x− h)2.
Mathematics 4 () A Discussion of Different Functions June 27, 2012 4 / 14
Quadratic Functionsf(x) = ax2 + bx+ c
Quadratic Function
A quadratic function has the form f(x) = ax2 + bx+ c wherea, b, c ∈ R, a 6= 0.
The graph of a quadratic function is a parabola. The graph opensup if a > 0 and opens down when a < 0.
The vertex of a parabola is given by the vertex equation(−b2a
, f
(−b2a
)).
The vertex can also be determined by using completing the squareand transforming the equation into the vertex form of the quadraticequation: (y − k) = a (x− h)2.
Mathematics 4 () A Discussion of Different Functions June 27, 2012 4 / 14
Quadratic Functions
Example:
Find the vertex (use completing the square), zeros, and graph off(x) = −2x2 + 8x− 5:
Mathematics 4 () A Discussion of Different Functions June 27, 2012 5 / 14
Quadratic Functionsf(x) = ax2 + bx+ c
Quadratic Function
The zeros of a quadratic function can be solved by letting f(x) = 0and 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} whenthe graph opens down.
Mathematics 4 () A Discussion of Different Functions June 27, 2012 6 / 14
Quadratic Functionsf(x) = ax2 + bx+ c
Quadratic Function
The zeros of a quadratic function can be solved by letting f(x) = 0and 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} whenthe graph opens down.
Mathematics 4 () A Discussion of Different Functions June 27, 2012 6 / 14
Quadratic Functionsf(x) = ax2 + bx+ c
Quadratic Function
The zeros of a quadratic function can be solved by letting f(x) = 0and 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} whenthe graph opens down.
Mathematics 4 () A Discussion of Different Functions June 27, 2012 6 / 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:
Mathematics 4 () A Discussion of Different Functions June 27, 2012 7 / 14
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 theleft.
Mathematics 4 () A Discussion of Different Functions June 27, 2012 8 / 14
Absolute Value Functionsf(x) = a |x− h|+ k
Absolute Value Function
An absolute value function has the form f(x) = a |x− h|+ k wherea ∈ R, a 6= 0.
The graph of an absolute value function forms the shape of a V. Thegraph 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 aand −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
Absolute Value Functionsf(x) = a |x− h|+ k
Absolute Value Function
An absolute value function has the form f(x) = a |x− h|+ k wherea ∈ R, a 6= 0.
The graph of an absolute value function forms the shape of a V. Thegraph 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 aand −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
Absolute Value Functionsf(x) = a |x− h|+ k
Absolute Value Function
An absolute value function has the form f(x) = a |x− h|+ k wherea ∈ R, a 6= 0.
The graph of an absolute value function forms the shape of a V. Thegraph 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 aand −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
Absolute Value Functionsf(x) = a |x− h|+ k
Absolute Value Function
An absolute value function has the form f(x) = a |x− h|+ k wherea ∈ R, a 6= 0.
The graph of an absolute value function forms the shape of a V. Thegraph 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 aand −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
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:
Mathematics 4 () A Discussion of Different Functions June 27, 2012 10 / 14
Absolute Value Functions
Example:
Given the graph below of the previousfunction f(x) = 2 |x+ 3| − 5, find the
equation of the function for thefollowing cases:
1 The graph is moved two units tothe left.
2 The graph is then moved 4 unitsup.
3 The direction of the graph is theninverted.
4 The slopes of the legs are thenreduced to 0.5 and −0.5.
Mathematics 4 () A Discussion of Different Functions June 27, 2012 11 / 14
Absolute Value Functions
Example:
Given the graph below of the previousfunction f(x) = 2 |x+ 3| − 5, find the
equation of the function for thefollowing cases:
1 The graph is moved two units tothe left.
2 The graph is then moved 4 unitsup.
3 The direction of the graph is theninverted.
4 The slopes of the legs are thenreduced to 0.5 and −0.5.
Mathematics 4 () A Discussion of Different Functions June 27, 2012 11 / 14
Absolute Value Functions
Example:
Given the graph below of the previousfunction f(x) = 2 |x+ 3| − 5, find the
equation of the function for thefollowing cases:
1 The graph is moved two units tothe left.
2 The graph is then moved 4 unitsup.
3 The direction of the graph is theninverted.
4 The slopes of the legs are thenreduced to 0.5 and −0.5.
Mathematics 4 () A Discussion of Different Functions June 27, 2012 11 / 14
Absolute Value Functions
Example:
Given the graph below of the previousfunction f(x) = 2 |x+ 3| − 5, find the
equation of the function for thefollowing cases:
1 The graph is moved two units tothe left.
2 The graph is then moved 4 unitsup.
3 The direction of the graph is theninverted.
4 The slopes of the legs are thenreduced to 0.5 and −0.5.
Mathematics 4 () A Discussion of Different Functions June 27, 2012 11 / 14
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.
Mathematics 4 () A Discussion of Different Functions June 27, 2012 12 / 14
The Square Root Function
Given the square root function f(x) =√x, whose graph is shown below:
f(x) =√x
1 Determine the domain andrange.
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 Different Functions June 27, 2012 13 / 14
The Square Root Function
Given the square root function f(x) =√x, whose graph is shown below:
f(x) =√x
1 Determine the domain andrange.
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 Different Functions June 27, 2012 13 / 14
The Square Root Function
Given the square root function f(x) =√x, whose graph is shown below:
f(x) =√x
1 Determine the domain andrange.
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 Different Functions June 27, 2012 13 / 14
The Square Root Function
Given the square root function f(x) =√x, whose graph is shown below:
f(x) =√x
1 Determine the domain andrange.
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 Different Functions June 27, 2012 13 / 14
The Square Root Function
Given the square root function f(x) =√x, whose graph is shown below:
f(x) =√x
1 Determine the domain andrange.
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 Different Functions June 27, 2012 13 / 14
The Square Root Function
Given the graph of the square root function below, find the equation ofthe function.
Mathematics 4 () A Discussion of Different Functions June 27, 2012 14 / 14