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