Transcript Functions III

Function II
axis
Y-axis
Y-axis
y = x2 - 2x - 8
y = mx + b
b
X-axis
X-axis
-b
m
1
-2
4
-8
(1,-9)
Functions: Domain and Range
By Mr Porter
Definitions
Function:
A function is a set of ordered pair in which no two ordered pairs have the same
x-coordinate.
Domain
The domain of a function is the set of all x-coordinates of the ordered pairs.
[the values of x for which a vertical line will cut the curve.]
Range
The range of a function is the set of all y-coordinates of the ordered pairs.
[the values of y for which a horizontal line will cut the curve]
Note: Students need to be able to define the domain and range from the equation of a curve or function. It is encourage
that student make sketches of each function, labeling each key feature.
Linear Functions
Any equation that can be written in the
• General form ax + by + c = 0
• Standard form y = mx + b
Examples
a) y = 3x + 6
Y-axis
b) 2x + 3y = 12
Y-axis
x-intercept at y = 0
0 = 3x + 6
x = -2
y-intercept at x = 0
y= 6
Sketching Linear Functions.
Find the x-intercep at y = 0
And the y-intercept at x = 0.
y = 3x + 6
x-intercept at y = 0
2x = 12
x= 6
y-intercept at x = 0
3y = 12
y= 4
6
X-axis
-2
4
X-axis
6
Every vertical line will cut 2x+3y=12.
Every vertical line will cut y = 3x + 6.
Every horizontal line will cut 2x+3y=12 6
Every horizontal line will cut y = 3x + 6
Domain : All x  R , real numbers
Domain : All x  R , real numbers
Range : All y  R , real numbers
Range : All y  R , real numbers


Special Lines
Examples
Equation of a vertical line is:
i) x = a
Y-axis
ii) x - a = 0
(4,5)
X-axis
Domain: x = 4
Range: all y in R
(a,b)
Sketch
a) x = 4
x=4
Vertical Lines: x = a
- these are not functions, as the first
element in any ordered pair is (a, y)
Y-axis
b) x + 2 = 0
Y-axis
x = -2
x=a
X-axis
X-axis
Domain: x = a
Range: all y in R
(-2,-6)
Domain: x = -2
Range: all y in R
Special Lines
Examples
Horizontal Lines: y = a
- these are functions, as the first
element in any ordered pair is (x, a)
Equation of a horizontal line is:
i) y = a
Y-axis
ii) y - a = 0
Sketch
a) y = 3
Y-axis
(-5,3)
y=3
X-axis
Domain: all x in R
Range: y = 3
(a,b)
y=a
X-axis
b) y + 6 = 0
Y-axis
X-axis
Domain: all x in R
Range: y = b
y = -6
Domain: all x in R
Range: y = -6
(2,-6)
Parabola: y = ax2 +bx + c
The five steps in sketching a parabola
function:
1) If a is positive, the parabola is
concave up.
If a is negative, the parabola is
concave down.
2) To find the y-intercept, put x = 0.
3) To find the x-intercept, form a
quadratic and solve
ax2 + bx + c = 0
* factorise
* quadratic formula
b
4) Find the axis of symmetry by x 
2a
5) Use the axis of symmetry x-value
to find the y-value of the vertex, h

Domain: all x in R
Range: y ≥ h for a > 0
Range: y ≤ h for a < 0
Example
Sketch y = x2 + 2x - 3, hence, state its
domain and range.
1) For y = ax2 + bx + c
a = 1, b = +2, c = -3 . Concave-up a = 1
2) y-intercept at x = 0, y = -3
3) x-intercept at y = 0, (factorise )
(x - 1)(x + 3) = 0
x = +1 and x = - 3.
b (2)

4) Axis of symmetry at x 
= -1
2a 2(1)
5) y-value of vertex: y = (-1)2 +2(-1) - 3
y = -4
Y-axis

X-axis
-3
Domain: all x in R
Range: y ≥ -4
-1
1
-3
(-1,-4)
Parabola: y = ax2 +bx + c
The five steps in sketching a parabola
function:
1) If a is positive, the parabola is
concave up.
If a is negative, the parabola is
concave down.
2) To find the y-intercept, put x = 0.
3) To find the x-intercept, form a
quadratic and solve
ax2 + bx + c = 0
* factorise
* quadratic formula
b
4) Find the axis of symmetry by x 
2a
5) Use the axis of symmetry x-value
to find the y-value of the vertex, h

Domain: all x in R
Range: y ≥ h for a > 0
Range: y ≤ h for a < 0
Example
Sketch y = –x2 + 4x - 5, hence, state its
domain and range.
1) For y = ax2 + bx + c, a = -1, b = +4,
c = -5. Concave-down a = -1
2) y-intercept at x = 0, y = -5
3) x-intercept at y = 0, NO zeros by
Quadratic formula.
4) Axis of symmetry at x 
b (4)

= +2
2a 2(1)
5) y-value of vertex: y = -(2)2 +4(2) - 5
y = -1
Y-axis

X-axis
2
(2,-1)
Domain: all x in R
Range: y ≤ -1
-5
Worked Example 1: Your task is to plot the key features of the given parabola, sketch
the parabola, then state clearly its domain and range.
Sketch the parabola y = x2 - 2x - 8, hence state clearly its domain and range.
Y-axis
axis
The five steps in sketching a
parabola function:
1) If a is positive, the
parabola is concave up. y = x2 - 2x - 8
If a is negative, the
parabola is concave down.
2) To find the y-intercept,
put x = 0.
3) To find the x-intercept,
form a quadratic and solve
ax2 + bx + c = 0
* factorise
X-axis
* quadratic formula
4)Find the axis 1of symmetry by
-2
4
b
x
2a
5) Use the axis of symmetry
x-value to find the y-value of
-8
the vertex,
h

(1,-9)
Step 1: Determine concavity: Up or Down?
For the parabola of the form y = ax2 + bx + c
a = 1 => concave up
Step 2: Determine y-intercept.
Let x = 0, y = -8
Step 3: Determine x-intercept.
Solve: x2 - 2x - 8 = 0
Factorise : (x - 4)(x + 2) = 0 ==> x = 4 or x = -2.
b (2)

1
x

Step 4: Determine axis of symmetry.
2(1)
2a
Step 5: Determine maximum or minimum y-value (vertex).
Domain all x in R
Range y ≥ -9
Substitute the value x = 1 into y = x2 - 2x - 8.


2
y = (1) - 2(1) - 8 = -9
Vertex at (1, -9)
Worked Example 2: Your task is to plot the key features of the given parabola, sketch
the parabola, then state clearly its domain and range.
Sketch the parabola f(x) = 15 - 2x - x2, hence state clearly its domain and range.
Step 1: Determine concavity: Up or Down?
For the parabola of the form f(x) = ax2 + bx + c
a = -1 => concave down
axis
The five Y-axis
steps in sketching a
parabola function:
1) If a is positive, the
parabola is concave up.
If a is negative, the
parabola is(1,16)
concave down.
2) To find the y-intercept,
15
put x = 0.
3) To find the x-intercept,
form a quadratic and solve
ax2 + bx + c = 0
* factorise
* quadratic formula
4)Find the axis of symmetry by
-5
3
b-1
x
2a
5) Use the axis of symmetry
x-value to find the y-value of
the vertex, h

f(x)=15 - 2x - x2
Domain: all x in R
Range: y ≤ 16
Step 2: Determine y-intercept.
Let x = 0, f(x) = +15
Step 3: Determine x-intercept.
X-axis
Solve: 15 - 2x - x2 = 0
Factorise : (3 - x)(x + 5) = 0 ==> x = 3 or x = -5.
b (2)

 1
x

Step 4: Determine axis of symmetry.
2(1)
2a
Step 5: Determine maximum or minimum y-value (vertex).
Substitute the value x = -1 into y = 15 - 2x - x2.


2
y = 15 - 2(-1) - (-1) = 16 Vertex at (1, 16)
Exercise: For each of the following functions:
a) sketch the curve
b) sate the largest possible domain and range of the function.
(ii) h(x) = 2x2 + 7x - 15
(i) f(x) = 5 - 2x
Domain: All x in R
Range: All y in R
Y-axis
Domain: All x in R
Range: All y ≥ -211/8
Y-axis
h(x) = 2x2 + 7x - 15
5
X-axis
X-axis
11/2
-13/4
-5
21/2
-15
f(x) = 5 - 2x
(iii) h(x) = x2 + 2x + 5
h(x) = x2 + 2x + 5
Y-axis
(-13/4 ,-211/8 )
(iv) g(x) = 5x + 4
Domain: All x in R
Range: All y ≥ 4
Domain: All x in R
Range: All y in R
Y-axis
4
X-axis
5
(-1 ,4)
-1
NO x-intercepts.
(try quadratic formula?)
X-axis
-4/5
g(x) = 5x + 4