Sketching Graphs of Functions: Techniques and Examples
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LO: To create the sketch of a function from
information given or transferring a graph
from screen to paper.
Creating the sketch of a function
6 October 2024
Sketching its graph.
In this lesson we will be working on Sketching the graph
of a function
If you are asked to sketch the graph of a function is to
represent by means of a diagram or graph (labelled as
appropriate).
Sketching the graph of a function
Every function has a particular behaviour and is
identified by a specific shape.
There are two ways to represent the graph of a
function.
Drawing its graph.
The sketch should give a general idea of the required
shape
or relationship and should include
relevant
features
.
Any maximum value
Sketching the graph of a function
All axes should be labelled.
Features that should be labelled:
Details to keep in mind when sketching a function.
Any minimum value
Axis intercepts
Zeros of functions or roots of equations
Equations of vertical and horizontal asymptotes
Identify starting and ending points
Axis of symmetry
(2, 0)
(0,
–10
)
.
(
–5, 0
)
.
Sketch the graph of the function
y
=
x
2
+ 3
x
– 10
.
y-intercept (0,
–10
)
.
We need some points to sketch the graph
(– 5, 0)
(2, 0)
x-intercepts
From the equation we
have the y-intercept.
We can find the zeros of
the function by factorizing
the equation and these are
the x-intercepts.
Sketching the graph of a function
0
=
x
2
+ 3
x
– 10
.
(x + 5)
(x
–
2)
0 =
(x + 5)
(x
–
2)
= 0
= 0
x = –5
x
=
2
These are the zeros of the function
The graph is a parabola
Equation of the axis of symmetry
x
=
p
+
q
2
-
5 + 2
2
-
1.5
=
=
(2, 0)
(0,
–10
)
.
(
–5, 0
)
.
Sketch the graph of the function
y
=
x
2
+ 3
x
– 10
.
We need some points to sketch the graph
Now we can calculate
the equation of the line
of symmetry.
The line of symmetry is
halfway between the
x
-intercepts.
y
= (-1.5)
2
+ 3(-1.5) – 10
This is the x-coordinate of the vertex
y
= -12.25
(-1.5, -12.25)
Sketching the graph of a function
Finding the
y
-coordinate
Vertex
The graph is a parabola
(
–1.25
,
–12.25
)
.
x
=
-
1.5
(1 , 0)
(0,
6
)
.
(
–3, 0
)
.
y-intercept (0,
6
)
.
(–3, 0)
x-intercepts
Sketch the graph of the function
y
= –2(
x
+ 3)(
x
– 1).
(1, 0)
We need some points to sketch the graph
From the equation we
have the y-intercept.
The graph is a parabola
–2
3
– 1
(
)
(
)
We can find the zeros of
the function by equating to
zero the equation and
these are the x-intercepts.
–2
(x + 3)
(x
–
1)
0 =
(x + 3)
(x
–
1)
= 0
= 0
x = –3
x
=
1
Sketching the graph of a function
(1 , 0)
(0,
6
)
.
(
–3, 0
)
.
Sketch the graph of the function
y
= -2(
x
+ 3)(
x
– 1).
y
= -2(-1+3)(-1 -1)
y
= 8
(-1, 8)
We need some points to sketch the graph
The graph is a parabola
Equation of the axis of symmetry
x
=
p
+
q
2
-
3 + 1
2
-
1
=
=
Now we can calculate
the equation of the line
of symmetry.
The line of symmetry is
halfway between the
x
-intercepts.
This is the x-coordinate of the vertex
Finding the
y
-coordinate
Vertex
Sketching the graph of a function
x
=
-
1
Use the GDC to sketch the graph of the function
y
=
x
3
–
x
2
– 7
x
– 1 for the domain –2 ≤
x ≤
3
Sketching the graph of a function
Turn on the GDC
Press Y=
Use the GDC to sketch the graph of the function
y
=
x
3
–
x
2
– 7
x
– 1 for the domain –2 ≤
x ≤
3
Sketching the graph of a function
Turn on the GDC
Press Y=
Type in the function
y
=
x
3
–
x
2
– 7
x
– 1
enter
Press
Graph
Use the GDC to sketch the graph of the function
y
=
x
3
–
x
2
– 7
x
– 1 for the domain –2 ≤
x ≤
3
Sketching the graph of a function
Turn on the GDC
Press 5
Graph
Type in the function
y
=
x
3
–
x
2
– 7
x
– 1
enter
Press
Graph
Press
Window
Use the GDC to sketch the graph of the function
y
=
x
3
–
x
2
– 7
x
– 1 for the domain –2 ≤
x ≤
3
Sketching the graph of a function
Turn on the GDC
Press 5
Graph
Type in the function
Set the domain
Xmin:
–2
Xmax:
3
Ymin:
–12
Ymax:
5
enter
enter
enter
enter
Press
graph
y
=
x
3
–
x
2
– 7
x
– 1
enter
Press
Graph
Press
Window
Set the range
Use the GDC to sketch the graph of the function
y
=
x
3
–
x
2
– 7
x
– 1 for the domain –2 ≤
x ≤
3
Sketching the graph of a function
Press 2nd
Calc
Maximum point:
(-1.2, 4.2)
We need some
points to sketch
the graph
Press 4
maximum
(-1.2, 4.2)
Drag the cursor to the
left of a maximum
enter
Drag the cursor to the
right of a maximum
enter
enter
Press 2nd
Calc
Use the GDC to sketch the graph of the function
y
=
x
3
–
x
2
– 7
x
– 1 for the domain –2 ≤
x ≤
3
Sketching the graph of a function
We need some
points to sketch
the graph
Press 3
minimum
(-1.2, 4.2)
Minimum point:
(1.9, -11.1)
(1.9, -11.1)
Drag the cursor to the
left of a minimum
enter
Drag the cursor to the
right of a minimum
enter
enter
Press 2nd
Calc
Use the GDC to sketch the graph of the function
y
=
x
3
–
x
2
– 7
x
– 1 for the domain –2 ≤
x ≤
3
Sketching the graph of a function
(-1.2, 4.2)
For the x-intercept:
(1.9, -11.1)
X-intercept: (-0.15, 0)
(-0.15, 0)
(-2, 1)
x:
–2
For the starting
point:
Starting point: (-2, 1)
enter
Press 2:
zero
Drag the cursor to the
left of a zero
enter
Drag the cursor to the
right of a zero
enter
enter
Press 1:
value
Press 2nd
Calc
Press 2nd
Calc
Use the GDC to sketch the graph of the function
y
=
x
3
–
x
2
– 7
x
– 1 for the domain –2 ≤
x ≤
3
Sketching the graph of a function
(-1.2, 4.2)
(1.9, -11.1)
(0, -1)
(-0.15, 0)
(-2, 1)
(3, -4)
Join the points
with a smooth
curve
x:
3
For the ending
point:
Ending point: (3, -4)
enter
Press 1:
value
x:
0
For the y-intercept:
Y-intercept: (0, -1)
enter
Press 1:
value
Press 2nd
Calc
Thank you for using resources from
https://www.mathssupport.org
If you have a special request, drop us an email
info@mathssupport.org
For more resources visit our website
The art of sketching graphs of functions involves representing specific shapes and behaviors through labeled diagrams. This lesson highlights key details to keep in mind when sketching functions, such as labeling axes, maximum and minimum values, intercepts, symmetry, and asymptotes. Two examples of sketching functions, including a parabola and a quadratic function, are explained step by step, demonstrating how to find key points like intercepts, line of symmetry, and vertex to accurately depict the function's graph.
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Presentation Transcript
6 October 2024 Creating the sketch of a function LO: To create the sketch of a function from information given or transferring a graph from screen to paper. www.mathssupport.org
Sketching the graph of a function Every function has a particular behaviour and is identified by a specific shape. There are two ways to represent the graph of a function. Drawing its graph. Sketching its graph. In this lesson we will be working on Sketching the graph of a function If you are asked to sketch the graph of a function is to represent by means of a diagram or graph (labelled as appropriate). The sketch should give a general idea of the required shape or relationship and should include relevant features. www.mathssupport.org
Sketching the graph of a function Details to keep in mind when sketching a function. All axes should be labelled. Features that should be labelled: Any maximum value Any minimum value Axis intercepts Axis of symmetry Zeros of functions or roots of equations Equations of vertical and horizontal asymptotes Identify starting and ending points www.mathssupport.org
Sketching the graph of a function Sketch the graph of the function y = x2 + 3x 10. We need some points to sketch the graph The graph is a parabola y From the equation we have the y-intercept. 7 6 5 4 y-intercept (0, 10). 3 2 We can find the zeros of the function by factorizing the equation and these are the x-intercepts. 0 = x2 + 3x 10. (x + 5) (x 2) 0 = (x + 5) = 0 x = 5 These are the zeros of the function ( 5, 0). 1 (2, 0) x 0 -1 -10 1 2 3 4 5 6 7 9 10 -9 -8 -7 -6 -5 -4 -3 -2 8 -1 -2 -3 -4 -5 -6 -7 -8 (x 2) = 0 x = 2 -9 -10 (0, 10). -11 ( 5, 0) (2, 0) x-intercepts -12 -13 www.mathssupport.org
Sketching the graph of a function Sketch the graph of the function y = x2 + 3x 10. We need some points to sketch the graph The graph is a parabola y 7 Now we can calculate the equation of the line of symmetry. The line of symmetry is halfway between the x-intercepts. 6 x = -1.5 5 4 3 2 ( 5, 0). 1 (2, 0) x 0 -1 -10 1 2 3 4 5 6 7 9 10 -9 -8 -7 -6 -5 -4 -3 -2 8 -1 -2 Equation of the axis of symmetry p + q 2 This is the x-coordinate of the vertex Finding the y-coordinate -3 -4 -5 + 2 2 -1.5 x = = = -5 -6 -7 -8 -9 y = (-1.5)2 + 3(-1.5) 10 y = -12.25 (-1.5, -12.25) Vertex -10 (0, 10). -11 ( 1.25, 12.25). -12 -13 www.mathssupport.org
Sketching the graph of a function Sketch the graph of the function y = 2(x + 3)(x 1). We need some points to sketch the graph The graph is a parabola 2 3 1 y From the equation we have the y-intercept. () () 8 7 (0, 6). 6 5 4 y-intercept (0, 6). 3 2 We can find the zeros of the function by equating to zero the equation and these are the x-intercepts. 2(x + 3)(x 1) 0 = (x + 3) = 0 x = 3 1 ( 3, 0). (1 , 0) x -1 -3 -6 -5 0 -10 2 3 4 5 6 7 9 10 -9 -8 -7 -4 -2 8 1 -2 -3 -4 -5 -6 (x 1) = 0 -7 -8 x = 1 -9 -10 -11 x-intercepts (1, 0) ( 3, 0) -12 -13 www.mathssupport.org
Sketching the graph of a function Sketch the graph of the function y = -2(x + 3)(x 1). We need some points to sketch the graph The graph is a parabola y 8 Now we can calculate the equation of the line of symmetry. The line of symmetry is halfway between the x-intercepts. 7 (0, 6). 6 5 4 3 2 1 ( 3, 0). (1 , 0) x -1 -3 -6 -5 0 -10 2 3 4 5 6 7 9 10 -9 -8 -7 -4 -2 8 1 -2 Equation of the axis of symmetry p + q 2 This is the x-coordinate of the vertex Finding the y-coordinate -3 -3 + 1 2 -1 x = = = -4 -5 -6 -7 -8 -9 y = -2(-1+3)(-1 -1) y = 8 (-1, 8) Vertex -10 x = -1 -11 -12 -13 www.mathssupport.org
Sketching the graph of a function Use the GDC to sketch the graph of the function y = x3 x2 7x 1 for the domain 2 x 3 y 5 Turn on the GDC Press Y= 4 3 2 1 x 3 2 1 1 2 3 -1 -2 -3 -4 -5 --6 -7 -8 -9 -10 -11 www.mathssupport.org
Sketching the graph of a function Use the GDC to sketch the graph of the function y = x3 x2 7x 1 for the domain 2 x 3 y 5 Turn on the GDC Press Y= Type in the function y = x3 x2 7x 1 enter Press 4 3 2 1 x 3 2 1 1 2 3 -1 Graph -2 -3 -4 -5 --6 -7 -8 -9 -10 -11 www.mathssupport.org
Sketching the graph of a function Use the GDC to sketch the graph of the function y = x3 x2 7x 1 for the domain 2 x 3 y 5 Turn on the GDC Press 5 Type in the function y = x3 x2 7x 1 enter Press Press 4 Graph 3 2 1 x 3 2 1 1 2 3 -1 Graph Window -2 -3 -4 -5 --6 -7 -8 -9 -10 -11 www.mathssupport.org
Sketching the graph of a function Use the GDC to sketch the graph of the function y = x3 x2 7x 1 for the domain 2 x 3 y 5 Turn on the GDC Press 5 Type in the function y = x3 x2 7x 1 enter Press Press 4 Graph 3 2 1 x 3 2 1 1 2 3 -1 Graph Window -2 -3 Set the domain Xmin: 2 Xmax: 3 Set the range -4 enter enter -5 --6 -7 -8 -9 -10 -11 enter enter Ymin: 12 Ymax: 5 Press graph www.mathssupport.org
Sketching the graph of a function Use the GDC to sketch the graph of the function y = x3 x2 7x 1 for the domain 2 x 3 We need some points to sketch the graph y 5 (-1.2, 4.2) 4 3 2 Press 2nd Calc Press 4 maximum Drag the cursor to the left of a maximum enter Drag the cursor to the right of a maximum enter enter 1 x 3 2 1 1 2 3 -1 -2 -3 -4 -5 Maximum point: (-1.2, 4.2) Press 2nd Calc --6 -7 -8 -9 -10 -11 www.mathssupport.org
Sketching the graph of a function Use the GDC to sketch the graph of the function y = x3 x2 7x 1 for the domain 2 x 3 We need some points to sketch the graph y 5 (-1.2, 4.2) 4 3 2 Press 3 minimum Drag the cursor to the left of a minimum enter Drag the cursor to the right of a minimum enter enter 1 x 3 2 1 1 2 3 -1 -2 -3 -4 Minimum point: (1.9, -11.1) Press 2nd Calc -5 --6 -7 -8 -9 -10 -11 www.mathssupport.org (1.9, -11.1)
Sketching the graph of a function Use the GDC to sketch the graph of the function y = x3 x2 7x 1 for the domain 2 x 3 For the x-intercept: Press 2: zero Drag the cursor to the left of a zero enter Drag the cursor to the right of a zero enter enter y 5 (-1.2, 4.2) 4 3 2 (-2, 1) 1 x (-0.15, 0) 3 2 1 1 2 3 -1 -2 X-intercept: (-0.15, 0) Press 2nd Calc -3 -4 For the starting point: Press 1: value -5 --6 -7 -8 -9 -10 -11 enter x: 2 Starting point: (-2, 1) Press 2nd Calc www.mathssupport.org (1.9, -11.1)
Sketching the graph of a function Use the GDC to sketch the graph of the function y = x3 x2 7x 1 for the domain 2 x 3 For the ending point: Press 1: value y 5 (-1.2, 4.2) 4 3 2 enter x: 3 Ending point: (3, -4) Press 2nd Calc (-2, 1) 1 x (-0.15, 0) 3 2 1 1 2 3 (0, -1) -1 For the y-intercept: Press 1: value -2 -3 enter x: 0 Y-intercept: (0, -1) (3, -4) -4 -5 Join the points with a smooth curve --6 -7 -8 -9 -10 -11 www.mathssupport.org (1.9, -11.1)
Thank you for using resources from A close up of a cage Description automatically generated For more resources visit our website https://www.mathssupport.org If you have a special request, drop us an email info@mathssupport.org www.mathssupport.org
