FUNCTIONS AND GRAPHS
THE STANDARD FORM AND SKETCH OF EACH TYPE OF GRAPH
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2
Type
Straight
line
Standard form
y = mx + c
Linear
m = gradient
c = y-intercept
Parabola
y = a(x+p)2 + q
Quadratic
Graph
𝑚+
𝑚−
𝑎+
𝑎−
p<0
a = shape
p = left and right
q = up and down
p>0
3
Hyperbola
𝑦=
𝑎
+𝑞
𝑥+𝑝
𝑎+
a = which 2
quadrants
p = left and right x
– asymptote
q>0
q<0
p<0
q = up and
down
y – asymptote
4
Exponential
𝑎−
p>0
𝑦 = 𝑏𝑎 𝑥+𝑝 + 𝑞
p = influence on
shape and xintercept
𝑞 = y-asymptote
up and down
b = above or
below asymptote
a>1
0 <a < 1
q>0
q<0
PARABOLA: From 𝑦 = 𝑎(𝑥 + 𝑝)2 + 𝑞 𝑦 = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐
Remove the brackets
From 𝑦 = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 𝑦 = 𝑎(𝑥 + 𝑝)2 + 𝑞
Complete the square
1
NB: Is often used to determine a maximum or minimum value in a real life situation.
COMPARISON BETWEEN 2 FORMULA
𝒚 = 𝒂(𝒙 + 𝒑)𝟐 + 𝒒
𝑦 −intercept
𝑥 − intercept
Symmetrical axis
Turning point
Let x = 0 , solve for y
Let y = 0, solve for x
𝑥 = −𝑝
(x;y) = (−p;q)
𝒚 = 𝒂𝒙𝟐 + 𝒃𝒙 + 𝒄
c -value
Let y = 0, factorize or use
formula, solve for x
−𝑏
𝑥=
2𝑎
(x;y) = (
−𝑏
2𝑎
;
4𝑎𝑐−𝑏2
4𝑎
)
Or subst the x-value in
equation to find the y-value.
HOW TO DRAW THE DIFFERENT GRAPHS ON SCALE
When starting you:
Draw in the asymptotes.
Then plot the critical values – x-intercepts, y-intercept, turning points.
Connect all points.
Use your calculator to draw the graph (hyperbola or exponential) to get the
shape accurate.
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•
•
•
•
•
Press MODE: choose option 3 or 7 (table)
Type in the graph [ For a “X” , alpha X(on the “)” ]
START: –5 =
END: 5 =
STEP: 1 =
You will get a table with x and y-values
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HOW TO SET UP AN EQUATION FOR EACH GRAPH
Method:
~ Identify the type of graph
~
~
~
~
Start with the standard form
Substitute asymptotes ( hyperbola and exponential)
Substitute y-intercept ( straight line and parabola )
Determine the a-value by going through any point on the graph
Parabola: Depends on the given information
If you have a turning point and a point
start with y = a(x + p)²+q
subst (−p ; q)
to solve for a – subst the point
If there are two x-intercepts and a
point:
start with y = a(x – x1)(x – x2)
subst the 2 x- intercepts
to solve for a – subst the point
If you have two random points:
Set up 2 equations going through
the two points, using y = ax2+bx+c
Information on c-value will be
given, directly or indirectly.
Solve a and b simultaneously.
PROPERTIES OF GRAPHS : DOMAIN AND RANGE, ASYMPTOTES,
SYMMETRICAL AXES
The domain can be defined as all the valid x-values of a graph.
The range can be defined as all the valid y-values of a graph.
Asymptotes are lines that are not valid on a graph.
Symmetrical axis are lines that gives you a mirror image of the graph.
Domain
Range
Straight Line
𝑥∈𝑅
𝑦∈𝑅
Parabola
𝑥∈𝑅
Hyperbola
Exponential
𝑥∈𝑅/
−p
𝑥∈𝑅
Asymptotes
Symmetrical axes
𝑦=𝑥
None
𝑦 ∈ [𝑞; ∞)
𝑦 ∈ (−∞; 𝑞)
None
𝑦 ∈ 𝑅/𝑞
𝑥 = −𝑝 and
𝑦=𝑞
𝑦 ∈ [𝑞; ∞)
𝑦=𝑞
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𝑥=
−𝑏
2𝑎
or 𝑥 = −𝑝
𝑦 = ±𝑥 ± 𝑐
Or 𝑦 = ±(𝑥 + 𝑝) + 𝑞
None
HOW TO DO READINGS FROM THE GRAPHS AND INTERPRETATIONS
Type of question
Write in coordinate form
To determine a x-intercept
To determine a y-intercept
Write down equations of asymptotes
Vertical lengths
Horizontal lengths
Lengths between two points
Maximum or minimum length between
two graphs
Method to answer
(x;y)
Let y = 0
Let x = 0
x = − p , y = q (hyperbola)
y = q (exponential)
x-values are fixed, difference in y-values
y-values are fixed, difference in x-values
Top graph – bottom graph @ certain x-value
Or determine the coordinate of each point and
get the difference between the y-values.
Max/Min AB = Top – Bottom
Create new quadratic equation
Determine x-value with 𝑥 =
−𝑏
2𝑎
Substitute in new quadratic equation to get
max/min value.
To rewrite y = ax2 + bx + c to
y = a(x + p)2 + q
Equation of axis of symmetry:
parabola
Equations of axis of symmetry :
hyperbola
f(x) = 0
Complete the square
f(x) > 0 or ≥ 0
f(x) above the x-axis, always start x ∈ …
f(x) < 0 or ≤ 0
f(x) below the x-axis, always start x ∈ …
f(x) > g(x)
f(x) above g(x)
f(x) = g(x) or f(x) – g(x) = 0
[when two graphs cross each other ]
f(x) . g(x) > 0
Equate the two equations and solve for x
f(x). g(x) < 0
Product of two graphs is negative
𝑥=
−𝑏
2𝑎
or 𝑥 = −𝑝
𝑦 = 𝑥 + 𝑐 or 𝑦 = −𝑥 + 𝑐
or 𝑦 = ±(𝑥 + 𝑝) + 𝑞
Intercepts at the x-axis
Product of two graphs is positive
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Average gradient
𝑚=
𝑓(𝑥1 ) − 𝑓(𝑥2 )
𝑥1 − 𝑥2
Domain
(has to do with the valid x-values)
Range
(has to do with the valid y-values)
Reflection in the y-axis
x∈…
Reflection in the x-axis
y – values signs change → −f(x)
Reflection in the y = x line
x and y values changes around – inverse
Movements of the graph up and down
q-value changes → f(x) ± q
Movements of the graph left and right
p-value changes (opp signs) → f(x ± p)
Increasing values of a graph
Decreasing values of a graph
Both x and y-values go bigger / smaller
X-values go up, y-values go down or the other
way round.
y∈…
x – values signs change → f(−x)
Remember : Round brackets excludes a value for eg. x ∈ ( … )
Block brackets includes a value for eg. x ∈ [ … ]
Infinity - ∞ - is not a number. It always has a round bracket
f(x) = y means the same thing. Eg f(x) = x + 4 , means y = x + 4
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