REVISION: FUNCTIONS
10 JUNE 2013
Lesson Description
In this lesson we revise:
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the effects of a, p and q on the graphs of the form:
2
y = a(x + p) + q
y = a.b
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
(x + p)
+q
how to find the equation of various functions from a given graph
how to find the average gradient between two points on a graph
Key Concepts
The effect of changing p and q
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If q > 0 the graph moves vertically up by 'q' units
If q < 0 the graph moves vertically down by 'q' units
If p > 0 then the graph shifts horizontally to the left
If p < 0 then the graph shifts horizontally to the right
The effect of changing a

Multiplying 'x' by 'a' will stretch or compress the graph vertically by the factor 'a'
o Example (0 < a < 1 compresses the graph)
o Example (a < 0 reflects and stretches the graph)
Graphs of the form: y = a(x + p)2 + q
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If a > 0 then we have a minimum turning point (smile)
If a < 0 then we have a maximum turning point (frown)
If q > 0 then the t.p is above the x axis
If q < 0 then the t.p is below the x axis
p affects the horizontal shift of the graph and the axis of symmetry:
If p > 0 then the t.p is on the left of the y axis
If p < 0 then the t.p is on the right of the y axis
The axis of symmetry is x = -p
Graphs of the form:
Recall: An increasing function-as the x values increase, the y values increase.
A decreasing function-as the x values increase, the y values decrease.
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If a > 0 then the hyperbola is a decreasing function
If a < 0 then the hyperbola is an increasing function
The value of q determines the vertical shift of the graph and also gives the horizontal
asymptote (y = q)
p affects the horizontal shift of the graph and gives the position of the vertical asymptote
(x = -p)
The axes of symmetry are y = x + p + q and y = -(x + p) + q
Graphs of the form: y = a.b(x+p) + q
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There is a horizontal asymptote at y = q
If a > 0 the graph will be decreasing for 0 < b < 1 and for b > 1 the graph is increasing
If a < 0 and 0 < b < 1, the standard graph is reflected about the x axis and is now an
increasing function
If a < 0 and b > 1, the standard graph is reflected about the x axis and is now a decreasing
function.
Finding the Equation
For a Parabola
Case 1: Turning point and another point given
2
Step 1 Use y = a(x + p) + q and substitute the coordinates of the turning point.
Step 2: Find the value of a by substituting the coordinates of another point on the graph into the
equation.
Case 2: The x intercepts and another point given
Step 1: Use y = a(x - x1)(x - x2) where x1 and x2 are the x intercepts
Step 2: Find the value of a by substituting the coordinates of another point on the graph into the
equation.
For a Hyperbola
Step 1: The position of the asymptotes gives us p and q in
Step 2: To find the value of a, substitute any point on the graph into the equation.
Questions
Question 1
(Adapted from Clever Keeping Maths Simple Grade 11 Learner's Book. Page 175, Ex5.21 no. 3)
The equations of the graphs shown in the sketch are
and
a.)
b.)
c.)
d.)
Give the coordinates of D, the y-intercept of both graphs.
Calculate the coordinates of A and B, the x-intercepts of g.
Determine the coordinates of C, the turning point of g.
Calculate the coordinates of E, one of the points of intersection of f and g.
Question 2
(Clever Keeping Maths Simple, Macmillan, Grade 11 Learner's Book. Page 165, Ex 5.19 no 6)
Find the equation of the following graph:
y=b
x+p
+q
Question 3
(Clever Keeping Maths Simple, Macmillan, Grade 11 Learner's Book. Page 177, Ex. 5.21 no. 6)
2
The diagram represents the graphs of y = f(x) = -(x – 2) + 4 and y = g(x) = 2
x+1
– 4.
Calculate the following:
a.)
b.)
c.)
d.)
e.)
OE
OD
OC
BC
FG
Question 4
(Adapted from Clever Keeping Maths Simple, Macmillan, Grade 11, Example 1, Page 167)
2
a.) Calculate the average gradient of the graph y = x - 4 between x=1 and x=3
b.) For which values of x is the graph
i.
increasing
ii.
decreasing