Dr J Frost ([email protected]) Last modified: 15 th February 2015.
GCSE: Curved Graphs Dr J Frost ([email protected]) Last modified: 31 st December 2014...
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Transcript of GCSE: Curved Graphs Dr J Frost ([email protected]) Last modified: 31 st December 2014...
GCSE: Curved Graphs
Dr J Frost ([email protected])
Last modified: 31st December 2014
GCSE Revision Pack Reference: 94, 95, 96, 97, 98
GCSE Specification
Plot and recognise quadratic, cubic, reciprocal, exponential and circular functions.
Plot and recognise trigonometric functions and , within the range -360Β° to +360Β°
Use the graphs of these functions to find approximate solutions to equations, eg given x find y (and vice versa)
Find the values of p and q in the function given coordinates on the graph of
βGiven that and are points on the curve , find the value of and .βThe graph shows .
Determine the coordinate of point .
The diagram shows the graph of y = x2 β 5x β 3(a) Use the graph to find estimates for the solutions of
(i) x2 β 5x β 3 = 0(ii) x2 β 5x β 3 = 6
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2
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Quadratic
When
The line for a quadratic equation is known as a parabola.
? ?
?
When
Skill #1: Recognising Graphs
Reciprocal
When
?
The lines x = 0 and y = 0 are called asymptotes.! An asymptote is a straight line which the curve approaches at infinity.
?
?
When
You donβt need to know this until A Level.
Skill #1: Recognising Graphs
Exponential
π¦=πΓππ₯
x
y
π
The y-intercept is because .(unless , but letβs not go there!)
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Skill #1: Recognising Graphs
?
π₯
π¦
5-5
5
-5
The equation of this circle is:
x2 + y2 = 25
The equation of a circle with centre at the origin and radius r is:
?
Circle
Skill #1: Recognising Graphs
Quickfire Circles
1-1
1
-1
x2 + y2 = 1
3-3
3
-3
x2 + y2 = 9
4-4
4
-4
x2 + y2 = 16
8-8
8
-8
x2 + y2 = 64
10-10
10
-10
x2 + y2 = 100
6-6
6
-6
x2 + y2 = 36
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Card Sort Match the graphs with the equations.
A B C D
E F G H
I J K L
x) y = x2 + x - 2
ix) y = 2x3
i) y = 5 - 2x2 vii) y=-2x3 + x2 + 6xiv) y = 3/x
iii) y = -3x3
viii) y = -2/xii) y = 4x
xii) y = 2x β 3
v) y = x3 β 7x + 6 xi) y = sin (x)
vi)
A: quadraticB: cubicC: quadraticD: cubicE: cubicF: reciprocalG: cubicH: reciprocalI: exponentialJ: linearK: sinusoidalL: fictional
Equation types:
????????????
Click to reveal answers.
90 180 270 360
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1
-1
π¦=sin π₯0 90 180 270 360
0 1 0 -1 0? ? ? ? ?
Click to brosketch
Skill #2: Plotting and recognising trig functions.
Quickfire Coordinates
π¦=sin π₯ π¦=cos π₯ π¦=sin π₯ π¦=cos π₯
π΄ (270 ,β1 ) π΅ (90 ,0 ) πΆ (360 ,0 ) π· (0 ,1 )
π¦=sin π₯ π¦=cos π₯ π¦=sin π₯ π¦=cos π₯
πΈ (180 ,0 ) πΉ (180 ,β1 ) πΊ (90,1 ) π» (270 ,0 )
? ? ? ?
? ? ? ?
π΄π΅ πΆ
π·
πΈ
πΉ
πΊ π»
SKILL #3: Using graphs to estimate values
The diagram shows the graph of y = x2 β 5x β 3
a) Find the exact value of when .
b) Use the graph to find estimates for the solutions of
(i) x2 β 5x β 3 = 0
(ii) x2 β 5x β 3 = 6
Bro Tip for (b): Look at what value has been substituted into the equation in each case.
a)
b) i) When , then using graph, roughly ii)
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?
?
Test Your Understanding
The graph shows the line with equation
Find estimates for the solutions of the following equations:
i)
ii) ?
?
90 180 270 360
1
-1
Suppose that Using the graph, find the other solution to
Using a Trig Graph
Q
π=πππΒ°1
β2 ?
ππ πππ
Suppose that Using the graph, find the other solution to Q
π=πππΒ°?
We can see by symmetry that the difference between 0 and 45 needs to be the same as the difference between and 180.
90 180 270 360
1
-1
Test Your UnderstandingThe graph shows the line with equation a) Given that , find the other solution to
b) Given that , find the other solution to ?
?
Exercise 1 (on provided sheet)Identify the coordinates of the indicated points.
π¦=sin π₯π΄
π΅π¦=3Γ2π₯
πΆ
π₯2+π¦2=9
π·
π¦=4π₯
1
πΈ
π¨ (ππ ,π ) π© (πππ ,π )
1
???
??
2 Which of these graphs could have the equation ?
a b c
c, because a is the wrong way up (given term has positive coefficient) and b has the wrong y-intercept.?
Match the graphs to their equations.
i. Eii. Biii. Fiv. Cv. Dvi. A
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Exercise 17 Using the cos graph below, and given
that , find all solutions to (other than 45).
Given that , find all solutions to
[Hard] Given , again using the graph, find all solutions to ?
?
b
c
a
The graph shows .
Use the graph to estimate the solution(s) to:i) ii) iii)
The graph shows the line with equation
Use the graph to estimate the solution(s) to: i) ii) iii) By drawing a suitable line onto the graph, estimate the solutions to
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???
???
?
Exercise 1
i) Given , determine all solutions to
ii) Given , determine all solutions to
iii) [Harder] Given , determine the two solutions to (note the minus)
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?
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(1,7)
(3,175)
The graph shows two points (1,7) and (3,175) on a line with equation:
Determine and (where and are positive constants).
Answer:
Dividing:
Substituting back into 1st equation:?
SKILL #4: Finding constants of
Bro Hint: Substitute the values of the coordinates in to form two equations. Youβre used to solving simultaneous equations by elimination β either adding or subtracting. Is there another arithmetic operation?
Test Your Understanding
Given that and are points on the curve , find the value of and .
Given that and are points on the curve where and are positive constants, find the value of and .
Q
N
?
?
Exercise 1 (continued)
Given that the points and lie on the exponential curve with equation , determine and .
Given that the points and lie on the exponential curve with equation , determine and .
Given that the points and lie on the exponential curve with equation , determine and .
Given that the points and lie on the exponential curve with equation , determine and .
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