1. GCSE: Curved Graphs

2. GCSE Specification
Plot and recognise quadratic, cubic, reciprocal, exponential and circular functions.
Plot and recognise trigonometric functions 𝑦= sin 𝑥 and 𝑦= cos 𝑥 , within the range -360° to +360°
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 𝑦=𝑝 𝑞 𝑥 given coordinates on the graph of 𝑦=𝑝 𝑞 𝑥 1 3
The diagram shows the graph of y = x2 – 5x – 3
(a) Use the graph to find estimates for the solutions of
(i) x2 – 5x – 3 = 0
(ii) x2 – 5x – 3 = 6 2 4
“Given that 2,6 and 5,162 are points on the curve 𝑦=𝑘 𝑎 𝑥 , find the value of 𝑘 and 𝑎.”
The graph shows 𝑦= cos 𝑥 . Determine the coordinate of point 𝐴.

3. Skill #1: Recognising Graphs
Linear
𝒚=𝒂𝒙+𝒃
When 𝑎>0
𝒚=𝒂𝒙+𝒃
When 𝑎<0 ? ? ?
The line is known as a straight line.

4. Skill #1: Recognising Graphs
Quadratic
𝑦=𝑎 𝑥 2 +𝑏𝑥+𝑐
When 𝑎>0
𝑦=𝑎 𝑥 2 +𝑏𝑥+𝑐
When 𝑎<0 ? ?
The line for a quadratic equation is known as a parabola. ?

5. Skill #1: Recognising Graphs
Cubic
𝑦=𝑎 𝑥 3 +𝑏 𝑥 2 +𝑐𝑥+𝑑
When 𝑎>0
𝑦=𝑎𝑥3
When 𝑎>0 y ? ? x
𝑦=𝑎 𝑥 3
When 𝑎<0
𝑦=𝑎 𝑥 3 +𝑏 𝑥 2 +𝑐𝑥+𝑑
When 𝑎<0 y ? ? x

6. Skill #1: Recognising Graphs
Reciprocal
𝑦= 𝑎 𝑥
When 𝑎>0
𝑦= 𝑎 𝑥
When 𝑎<0 ? ? ?
The lines x = 0 and y = 0 are called asymptotes.
! An asymptote is a straight line which the curve approaches at infinity.
You don’t need to know this until A Level.

7. Skill #1: Recognising Graphs
Exponential
𝑦=𝑎× 𝑏 𝑥 ? y 𝑎 x
The y-intercept is 𝑎 because 𝑎× 𝑏 0 =𝑎×1=𝑎.
(unless 𝑎 = 0, but let’s not go there!) ?

8. Skill #1: Recognising Graphs
Circle
The equation of this circle is:
x2 + y2 = 25 𝑦 ? 5
The equation of a circle with centre at the origin and radius r is: 𝑥 2 + 𝑦 2 = 𝑟 2 𝑥 -5 5 -5

9. Quickfire Circles ? ? ? ? ? ? x2 + y2 = 1 x2 + y2 = 9 x2 + y2 = 163 4 -1 1 -3 3 -4 4 -1 -3 -4 ? ?
x2 + y2 = 1
x2 + y2 = 9
x2 + y2 = 16 ? ? 8 10 6 -8 8
-10 10 -6 6 -8
-10 -6 ?
x2 + y2 = 64
x2 + y2 = 100
x2 + y2 = 36

10. Matching Activity Match the graphs with the equations. A B C D E F G H
Equation types:
A: quadratic
B: cubic
C: quadratic
D: cubic
E: cubic
F: reciprocal
G: cubic
H: reciprocal
I: exponential
J: linear
K: Trig Function ? ? ? ? ? E F G H ? ? ? ? ? ? I J K
i) y = 5 - 2x2
iv) y = 3/x
vii) y=-2x3 + x2 + 6x
x) y = x2 + x - 2
ii) y = 4x
v) y = x3 – 7x + 6
viii) y = -2/x
xi) y = sin (x)
iii) y = -3x3
ix) y = 2x3
xii) y = 2x – 3

12. 𝑦= sin 𝑥 ? ? ? ? ? Skill #2: Plotting and recognising trig functions. 8 1 -1
𝑦= sin 𝑥 𝑥 90
180
270
360 𝑦 1 -1 ? ? ? ? ?
Get students to sketch axes and tables in their books.
Skill #2: Plotting and recognising trig functions.
Click to sketch

13. Test Your Understanding? ?

14. 8 1 -1
𝑦= cos 𝑥 𝑥 90
180
270
360 𝑦 1 -1 ? ? ? ? ?
Get students to sketch axes and tables in their books.
Click to brosketch

15. Quickfire Coordinates
𝑦= sin 𝑥
𝑦= cos 𝑥
𝑦= sin 𝑥
𝑦= cos 𝑥 𝐷 𝐵 𝐶 𝐴
𝐴 270, −1 ?
𝐵 90, 0 ?
𝐶 360, 0 ?
𝐷 0, 1 ?
𝑦= sin 𝑥
𝑦= cos 𝑥
𝑦= sin 𝑥
𝑦= cos 𝑥 𝐺 𝐸 𝐻 𝐹
𝐸 180, 0 ?
𝐹 180,−1 ?
𝐺 90,1 ?
𝐻 270, 0 ?

16. SKILL #3: Using graphs to estimate values
The diagram shows the graph of y = x2 – 5x – 3
a) Find the exact value of 𝑦 when 𝑥=−2.
b) Use the graph to find estimates for the solutions of
(i) x2 – 5x – 3 = 0
(ii) x2 – 5x – 3 = 6
Tip for (b): Look at what value has been substituted into the equation in each case.
𝑦= −2 2 −5 −2 −3 =11
i) When 𝑦=0, then using graph, roughly 𝒙=−𝟎.𝟓 𝒐𝒓 𝒙=𝟓.𝟓 ii) 𝒙=−𝟏.𝟒 𝒐𝒓 𝒙=𝟔.𝟒 ? ? ?

17. Test Your Understanding
The graph shows the line with equation 𝑦= 𝑥 2 +𝑥−12
Find estimates for the solutions of the following equations:
i) 𝑥 2 +𝑥−12=5
𝒙=−𝟒.𝟔 𝒐𝒓 𝒙=𝟑.𝟔
ii) 𝑥 2 +𝑥−12=−7
𝒙=−𝟐.𝟖 𝒐𝒓 𝒙=𝟏.𝟖 ? ?

18. Using a Trig Graph Q ? Q ? 𝒙=𝟏𝟑𝟓° 𝒙=𝟑𝟑𝟎° Suppose that sin 45 = 1 2
Using the graph, find the other solution to sin 𝑥 = 1 2 1
1 2
𝒙=𝟏𝟑𝟓° ? 𝟒𝟓
𝟏𝟑𝟓 -1
We can see by symmetry that the difference between 0 and 45 needs to be the same as the difference between 𝑥 and 180. Q
Suppose that sin 210 =− 1 2
Using the graph, find the other solution to sin 𝑥 =− 1 2
𝒙=𝟑𝟑𝟎° ?

19. Test Your Understanding 1 -1
The graph shows the line with equation 𝑦= cos 𝑥
Given that cos 60 = 1 2 , find the other solution to cos 𝑥 = 1 2 𝒙=𝟑𝟎𝟎°
Given that cos 150° =− , find the other solution to cos 𝑥 =− 𝒙=𝟐𝟏𝟎° ? ?
Get students to sketch axes and tables in their books.

20. Exercise 1 (on provided sheet)
Identify the coordinates of the indicated points.
Match the graphs to their equations. 1 3 𝐴
𝑦=3× 2 𝑥
𝑦= sin 𝑥 𝐶 𝐵
𝑦= 4 𝑥
𝑥 2 + 𝑦 2 =9 𝐸 𝐷 1
𝑨 𝟗𝟎,𝟏 𝑩 𝟏𝟖𝟎,𝟎 𝑪 𝟎,𝟑 𝑫 −𝟑,𝟎 𝑬 𝟏,𝟒 ? ? ? ? ?
Which of these graphs could have the equation 𝑦= 𝑥 3 −2 𝑥 2 +3? 2
i. 𝑦=4 sin 𝑥 E ii. 𝑦=4 cos 𝑥 B iii. 𝑦= 𝑥 2 −4𝑥 F iv. 𝑦=4× 2 𝑥 C v. 𝑦= 𝑥 D vi. 𝑦= 4 𝑥 A ? a b c
c, because a is the wrong way up (given 𝒙 𝟑 term has positive coefficient) and b has the wrong y-intercept. ?

21. Exercise 1 (on provided sheet)4 ? ? ? ?
Reveal

22. Exercise 1
The graph shows 𝑦= 𝑥 2 −𝑥−2.
Use the graph to estimate the solution(s) to: i) 𝑥 2 −𝑥−2= 𝒙=−𝟐 𝒐𝒓 𝟑 ii) 𝑥 2 −𝑥−2=− 𝒙≈−𝟎.𝟔 𝒐𝒓 𝟏.𝟔 iii) 𝑥 2 −𝑥−2= 𝒙≈−𝟐.𝟓 𝒐𝒓 𝟑.𝟓 5 7
Using the cos graph below, and given that cos 45 = , find all solutions to cos 𝑥 = (other than 45).
𝒙=𝟑𝟏𝟓° a ? ? ? ?
Given that cos 30 = , find all solutions to cos 𝑥 =
𝒙=𝟑𝟑𝟎°
[Hard] Given cos 60 = 1 2 , again using the graph, find all solutions to 𝑐𝑜𝑠 𝑥 =− 1 2
𝒙=𝟏𝟐𝟎°, 𝟐𝟒𝟎° 6
The graph shows the line with equation 𝑦=6+2𝑥− 𝑥 2
Use the graph to estimate the solution(s) to: i) 6+2𝑥− 𝑥 2 = 𝒙≈−𝟏.𝟔𝟓 𝒐𝒓 𝟑.𝟐𝟓 ii) 6+2𝑥− 𝑥 2 = 𝒙≈−𝟎.𝟕 𝒐𝒓 𝟐.𝟕 iii) By drawing a suitable line onto the graph, estimate the solutions to 6+2𝑥− 𝑥 2 =𝑥 𝒙≈−𝟏.𝟓𝟔 𝒐𝒓 𝟐.𝟓𝟔 b ? c ? ? ? ?

23. Exercise 1 8
Given sin 60 = , determine all solutions to sin 𝑥 = 𝒙=𝟏𝟐𝟎 (,𝟔𝟎)
Given sin 30 = 1 2 , determine all solutions to sin 𝑥 = 1 2 𝒙=𝟏𝟓𝟎 (,𝟑𝟎)
[Harder] Given sin 45 = , determine the two solutions to sin 𝑥 =− (note the minus) 𝒙=𝟐𝟐𝟓°, 𝟑𝟏𝟓° ? ? ?

24. SKILL #4: Finding constants of 𝑦=𝑎× 𝑏 𝑥
The graph shows two points (1,7) and (3,175) on a line with equation:
𝒚=𝒌 𝒂 𝒙
Determine 𝑘 and 𝑎 (where 𝑘 and 𝑎 are positive constants).
(3,175)
Answer:
𝟕=𝒌 𝒂 𝟏 𝟏𝟕𝟓=𝒌 𝒂 𝟑
Dividing:
𝟐𝟓= 𝒂 𝟐 𝒂=𝟓
Substituting back into 1st equation: 𝒌= 𝟕 𝟓
(1,7)
Hint: Substitute the values of the coordinates in to form two equations. You’re used to solving simultaneous equations by elimination – either adding or subtracting. Is there another arithmetic operation?

25. Test Your Understanding
Given that 2,6 and 5,162 are points on the curve 𝑦=𝑘 𝑎 𝑥 , find the value of 𝑘 and 𝑎.
6=𝑘 𝑎 =𝑘 𝑎 5 →27= 𝑎 3 𝒂=𝟑 𝒌= 𝟔 𝟑 𝟐 = 𝟐 𝟑
Given that 3, 45 and 1, are points on the curve 𝑦= 𝑎 2 𝑏 𝑥 where 𝑎 and 𝑏 are positive constants, find the value of 𝑎 and 𝑏.
45= 𝑎 2 𝑏 = 𝑎 2 𝑏 →25= 𝑏 2 𝒃=𝟓 𝒂= 𝟒𝟓 𝒃 𝟑 = 𝟒𝟓 𝟏𝟐𝟓 = 𝟗 𝟐𝟓 = 𝟑 𝟓 Q ? N ?

26. Exercise 1 (continued) ? ? ? ? 1 3 4* 2
Given that the points (1,6) and 4,48 lie on the exponential curve with equation 𝑦=𝑏× 𝑎 𝑥 , determine 𝑎 and 𝑏.
𝟔=𝒃𝒂 𝟒𝟖=𝒃 𝒂 𝟒 →𝟖= 𝒂 𝟑 𝒂=𝟐 𝒃=𝟑
Given that the points (2,48) and 5,3072 lie on the exponential curve with equation 𝑦=𝑏× 𝑎 𝑥 , determine 𝑎 and 𝑏.
𝒂=𝟒, 𝒃=𝟑
Given that the points (1,3) and 3,108 lie on the exponential curve with equation 𝑦=𝑏× 𝑎 𝑥 , determine 𝑎 and 𝑏.
𝒂=𝟑, 𝒃= 𝟏 𝟐
Given that the points (3, 1 72 ) and 7, lie on the exponential curve with equation 𝑦= 𝑏 2 𝑎 𝑥 , determine 𝑎 and 𝑏.
𝒂= 𝟏 𝟐 ,𝒃= 𝟏 𝟑 1 3 ? ? 4* 2 ? ?