GCSE: Curved Graphs

Dr J Frost (jfrost@tiffin.kingston.sch.uk)

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 -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 given coordinates on the graph of

“Given that and are points on the curve , find the value of and .”The graph shows .

Determine the coordinate of point .

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

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Skill #1: Recognising GraphsLinear

When

The line is known as a straight line.

? ?

?

When

Quadratic

When

The line for a quadratic equation is known as a parabola.

? ?

?

When

Skill #1: Recognising Graphs

Cubic

When When

When

x

y

x

y

?

?

?

?

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 don’t need to know this until A Level.

Skill #1: Recognising Graphs

Exponential

𝑦=𝑎×𝑏𝑥

x

y

𝑎

The y-intercept is because .(unless , but let’s not go there!)

?

Skill #1: Recognising Graphs

?

𝑥

𝑦

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 – 3

v) y = x3 – 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

𝑦=sin 𝑥0 90 180 270 360

0 1 0 -1 0? ? ? ? ?

Click to brosketch

Skill #2: Plotting and recognising trig functions.

Test Your Understanding

90 1

180 0

?

?

90 180 270 360

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1

-1

𝑦=cos 𝑥0 90 180 270 360

1 0 -1 0 1? ? ? ? ?

Click to brosketch

Quickfire Coordinates

𝑦=sin 𝑥 𝑦=cos 𝑥 𝑦=sin 𝑥 𝑦=cos 𝑥

𝐴 (270 ,−1 ) 𝐵 (90 ,0 ) 𝐶 (360 ,0 ) 𝐷 (0 ,1 )

𝑦=sin 𝑥 𝑦=cos 𝑥 𝑦=sin 𝑥 𝑦=cos 𝑥

𝐸 (180 ,0 ) 𝐹 (180 ,−1 ) 𝐺 (90,1 ) 𝐻 (270 ,0 )

? ? ? ?

? ? ? ?

𝐴𝐵 𝐶

𝐷

𝐸

𝐹

𝐺 𝐻

SKILL #3: Using graphs to estimate values

The diagram shows the graph of y = x2 – 5x – 3

a) Find the exact value of when .

b) Use the graph to find estimates for the solutions of

(i) x2 – 5x – 3 = 0

(ii) x2 – 5x – 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

𝒙=𝟏𝟑𝟓°1

√2 ?

𝟒𝟓 𝟏𝟑𝟓

Suppose that Using the graph, find the other solution to Q

𝒙=𝟑𝟑𝟎°?

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.

𝑦=sin 𝑥𝐴

𝐵𝑦=3×2𝑥

𝐶

𝑥2+𝑦2=9

𝐷

𝑦=4𝑥

1

𝐸

𝑨 (𝟗𝟎 ,𝟏 ) 𝑩 (𝟏𝟖𝟎 ,𝟎 )

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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-15 -7 -6 1

Reveal

? ?? ?

Exercise 1 (on provided sheet)

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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 ?

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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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(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. You’re used to solving simultaneous equations by elimination – 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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