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Graphs Flashcards

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1
Q

What is the name of a line that runs all the way through a circle and continues through it?

A

A secant line

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2
Q

What is the solution to: sketch the graph of a>0, b<0 where y=ax+b

A

a>0 so ‘a’ is a positive integer, thus the gradient is positive
b<0 so ‘b’ is a negative integer, thus the y-intercept is negative
(Answer is any graph that fits these conditions)

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3
Q

When y=k(x-a)^2 what effect does ‘k’ have on the roots?

A

No effect

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4
Q

What effect does the ‘2’ have on the roots for this equation:
y = 2(4x-3)(8x+1) ?

A

No effect

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5
Q

What’s the value of root 2 as a power of 8?

A

8^1/6

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6
Q

What is the value of 1/(x^-1) ?

A

x

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7
Q

What is the fewest number of real roots a cubic graph can have?

A

1

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8
Q

Can a cubic graph have 2 roots?

A

Yes, it can have a repeated root at one point (where the graph has a turning point there) and another root somewhere else

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9
Q

How do you name the quadrants in a graph?

A

1 is the top right quadrant, the others are named up to 4 in a clockwise direction

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10
Q

When mapping y=x^3 onto y=-x^3 what is the transformation?

A

Reflection in the x-axis

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11
Q

In y = 1/8(2-x)(3-x) what effect does 1/8 have on the gradient of the graph?

A

The gradient becomes shallower

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12
Q

In y = -(x+3)(x-2) what effect does the minus sign have on the roots?

A

No effect it only refers to the orientation of the graph (-x^3 shape rather than x^3)

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13
Q

How do you describe the mapping of y = 1/x onto y = (2/x) -3 ?

A

A stretch parallel to the y-axis first (represented by the 2) then a translation of the graph by the vector zero, -3

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14
Q

Simplify y = (x+5) / (x+3)

A

y = (x+3)+2 / (x+3) = 1 + (2 / (x+3))

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15
Q

What does the domain of a graph refer to

A

The set of all possible input values of x for which the function is defined. (To find it, look at the graph’s horizontal extent—from the furthest left point to the furthest right point—to see which x values are covered)

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16
Q

What is the range of a graph?

A

The set of all possible y-values it covers, from its lowest to its highest point. (To find the range, you look at the graph vertically, scanning from the bottom of the graph upwards to determine the minimum and maximum y-values the function reaches.)

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17
Q

How is y=1/x mapped onto y=-1/x ?

A

A reflection in the x-axis

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18
Q

How is y=1/x mapped onto y=1/-x ?

A

A reflection in the y-axis

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19
Q

Use polynomial division to turn y = (2x-3) / (x+4) into a form that can be plotted onto a graph more easily

A

2 + (a remainder of ) 11 / (x+4) is the answer

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20
Q

Simplify y = (3x+1) / x

A

(3x/x) + (1/x) = 3 + 1/x

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21
Q

What does the 2 in y=2/x do to the shape of the graph compared to y=1/x ?

A

Causes a stretch by scale factor 2 parallel to the y-axis (the curve of the graph gets further away from the intersection of the asymptotes!!)

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22
Q

What is the transformation mapping y=x^2 onto y+3=((x+1)^2)-9

A

y=x^2 mapped onto y=((x+1)^2)-12 is a translation by the vector [-1,-12]

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23
Q

How is y=log(x) mapped onto y=log((x+2)/1000) ?

A

log((x+2)/1000) is the same as log(x+2) - 3 so it’s a translation by the vector [-2,-3]

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24
Q

Write the equation of the image of y=x^2+6x after a translation by vector [-2,3]

A

y=(x+2)^2+6(x+2)+3

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25
If you have 3 points (A,B &C) plotted on a graph and you construct the perpendicular bisectors of AB & BC, then find where the bisecting lines intersect (P). What is P?
The centre of the circle that forms if you connect A, B & C (if this doesn’t make sense there’s a diagram in the yellow book)
26
What is the technical name for a parallelogram?
A rhomboid
27
What is true of opposite angles in a rhomboid?
They’re equal
28
What is true of the side lengths of a rhombus?
They’re equal (it’s like a tilted square)
29
If you connected the opposite corners of a square what would be true of the angles of intersection?
They would all be 90 degrees (orthogonal lines)
30
What are the 4 examples of parallelograms?
1) Rhombus 2) Rhomboid 3) Square 4) Rectangle (Parallelograms have opposite sides which are parallel)
31
If you drew a shape inside a circle with its vertices touching the circle what would happen to its inner angles as the number of sides increased towards infinity? What does this suggest about how many sides a circle has?
The inner angles would tend towards 180 degrees (but never reach it otherwise you would have a linear shape not a circle). This tells you a circle has an infinite number of sides.
32
x^2 + y^2 = 25, what is the equation of a circle with the same centre but twice the diameter.
Radius = 5 Doubled radius = 5x2 =10 10^2 =100 so **y^2 + x^2 = 100**
33
How is the graph x^2 + y^2 = 36 mapped onto (x+2)^2 + (y-10)^2 = 36
Translation by vector [-2,10]
34
What is the centre and radius of x^2 + y^2 = 64
Centre = origin (0,0) Radius = 8
35
What is the centre and radius of (x+8)^2 + (y-10)^2 = 10^2
Centre = (-8,10) Radius = 10
36
How is (x+5)^2 + (y-8)^2 = 1 mapped onto (x+1)^2 + (y-9)^2 = 1
Translation by vector [4,1]. **An easy way of working this out is to find the centres of both circle equations and find the translation between them. (-5,8) onto (-1,9) = 4 across, 1 up**
37
If the answer to an equation for calculating a circle’s area is greater than the radius^2 is the point inside, outside or on the circle?
Outside
38
Complete the circle theorem: if
A diameter of the circle with A, B and C on the circumference
39
Solve 2log3,(x-5) - log3,(2x-13) = 1
1) log3,(x-5)^2 - log3,(2x-13) = 1 2) log3,(x-5)^2/(2x-13) = 1 3) 3 = (x-5)^2/(2x-13) 4) x^2-10x+25 = 6x-39 5) x^2-16x+64 = 0 6) (x-8)^2 = 0, **x=8**
40
What are the roots and y-intercept of y = (3x-1)(2x+5) ?
Y-intercept = (0,-5) Roots = **(1/3,0) & (-5/2,0)**
41
If g(x) = 10^x work out the coordinates of intersection of y = g(x+3)-3 with the coordinate axis
1) y = 10^(x+3) -3 2) intercepts x-axis at ((log10,3)-3,0) 3) intercepts y-axis at (0,997)
42
Describe the transformation that maps y = x^2 +5 onto y = 2x^2 +5
Stretch, parallel to the x-axis, factor 1/√2 ((√2)^2 gives you the 2x, then because it’s a stretch parallel to the x-axis you need the reciprocal!)
43
The curve y = log2(x) passes through the point (a,3) state the value of a
3 = log2(a) 2^3 = a a = 8
44
How is the graph of y = f(x) mapped onto y = g(x) when f(x) = -x^3 +2x^2 +5x-10 and g(x) = x^3 +2x^2 -5x -10
Reflection in the y-axis
45
How is the graph of y = f(x) mapped onto y = h(x) when f(x) = -x^3 +2x^2 +5x-10 and h(x) = -8x^3 +8x^2 +10x -10
Stretch by scale factor 1/2, parallel to the x-axis because x has been replaced by 2x
46
How is the graph of xy=1 mapped onto x(y+2)=1?
Translation by vector [0,-2]
47
How is the graph of xy=1 mapped onto xy-y=1?
Rearrange xy-y=1 to get y(x-1)=1 Translation by vector [1,0]
48
How is x^2+y^2=9 mapped onto x^2+1/4y^2=9
Stretch, parallel to y-axis, factor 2
49
How is y=9^x mapped onto y=3^x?
Stretch, parallel to x-axis, factor 2
50
The perimeter of a rectangle is 20cm , the length of one of its diagonals is 3√6cm work out the dimensions of the rectangle in surd form
x^2 + y^2 = (3√6)^2 =54 2x+2y = 20, x+y=10 Substitute into top equation gives you x= 5±√2
51
State the turning point of the graph y=5-3(x-8)^2
Same as y=-3(x-8)^2 +5 so **(8,5)**
52
Write the interval and set notation for this inequality: -3≤x<1
Set: {x:x∈ℝ,-3≤x<1} (colon can be replaced with a straight line) Interval: x∈[-3,1)
53
What does the straight line after the x mean in set notation?
‘Such that’
54
What are the possible solutions to -2
∅ (ℤ means positive integers of which there are none between -2 and -1)
55
What does a square bracket in interval notation denote?
That the number is included so the symbol must be (great/ less than) or **equal** to
56
What does a curved bracket in interval notation denote?
That the number is not included e.g. infinity which is never included as it’s not a number
57
For the graph y=x^2+kx+5 what values of k are there no real roots? Write in set notation
b^2-4ac<0 means no real roots k^2-20<0 (k+√20)(k-√20)<0 so roots are √20 and -√20 and because the quadratic is less than zero you want the bit under the x-axis (between the roots) The inequality is -√20
58
Is zero in ℤ and ℕ?
Yes
59
Describe how the set notation (letters) are linked in terms of concentric circles
ℕ is in ℤ which is in ℚ which is in ℝ
60
Is 5 in the naturals, reals, rationals and integers?
Yes (it can be expressed as 5/1)
61
How do you describe 2 unrelated set/interval notations?
Using the union symbol: U
62
Find x for 2-3x ≥ 17
-3x ≥ 15 x ≤ -5
63
Give the inequality for (x-5)(x+5)<0 and (x-5)(x+5)≤0
1) -5
64
Write x^2>9 as an inequality
x^2-9>0 (x-3)(x+3)>0 **-3>x, 3
65
Write the interval notation for: {x:x∈ℝ,x≤-1} U {x:x∈ℝ,x>2}
**x∈(-∞,-1] U x∈(2,∞)** (remember square versus curved brackets and the union symbol!)
66
What should a sketch of y>x^2 +4x -12 look like?
1) the quadratic should be drawn with a **dashed line** as the sign is < not less than **or equal to** (which would be a solid line) 2) everything above the dashed line should be shaded and labelled with the letter ‘R’ (stands for region) 3) points of intersection labelled (-6,0), (2,0) & (0,-12)
67
What is the y-intercept and x-intercept of this graph: y= (3^x)-5 ?
y-intercept = **(0,-4)** x-intercept = 5=(3^x), x= log3(5), **(log3(5),0)**
68
What are the y-intercept, x-intercept and asymptote of this graph: y= log2(x+4)?
y-intercept = log2(4), 2^x=4, x=2 **(0,2)** x-intercept = 2^0=x+4, 1=x+4, x=-3, **(-3,0)** Asymptote = **x=-4** as it’s a translation by vector [-4,0]
69
Fully factorise: 3x^3-13x^2-18x+40, 2 factors of the quadratic are (x+2) and (x-5)
(x-5)(x+2)(3x-4) you can work this out by doing polynomial division twice (it’s -4 because the numbers at the ends must multiply to make the y-intercept of +40 and -5x2x-4=40)
70
A circle has equation x^2+y^2+22x-30y+k=0. In the case where the circle doesn’t intersect either axis show that 225 < k < 346
(x+11)^2 + (y-15)^2 -121 -225 +k=0 (x+11)^2 + (y-15)^2 = 346-k For this to be a circle then the radius is >0 hence 346-k>0 so k<346 Centre is (-11,15), radius <11 (because 11 is smaller than 15 we use this one as to not intersect the axis ‘r’ must be less than 11 & 15.) √346-k<11, 346-k<121 k>225 hence 225
71
It is given that the circles with equations: C1: (x-3)^2 + y^2 =1 C2: (x-5)^2 + (y-3)^2 =6 don’t touch. Point A lies on C1, point B lies on C2. Find the minimum distance between points A and B
C1 centre: (3,0), radius: 1 C2 centre: (5,3), radius: √6 Distance between circles is √(3-5)^2 + (0-5)^2 = √13 Minimum distance between A & B is √13 -1- √6 =0.156
72
Sketch the graph of: y = (x-2)^2 (x+1)(x+2)^2 Hence solve the inequality: y = (x-2)^2 (x+1)(x+2)^2 ≤ 0
On the sketch the roots should be at x=2,-1,-2 and the points where the graph is less than or equal to zero are less than -1 **and at 2**. So, x≤-1 and **x=2**
73
By sketching a suitable diagram and without solving the equation, state the number of real solutions to the equation: x^2-4x = -1/x
1) sketch the graph of y=x^2-4x 2) sketch the graph of y=-1/x on the same axis 3) the graphs intersect at 3 points!
74
The curve y=x^2+7x+7 is translated by [3,0]. Find the equation of the translated curve.
(x-3)^2+7(x-3)+7 x^2-6x+9+7x-21+7 x^2+x-5
75
How is ln(x) mapped onto ln(x)^2?
ln(x)^2= 2ln(x) Stretch parallel to the y-axis factor 2
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