Quadratics

Question 1

What are quadratic relationships?

How are they different to linear relationships

Answer 1

Quadratics similar to linear equations but instead of a straight line, they form a U or a ∩ shape on the graph

Question 2

What is the quadratic equation?

Answer 2

Y=ax^2+bx+c

Question 3

What are the 2 rules that allow Quadratics to be formed?

Answer 3

Quadratics can be formed when
1. Equations that have terms where the highest power is 2
2. All the powers are positive

As long as the equation follows the 2 rules, a quadratic can be formed

Question 4

What do quadratics form?

Answer 4

The U or ∩ shaped formed on the graph is called a parabola
Quadratics are the equation
Parabola is the U/∩ on the graph
Parabolas go on forever (draw arrows on the ends)

Question 5

What are the 3 features of a parabola?

Answer 5

1. Turning point/vertex
2. Axis of Symmetry
3. X/Y intercepts

Question 6

If the parabola opens upwards what is it called?

Answer 6

It’s called concave up and it will have a minimum value.
Convave up=Opens upwards
Minimum value=Turning point is the lowest value

Question 7

If the parabola opens downwards what is it called?

Answer 7

Its called concave down and it will have a maximum value
Concave down=Opens downwards
Maximum value=Turning point is the highest value

Question 8

How is the turning point/vertx written?

Answer 8

The turning point is marked by coordinates

Question 9

What is the axis of symmetry in parabolas?

Answer 9

The vertical line going through the middle of U or ∩. The axis of symmetry has the equation of x=? because it is a vertical line (e.g x=2)

Question 10

What is an abosolute value?

Answer 10

The absolute value is the positive version of something

Question 11

What is the absolute value of the coefficient -3x^2+9

^ means power so -3x to the power of 2

Answer 11

3x^2+9 is the abosolute value of the coefficient because the absolute value of -3 is 3

Question 12

What is the most basic parabola?

Answer 12

The most basic parabola is Y=x^2 where a=1 and b/c=0.

Basically its Y=x^2+0x+0 but because b/c=0 we don’t write it

Question 13

What is the vertical reflection of y=x^2?

Vertical reflection means reflecting across the x axis

Answer 13

Y= -x^2 is the vertical reflection because the x becomes negative

Vertical reflection means reflecting across the x axis

Question 14

If the parabola opens upwards what is the “a” value?

Answer 14

The “a” value will be positive if the parabola opens upwards

Question 15

If the parabola opens downwards what is the “a” value?

Answer 15

If the parabola opens downwards the “a” value will be negative

Question 16

What will the vertex be in the most basic parabolas y=ax^2/y=-ax^2

Vertex=Turning Point

Answer 16

The vertex will always be (0,0) in the most basic parabola

Vertex=Turning Point

Question 17

What is dilation?

Answer 17

Dilation streches or compresses a graph based on a scale factor

The scale factor is usually a coefficient in quadratics

Question 18

What is the dilation of y=ax^2

Answer 18

The dilation represents a vertical dilation of y=x^2 by a factor of a

a is the scale factor

Question 19

What would a dilation of y=x^2 by a factor of 3 do to the parabola?

Answer 19

The parabola will go narrower (making the parabola stretch vertically)

Question 20

What would a dilation of y=x^2 by a factor of 0.2 do to the parabola?

Answer 20

The parbola will become wider (making the parabola compressed)

Question 21

What would a dilation more than one do to the parabola?

Answer 21

Would make the parbola narrower (stretched vertically)

Also known as vertical enlargement

Question 22

What would a dilation less than one do to the parabola?

Answer 22

It would make the parabola wider (compressing it)

Also known as vertical compression

Question 23

Which graph is vertically enlarged?
1. y= -2x^2
2. y= -x^2

Answer 23

Y= -2x^2 because if a is higher than 1 (absolute value) the parabola will be vertically enlarged.
Y= -x^2 is vertically compressed

When talking about dilation use the abosolute value of a

Question 24

What is the quadratic form for vertical translation?

Answer 24

Y=ax^2+k is the form when a parabola is vertically translated

Question 25

How do you know if the parabola is translated up or down from the formula: Y=ax^2+k?

Answer 25

If k is bigger than 1:parabola translate up If k is lower than 1: Parabola translate down

Question 26

What is the turning point of a vertically translated parabola in the form y=ax^2+k?

Answer 26

The turning point will always be (0,k)

Question 27

What happens if y= -x^2+k?

Answer 27

Even thought the parabola is opening down the vertex will still go up.

Question 28

Find the equation of the quadratic that passes through (1,7) | In the form y=x^2+k (without k the quadratic can no be formed)

Answer 28

1. Sub x=1, y=7 2. 7=1^2+k 3. 7=1+k 4. 6=k 5. Y=x^2+6 | We put it into standard vertical translated quadratic form

Question 29

Find the equation of the quadratic that has a y-int of -4 and passes through (2,-16) | In the form y=ax^2+k (without k the quadratic can no be formed)

Answer 29

1. Sub: x=2, y=-16, k=-4 2. -16=a*2^2-4 3. -16=4a-4 4. -12=4a 5. -3=a 6. Equation = y=-3x^2-4 The y-int=k | We are trying to find a

Question 30

What is the quadratic equation when the parabola is translated horizontally?

Answer 30

Equations in the form y= a(x-h)^2 are horizontally translated

Question 31

How can you tell from the equation the horizontal translation

Answer 31

h determines the translation however whatever the sign of h is inside the bracket it is the opposite. | If h is positive in the bracket the real value is negative (moves left)

Question 32

Why in the horizontal translation of y=(x-h)^2 is the actual h value positive?

Answer 32

Because of the square the -h is being multiplied by itself and becomes a positive h (moves right)

Question 33

What is the axis of symmetry in a horizontally translated parbola?

Answer 33

The axis of symmetry is = to the value of h | X int = (h,0)

Question 34

If a=1 and h=3 what is the quadratic equation? | The parbola has been horizontally translated

Answer 34

If we sub h=3 into the equation it becomes y=(x-3)^2 | since a=1 it doesn't count

Question 35

If a=1 and h=-3 what is the equation? | The parbola has been horizontally translated

Answer 35

if we sub h=-3 the equation will be y=(x--3)^2 and that becomes y=(x+3)^2

Question 36

What is the value of h if h in the bracket (of the formula) is negative | In a horizontally translated parabola

Answer 36

The value of h is positive when a h is negative in the formula | Opposite signs

Question 37

What is the value of h if h in the bracket (of the formula) is positive | In a horizontally translated parabola

Answer 37

The value of h is positive when a h is negative in the formula | Opposite Signs

Question 38

Describe the translation of y=(x-7)^2

Answer 38

The parbola has been translated 7 units to the right because the real value of h is 7

Question 39

What is the horizontal translation from Y=(x+7)^2

Answer 39

The real h value is -7 because it is the opposite sign of h inside the bracket. Therefore the parabola is translated 7 units left

Question 40

What is the quadratic equation when the parabola is translated 9 units to the left? | In the form y=a(x-h)^2

Answer 40

h=-9 so so Y=1(x--9)^2 is equal to y=(x+9)^2

Question 41

What is the quadratic equation when the parabola is translated 11 units right and vertically reflected from the base y=x^2 | Vertically reflected from concave up (opens upwards)

Answer 41

Vertically reflected=a=-1 11 units to the right= h=11 Y=a(x-h)^2 becomes Y=-1(x-11)^2 | The veritcal reflection makes the parabola open downwards so a= -1

Question 42

Describe the equation y=(x+4)^2

Answer 42

The concave up parabola has been horizontally translated left by 4 units

Question 43

Descirbe the equation y=-x^2-6

Answer 43

The parabola has been reflected over the x-axis and translated 6 units down

Question 44

Describe the equation y=(x+5)^2+3

Answer 44

The parabola has been translated left by 5 and up by 3