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2022 ALGEBRA 2 > DECK 2: QUADRATIC TERMS INTRO > Flashcards

DECK 2: QUADRATIC TERMS INTRO Flashcards

1
Q

What is the shape of a quadratic graph called?

A

A parabola

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2
Q

What is the highest or lowest point (x,y) called on a parabola? It is on the AOS. It is sometimes written as (h,k)

A

The vertex.

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3
Q

Is the vertex of a parabola always on the axis of symmetry?

A

YES. The x value of the vertex is the axis of symmetry.

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4
Q

What is the axis of symmetry?

A

It is the vertical line that goes through the middle of the parabola- parabola folds on itself. THE VERTEX IS ON IT

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5
Q

What does VERTICAL mean?

A

Up and Down

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6
Q

What does the line x = 10 look like?

A

A vertical line through 10 on the x axis

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7
Q

What is the equation of a vertical line through 7 on the x axis? (visualize a vertical line through 7 on the x axis)

A

x = 7
All vertical lines have equation
“ X = some number”

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8
Q

Since the axis of symmetry is a vertical line, the equation of the axis of symmetry is always:

A

X = some number

you have to write “x = ? “

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9
Q

What does HORIZONTAL mean?

A

Left and right

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10
Q

Do all parabolas have x-intercepts? If so, how many?

A

NO - they may have
NONE, ONE or TWO.

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11
Q

Do all parabolas have y-intercepts? If so, how many?

A

YES, they all have EXACTLY ONE y intercept.

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12
Q

How can a parabola have no real roots (no x intercepts)?

A

If it doesn’t touch the x axis. Smiling parabola above, or frowning parabola below.

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13
Q

How can a parabola have only one real root (one x intercept)

A

If it touches the x-axis only at one point (the vertex) and “bounces off”

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14
Q

How can a parabola have two real roots (two x intercepts)

A

If it crosses twice.

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15
Q

How do you find the y intercept for any linear or quadratic function?

A

Set X=0 and solve for y

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16
Q

How do you find the x-intercepts of any linear or quadratic functions?

A

Set Y=0 and solve for x (this involves setting each factor = 0 and solving for x for each)

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17
Q

What are the three forms of quadratic functions?

A

standard, vertex and factored (intercept) forms

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18
Q

What form?
f(x)=ax 2 +bx+c

A

Standard

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19
Q

What is the standard form of a quadratic?

A

f(x)=ax 2 +bx+c

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20
Q

What form?
f(x) = a(x-h) 2 +k

A

Vertex form

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21
Q

What is the vertex form of a quadratic?

A

f(x) = a(x-h) 2 +k

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22
Q

What form?
f(x)=a(x-Z)(x-Z)

A

Intercept or Factored form

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23
Q

What is the intercept (factored) form of a quadratic?

A

f(x)=a(x-Z)(x-Z) where Z and Z are ZEROS, roots, x-intercepts

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24
Q

For any polynomial in factored form, how do you find the ROOTS, ZEROS, SOLUTIONS, X-INTERCEPTS?

A

SET THE FACTORS EQUAL TO ZERO AND SOLVE!
EX: f(x)=3 (x-5)(x+10) Roots are 5 and -10
because (x-5)=0 when x=5
and (x+10)=0 wnen x=-10
You can often do the math in your head. But if a factor is (3x-5), you would have to solve for
x in 3x-5=0

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25
Q

What does “a” tell us about the graph?

A

Two things. If it is + it smiles, if it is - it frowns AND if a>1 (bigger than 1) it gets narrow. If a <1 (smaller than 1) it is WIDE

26
Q

Is “a” the same for all three forms?

A

Yes, whatever “a” is in standard form, it will be the same value in factored and in vertex form. It always tells us + SMILE, -FROWN,
bigger than 1 it gets NARROW, and smaller than one it gets WIDE

27
Q

What if you have +a value?

A

Concave UP, smiling

28
Q

What if you have -a value?

A

Concave DOWN, frowning

29
Q

How do you know when a parabola is smiling?

A

when a is positive

30
Q

How do you know when a parabola is frowning?

A

when a is negative

31
Q

What are the three forms of a quadratic function and their equations?

A

Standard y=ax 2 +bx+c Vertex y=a(x-h) 2 +k Factored y=a(x+f)(x+g)

32
Q

What is the GIFT you get in standard form?

A

The y intercept. It is at C or (0,c)

33
Q

What is the GIFT that you get in vertex form?

A

The vertex (h,k). This also give you the AOS and the MAX/MIN

34
Q

What is the GIFT that you get in factored form?

A

The x intercepts - the ZEROS- the ROOTS.

35
Q

WHAT FORM and WHAT IS THE GIFT?
f(x)= -2x2-4x+9

A

Standard Form
GIFT
Y intercept at (0,9)

36
Q

WHAT FORM and WHAT IS THE GIFT?
f(x)= -2(x+7)2+11

A

Vertex Form
GIFT
Vertex at ( -7, 11)

37
Q

WHAT FORM and WHAT IS THE GIFT?
f(x)= -2(x+9)(x-11)

A

Intercept/Factored Form
GIFT
X-intercepts at -9 and 11
or (-9,0) and (11,0)

38
Q

f(x)=2x 2 -4x+6
What is the y intercept?

A

(0,6)

39
Q

f(x)=2(x-5)(x-3)
What are the x intercepts?

A

At 5 and at 3

40
Q

f(x)= 2(x+4) 2 -3
What is vertex?

A

(-4, -3)

41
Q

If you have a vertex, like (5.10), what else do you have?

A

The AOS and MAX/MIN.
The AOS is at x=5 and the max/min is y=10

42
Q

y=-2x 2 -4x+6
concave up or concave down?

A

a= -2 concave DOWN

43
Q

f(x)=2(x-5)(x-3)
concave up or concave down?

A

a=2 concave UP

44
Q

f(x)= -3(x+4) 2 -3
concave up or concave down?

A

a= -3 concave DOWN

45
Q

If we know the coordinates of the vertex is always (AOS, MAX/MIN), then if a vertex is at (5,12), WHAT OTHER INFORMATION DO WE KNOW?

A

We know the AOS is at x=5
and the max/min is at y=12

46
Q

f(x) = a(x-h) 2 + k
How do we know if it opens up or down?

A

by the sign of a, if its positive, up, negative , down

47
Q

If a parabola is concave up, smiling, will it have a MAX or MIN?

A

IF it is smiling, it goes up forever, so it has no max value IT ONLY HAS A MIN Y VALUE

48
Q

If a parabola is concave down, frowning, will it have a MAX or MIN?

A

If frowning, it goes down forever so it has no minimum, IT ONLY HAS A MAX Y VALUE

49
Q

f(x)= 2(x+4) 2 -3
What is vertex ?

A

at (-4, -3), SMILING so it opens upwards, -3 is a MIN value

50
Q

f(x)= (x+9)2-8
What is vertex

A

at (-9, -8)

51
Q

f(x)=2(x-5)(x-3)
What are the ROOTS?

A

5 and 3

52
Q

y=2(x-7)(x+3)
what are the coordinates of the x intercepts?

A

(7,0) and (-3,0)

53
Q

How do we know when to take opposites when working with quadratics?

A

usually when they are in parentheses

54
Q

Suppose a vertex of a parabola is at (5,10), what else do you know?

A

You know the AOS is at x=5
and the max/min is at y=10.

55
Q

If a vertex is at (8, -9) and frowning, what do you know?

A

You know the AOS is at x=8
and the MAX value is y=-9
FROWNS HAVE MAXIMUM HEIGHTS BUT GO DOWN FOREVER

56
Q

If a vertex is at (5, 12) and smiling, what do you know?

A

You know the AOS is at x=5
and the MIN value is y=12
SMILES HAVE MINIMUM HEIGHTS BUT GO UP FOREVER

57
Q

What is the word for a line that goes left and right?

A

Horizontal line

58
Q

What is the word for a line that goes up and down?

A

Vertical line

59
Q

What is the equation of a horizontal line through the y axis that is up 5

A

y=5

60
Q

What is the equation of the vertical line that is over to the left at -10 on the x axis?

A

x= -10

61
Q

what does “a” tell us in this quadratic?
f(x)= -5(x-3)2+13
TWO THINGS

A

It is - so frowning
5 makes it NARROW

62
Q

What does “a” tell us in this quadratic?
f(x)= 0.25(x-3)2+13
TWO THINGS

A

It is + so smiling,
and 0.25 makes it wide.

2022 ALGEBRA 2 (3 decks)

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