𝟐
𝒚= 𝒙
The Parent Quadratic
x intercept: (0,0)
y intercept: (0,0)
vertex: (0,0)
D: ( -∞ , ∞)
R: [0 , ∞)
𝒚 = (𝒙 − 𝟑)
𝟐
Move the vertex of the parent three
units RIGHT on X AXIS
x intercept: (3,0)
y intercept: (0,9)
vertex: (3,0)
D: ( -∞ , ∞)
R: [3 , ∞)
𝒚 = −(𝒙 − 𝟑)
𝟐
Move the vertex of the parent three
units RIGHT on X AXIS and reflect it
over the x axis.
x intercept: (3,0)
y intercept: (0,-9)
vertex: (3,0)
D: ( -∞ , ∞)
R: ( -∞ , 3]
𝟏
𝟐
𝒚 = (𝒙 + 𝟐)
𝟐
Move the vertex of the parent two
units LEFT on X AXIS and the ½ will
make it “wider”.
x intercept: (-2,0)
y intercept: (0,2)
vertex: (-2,0)
D: ( -∞ , ∞)
R: [ 0, ∞)
𝟐
𝒚 = (𝒙 + 𝟐) - 4
Move the vertex of the parent 3units
RIGHT on x axis and 4 units DOWN on
the y axis.
x intercept: (0,0) (-4,0)
y intercept: (0,0)
vertex: (-2,-4)
D: ( -∞ , ∞)
R: [4, ∞)
𝟐
𝒚= 𝒙 +4
Move the vertex of the parent 4 units
up on the y axis.
x intercept: none
y intercept: (0,4)
vertex: (0 ,4)
D: ( -∞ , ∞)
R: [4, ∞)
𝒚 = (𝒙 − 𝟑)(𝒙 + 𝟐)
Find the x intercepts at 3 and -2. Flips
up.
x intercept: (3,0) (-2,0)
y intercept: (0,-6)
vertex: (1/2, -6.25)
Add the x intercepts together and divided by 2 to find the x
coordinate of the vertex.
Substitute ½ into the equation to find y.
D: ( -∞ , ∞)
R: [-6.25 , ∞)
𝒚 = 𝒙(𝒙 − 𝟑)(𝒙 + 𝟐)
Find the x intercepts at 0, 3, and -2.
Flips up. It’s a CUBIC (note: 3 x’s)
x intercept: (3,0) (-2,0) (0,0)
y intercept: (0,0)
D: ( -∞ , ∞)
R: ( -∞ , ∞)
𝒚 = 𝟑(𝒙 − 𝟑)(𝒙 + 𝟐)
Find the x intercepts at 3 and -2. Flips
up. NOTE: The 3 in front does NOT
create an x-intercept!
x intercept: (3,0) (-2,0)
y intercept: (0,-18)
vertex: (1/2, -18.75)
Add the x intercepts together and divided by 2 to find the x
coordinate of the vertex.
Substitute ½ into the equation to find y.
D: ( -∞ , ∞)
R: ( -∞ , ∞)
𝒚=𝟒
Did you notice that this is a Horizontal
LINE???
x intercept: none
y intercept : (0, 4)
slope: 0
Note that slope of a vertical line is
undefined.
D: ( -∞ , ∞)
R: [4]
𝒚 = 𝟑(𝒙 − 𝟑)(𝒙 + 𝟐)
Find the x intercepts at 3 and -2. Flips
up. NOTE: The 3 in front does NOT
create an x-intercept!
x intercept: (3,0) (-2,0)
y intercept: (0,-18)
vertex: (1/2, -18.75)
Add the x intercepts together and divided by 2 to find the x
coordinate of the vertex.
Substitute ½ into the equation to find y.
D: ( -∞ , ∞)
R: ( -∞ , ∞)
𝑦= 𝑥
The Parent Absolute Value
x intercept: (0,0)
y intercept: (0,0)
vertex: (0,0)
“V” shaped with “arms” in quadrants
1 and 2
D: ( -∞ , ∞)
R: [0, ∞)
𝒚 = −(𝒙 − 𝟑)(𝒙 +
𝟐
𝟐)
Find the x intercepts at 3 and -2 and
-2. Flips into quadrants 2 and 4
because of the -1 in front. It’s a CUBIC.
It has a DOUBLE root of -2 OR we
could say it has “multiplicity of 2”.
Recall that a graph turns at the double
root—changes direction!
x intercept: (3,0) (-2,0) (-2,0)
y intercept: (0,6)
D: ( -∞ , ∞)
R: ( -∞ , ∞)
𝒚 = (𝒙 − 𝟐) +𝟖
Did you notice that this is a LINE???
y=x+6
x intercept: (−6,0)
y intercept : (0,6)
slope: 1
Note that it is easy to find the inflection point
when the cubic function is in this form.
D: ( -∞ , ∞)
R: ( -∞ , ∞)
𝒚 + 𝟏 = −𝟑(𝒙 − 𝟐)
Did you notice that this is a LINE???
5
( ,
3
x intercept:
0)
y intercept : (0, 5)
slope: -3
Note that it is easy to find the inflection point
when the cubic function is in this form.
D: ( -∞ , ∞)
R: ( -∞ , ∞)
𝑦 = 𝑥−2
Parent moves 2 units right on x axis. It
does NOT move off of 0 on the y axis.
x intercept: (2,0)
y intercept: (0,0)
vertex: (2,0)
“V” shaped with “arms” in quadrants
1 and 2
D: ( -∞ , ∞)
R: [ 2, ∞)
𝑦 =− 𝑥+2
Parent moves 2 units left on x axis. It
does NOT move off of 0 on the y axis.
It flips!
x intercept: (-2,0)
y intercept: (0,-2)
vertex: (-2,0)
“V” shaped with “arms” in quadrants
3 and 4
D: ( -∞ , ∞)
R: ( -∞ , 2]
𝑦 =− 𝑥+2 +8
Parent moves 2 units left on x axis and
8 units up on the y axis. Flips!
x intercept: (-10,0) (6,0)
y intercept: (0,6)
vertex: (-2,8)
“V” shaped with “arms” in quadrants
3 and 4
D: ( -∞ , ∞)
R: ( -∞ , 8]
𝑦 =3 𝑥+2 +8
Parent moves 2 units left on x axis and
8 units up on the y axis. The three in
front makes it “skinnier”.
x intercept: none
Can’t solve an absolute value that is equal to a negative value!
y intercept: (0,14)
vertex: (-2,8)
“V” shaped with “arms” in quadrants
1 and 2
D: ( -∞ , ∞)
R: [8 , ∞)
𝑦 = 𝑥+2 −9
Parent moves 2 units leftt on x axis and
9 units down on the y axis.
x intercept: (7, 0) (-11,0)
Remember: We must split absolute values
equations: -9 = x+2 and 9 = x+2
y intercept: (0,-7)
vertex: (-2,-9)
“V” shaped with “arms” in quadrants
1 and 2
D: ( -∞ , ∞)
R: [-9, ∞)
𝟑
𝒚 = (𝒙 + 𝟐)
Moves the cubic parent two units to
the right. In quadrants 1 and 3.
3
x intercept: ( −2,0)
y intercept : (0,8)
Inflection point: (-2, 0)
Note that it is easy to find the inflection point
when the cubic function is in this form.
D: ( -∞ , ∞)
R: ( -∞ , ∞)
𝟒
𝒚 = (𝒙 + 𝟐)
Moves the quartic parent two units to
the right. In quadrants 1 and 2.
x intercept: (-2,0)
y intercept : (0,16)
Inflection point: (-2, 0)
Note that a quartic is similar to a parabola but is wider at
the “bottom” and closer to the y axis past the points (1,1)
and (-1,1).
D: ( -∞ , ∞)
R: ( -∞ , ∞)
𝟑
𝒚 = (𝒙 + 𝟐) +𝟖
Moves the cubic parent two units to
the left on the x axis and 8 units up on
the y axis. In quadrants 1 and 3.
x intercept: (−4,0)
y intercept : (0,16)
Inflection point: (-2, 8)
Note that it is easy to find the inflection point
when the cubic function is in this form.
D: ( -∞ , ∞)
R: ( -∞ , ∞)
𝟓
𝒚 = (𝒙 + 𝟐) +𝟖
Very similar in graph to a cubic but it is
“wider” near the inflection point and
then nearer the y axis past the points (1,-1) and (1, 1).Moves the parent two
units to the right. In quadrants 1 and 3.
5
x intercept: (-2+ −8, 0)
y intercept : (0,40)
Inflection point: (-2, 8)
D: ( -∞ , ∞)
R: ( -∞ , ∞)
𝟑
𝒚 = (𝒙 + 𝟐) +𝟖
Moves the parent cubic 2 units left on
x axis and 8 units up on y axis
x intercept: (-4,0)
y intercept: (0,16)
inflection point: (-2,8)
D: ( -∞ , ∞)
R: ( -∞ , ∞)
𝑦=
𝑥
The Parent Square Root
x intercept: (0,0)
y intercept: (0,0)
Starting point: (0,0)
D: [0, ∞)
R: [0, ∞)
𝑦=
𝑥−4
Move parent 4 units right on x axis
x intercept: (4,0)
y intercept: none
Starting point: (4,0)
D: [4, ∞)
R: [0, ∞)
𝑦=
𝑥+4 -5
Move the parent 4 units to the left on
x axis and 5 units down on the y axis.
x intercept: (21,0)
y intercept: (0,-3)
Starting point: (-4,-5)
D: [-4, ∞)
R: [-5, ∞)
Note how the starting point dictates the “starting point” of the domain and range.
𝑦 =− 𝑥 +2
Move the parent no where on the x
axis and 2 units up on the y axis. And
flips
x intercept: (4,0)
y intercept: (0,2)
Starting point: (0,2)
D: [0, ∞)
R: (-∞,2)
Note how the starting point dictates the “starting point” of the domain and range.
𝒚 = 𝒙(𝒙 − 𝟑)(𝒙 + 𝟐)
Find the x intercepts at 0, 3, and -2.
Flips up. It’s a CUBIC (note: 3 x’s)
x intercept: (3,0) (-2,0) (0,0)
y intercept: (0,0)
D: ( -∞ , ∞)
R: ( -∞ , ∞)
𝒚 = 𝒙(𝒙 − 𝟑)(𝒙 + 𝟐)
Find the x intercepts at 0, 3, and -2.
Flips up. It’s a CUBIC (note: 3 x’s)
x intercept: (3,0) (-2,0) (0,0)
y intercept: (0,0)
D: ( -∞ , ∞)
R: ( -∞ , ∞)