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Algebra 22

Study note

Ava Deutsch

United States of America

DEFAULT

Algebra 2

Algebra 2 Chapter 2 Notes

Transformations

Shifting Functions

quadratic
• Things to remember:
Horizontal Translations
quadratic function: can be written in the form f(x) = a (xh) ³²+k
parabola: U-shaped graph of a
quadratic function
f(x)=x²
f(xh) = (xh)²
y = (2-n)²₂
hao
y=x²
ww
• Shifts left when h 40
• shifts right when h> 0
y = (x - 1)²₂
h>0
and axis reflection 1
The focus is in the interior.
of the parabolaand lies on
the axis of symmetry.
Vertical Translations
f(x)=x2
f(x) + k = 2² + k
y= 2² +k₁
minimum
The venex lies halfway-
between the focus and
the directrix
Tincreasing
√√x-
• Minimum value: fl.
• Domain: All seal numbers
• Range: y = f(-2)
• Decreasing to the left of x =
• Increasing to the right of x =
k> 0 44 1
2a
vertical stretch/shrink horizontal translation
y= 2² +k₁
k<0
shifts down whenk 0
• shifts up whenk> 0
increasing
y = x²
h
• vertex form: f(x)= a (x-h)² + k ². -vertical translation a‡0; vertex is (h, k)
• Maximumvalue: f(-4)
• Domain: All real numbers
symmetry
-|||
decreasing
• Range: y ≤ f(-2)
• Increasing to the left of x =
Decreasing to the right of x =
+ axis of
Chapter 2uations
Horizontal Stretches and Shrinks
f(x) = 2²
flax) = (ar) ²
The directrix is
perpendicular to the
axis of symmetry
y = (2x)²,
a>
y
X
• Transforming functions to extend horizontal distance: Find x-intercept, add distance, plug that number into original equation, take the absolute value of the
output, and then add that to original function.
V
• axis of symmetry: x=h (x-coordinate of vertex) • Standard form: flx)= ax²+bx+c
• minimum value: • maximum value "intercept form: a (x-p)(x-9); azo
a>o
a<o
x=- £
maximum
decreasing
• horizontal stretch laway from
yaxis) when <a<í
y = (ax)²,
• horizontal shrink (towardy-axis)
when a > I
•Writing Quadratic Equations:
• Given a point and vertex (h, k):
☆ Vertex Formiy-a(x-h) ² + k
O Given a
point and x-intercepts p and q:
Intercept Form: y = a (x-p)(x-q)
Given three points:
Write and solve a system of three equations
in three variables
• The axis of symmetry is halfway between
(p, 0) and (9, 0). So, the axis of symmetry
is x = ptg.
• The parabola opens up when a > 0 and
opens down when a <a
Properties of the Graph of f(x)= a(x - p)(x-q)
• Because f(p) = 0 and flq) = ap and
are the x-intercepts of the graph of
the function.
My
Focus: (0,p)
Directrix: y=-p
Vertical Stretches and Shrinks
f(x) = 2³
a-fix) = 02²
a>I'
vertex (0,0)
• Vertical sheich (away from
x-axis) when
• Vertical Shrink (toward savis)
focus;
10,02
Focus: (P,0)
Directrix. x= -P directrix
x-p
when <O<I
·Standard Equations of a Parabola with Vertex at the Origin
Vertical axis of symmetry (x = 0)
Equation y=x²
+ directrix"
y=-p
>O
Horizontal axis of symmetry (y=0)
Equation: x=²
/Focus:
(0, 0)
p>0
vertex
(90)
le, 9)
214
OKAS!
/y=alx-p)(x-a)
(9,0)
directri
y=-p
focus:
(P₁0)
P40
venex
(0, 0))
vertex
focus
(0, P)
x
directrix:
xz-p
PTO
Reflections in the x-Axis
f(x)=x2
-f(x) = -(2³) = -1²
ty Ay=x²
Aips over the x-vis
Properties of the Graph of f(x) = ax³ + bx + c
y = x² + b + c₂a> 0
th
x=
Reflections in the y-Axis
f(x) = x2
1(-2)= (-2)² = 2²
• The parabola opens up when a>0 and opens down when a < 0.
• The graph is narrower than the graph of f(x)=x² when |a|> I and wider
Whenja|<1.
connecting (x, f(x)) and (x₂, (x₂));
• Quadratic functions have constant second differences
• The axis of symmetry is ==- and the vertex is (-50 f(-5)).
• They-intercept is c. So, the point(0, c) is on the parabola
Horizontal axis of symmetry (y = k)
Equation: x==(-k)²³ th
Fox (h+p, A
Directriv: x=h-P
•Average rate of change of a function f from x, to x₂ is the slope of the line
f(x₂)-f(x)
x₂-X₁
you
y = 2² is its own reflection
in the y-axis.
Standard Equations of a Parabola with Vertex at (h₂ k)
Vertical axis of symmetry (x = h)
Equation: y = (2-h)² + k
Focus: (h₂k + p)
Directrix y=k-P
•quadratic regression: tool you can use to find a quadratic function
that best models a set of data
y=0x²+bx+c₂0 <0
#
y=k
y=k-ph W
x=h
*Ah, k+p)
P>0
x=h-P_
fin, k)
p>o
#
x=h
(h, k + p)
p=0
(h+P₂K)
Ex=h-p²
PRO
42=k

Algebra 2 Chapter 2 Notes

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Algebra 2 Chapter 2 Notes

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Knowunity is the # 1 ranked education app in five European countries

Knowunity was a featured story by Apple and has consistently topped the app store charts within the education category in Germany, Italy, Poland, Switzerland and United Kingdom. Join Knowunity today and help millions of students around the world.

4.9+

20 M

#1

950 K+

Still not sure? Look at what your fellow peers are saying...

I love this app so much [...] I recommend Knowunity to everyone!!! I went from a C to an A with it :D

The application is very simple and well designed. So far I have found what I was looking for :D

Love this App ❤️, I use it basically all the time whenever I'm studying