Lecture Notes for Sections 3.3 (Castillo

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Vertex Form
Forms of quadratics
• Factored form a(x-r1)(x-r2)
• Standard Form ax2+bx+c
• Vertex Form a(x-h)2+k
Each form gives you different information!
• Factored form a(x-r1)(x-r2)
– Tells you direction of opening
– Tells you location of x-intercepts (roots)
• Standard Form ax2+bx+c
– Tells you direction of opening
– Tells you location of y-intercept
• Vertex Form a(x-h)2+k
– Tells you direction opening
– Tells you the location of the vertex (max or min)
Direction of opening
• x2 opens up
Direction of opening
• ax2 stretches x vertically by a
– Here a is 1.5
Direction of opening
• ax2 stretches x vertically by a
– Here a is 0.5
– Stretching by a fraction is a squish
Direction of opening
• ax2 stretches x vertically by a
– Here a is -0.5
– Stretching by a negative causes a flip
Direction of opening
• a is the number in front of the x2
• The value a tells you what direction the
parabola is opening in.
– Positive a opens up
– Negative a opens down
• The a in all three forms is the same number
– a(x-r1)(x-r2)
– ax2+bx+c
– a(x-h)2+k
Factored form a(x-r1)(x-r2)
• a is the direction of opening
• r1 and r2 are the x-intercepts
– Or roots, or zeros
• Example: -2(x-2)(x+0.5)
– a is negative, opens down.
– r1 is 2, crosses the x-axis at 2.
– r2 is -0.5, crosses the x-axis at -0.5
Factored form a(x-r1)(x-r2)
• a is the direction of opening
• r1 and r2 are the x-intercepts
– Or roots, or zeros
• Example: -2(x-2)(x+0.5)
– a is negative, opens down.
– r1 is 2, crosses the x-axis at 2.
– r2 is -0.5, crosses the x-axis at -0.5
Standard form ax2+bx+c
• a is the direction of opening
• c is the y-intercept
– ƒ(0)=a02+b0+c=c
• Example: -2x2+3x+2
– Opens down
– Crosses through the point (0,2)
Standard form ax2+bx+c
• a is the direction of opening
• c is the y-intercept
– ƒ(0)=a02+b0+c=c
• Example: -2x2+3x+2
– Opens down
– Crosses through the point (0,2)
Vertex form
• Start with f(x)=x2
Vertex form
• Stretch/Flip if you want
– aƒ(x)=ax2
Vertex form
• Shift right by h
– aƒ(x-h)=a(x-h)2
h
Vertex form
• Shift up by k
– aƒ(x-h)+k=a(x-h)2+k
k
h
Vertex form
• Define a new function
– g(x)=a(x-h)2+k
(h,k)
Vertex form a(x-h)2+k
• a tells you direction of opening
• (h,k) is the vertex
(h,k)
Vertex form a(x-h)2+k
• a tells you direction of opening
• (h,k) is the vertex
• Example: -2(x-3/4)2+25/8
– Opens down
– Has vertex at (3/4, 25/8)
Vertex form a(x-h)2+k
• a tells you direction of opening
• (h,k) is the vertex
• Example: -2(x-3/4)2+25/8
– Opens down
– Has vertex at (3/4, 25/8)
(3/4, 25/8)
Switching between forms
Gives you a full picture
• Example:
ƒ(x)=-2(x-2)(x+0.5)
ƒ(x)=-2x2+3x+2
ƒ(x)=-2(x-3/4)2+25/8
are all the same function
– Opens down
– Crosses x axis at 2 and -0.5
– Crosses the y-axis at 2
– Has vertex at (3/4, 25/8)
Switching between forms
Gives you a full picture
• Example:
ƒ(x)=-2(x-2)(x+0.5)
ƒ(x)=-2x2+3x+2
ƒ(x)=-2(x-3/4)2+25/8
are all the same function
– Opens down
– Crosses x axis at 2 and -0.5
– Crosses the y-axis at 2
– Has vertex at (3/4, 25/8)
Consider the function f(x) = -3x2+2x-9. Which of the
following are true?
A) The graph of f(x) has a negative
y-intercept
B) f(x) has 2 real zeros.
C) The graph of f(x) attains a maximum value
D) Both (A) and (B) are true
E) Both (A) and (C) are true.
Consider the function f(x) = -3x2+2x-9. Which of the
following are true?
Standard form: ax2+bx+c.
a is negative: opens down. ƒ(x)
attains a maximum value. (C) is
true.
c is my y-intercept. c is
negative. My y-intercept is
negative. (A) is true.
E) Both (A) and (C) are true.
The Vertex Formula
• Remember the Quadratic formula
when ax + bx + c = 0
2
-b
b - 4ac
x=
±
2a
2a
2
What does the QF say?
+
-
b 2 - 4ac
2a
Is the distance you have to move
b 2 - 4ac
2a
b 2 - 4ac
2a
from the center left and right
to get to the roots
x=
-b
is the line of symmetry for the curve
2a
The Vertex Formula
when f (x) = ax 2 + bx + c
And you want to rewrite f (x) as f (x) = a(x - h)2 + k
-b
h=
2a
and k = f (h)
Example
when f (x) = -2x 2 + 3x + 2
And you want to rewrite f (x) as f (x) = a(x - h)2 + k
-3
3
h=
=
2(-2) 4
and k = f ( ) = -2 (
3
4
3
4
)
2
+ 3( 34 ) + 2
= -2 ( 169 ) + 94 + 2 = -98 + 188 + 168 = 258
So f (x) = -2 ( x -
3
4
)
2
+ 258
Given the function R(x)=(2x+6)(x-12), find an equation
for its axis of symmetry.
A)
B)
C)
D)
E)
x=-9
x=9
x=2
x=6
None of the above.
Given the function R(x)=(2x+6)(x-12), find an equation
for its axis (line) of symmetry.
• The roots are x=-3 and x=12.
• The axis of symmetry is halfway between the
roots.
• (12-3)/2=4.5, the number halfway between -3
and 12.
• x=4.5 is the axis of symmetry
• E) None of the above.
How to find an equation from vertex
and point
• A parabola passes has its vertex at (1,3) and
passes through the point (0,1). What is the
equation of this parabola?
How to find an equation from vertex
and point
• A parabola passes has its vertex at (1,3) and
passes through the point (0,1). What is the
equation of this parabola?
• (h,k)=(1,3)
• (x1,y1)=(0,1)
How to find an equation from vertex
and point
2
• A parabola
y = a(x - h) + k
passes has its
vertex at (1,3)
and passes
through the
point (0,1). What
is the equation
of this parabola?
• (h,k)=(1,3)
• (x1,y1)=(0,1)
y = a(x -1) + 3
2
But to be finished, I need to know a!
Use: My formula is true for every x,y including x1,y1
How to find an equation from vertex
and point
2
y = a(x - h) + k
• A parabola passes
has its vertex at
(1,3) and passes
through the point
(0,1). What is the
equation of this
parabola?
• (h,k)=(1,3)
• (x1,y1)=(0,1)
My formula is true
for every x,y; not
just x1,y1
y = a(x -1) + 3
2
y1 = a(x1 -1) + 3
2
1 = a(0 -1) + 3
-2 = a(1)
a = -2
2
y = -2(x -1) + 3
2
A quadratic function has vertex at (0,2) and passes through
the point (1,3). Find an equation for this parabola.
A)
B)
C)
D)
E)
y = (x+2)2
y = x2+3
y = x2+1
y = x2
None of the above
A quadratic function has vertex at (0,2) and passes through
the point (1,3). Find an equation for this parabola.
Generic Formula: y = a(x - h) + k
2
Plug in vertex: y = a(x - 0) + 2
2
y = ax 2 + 2
Find a: 3 = a12 + 2
1= a
Plug in a: y = 1x 2 + 2
y= x +2
2
E

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