Transcript Lecture Notes for Sections 3.3 (Castillo
Vertex Form
Forms of quadratics
• Factored form a(x-r 1 )(x-r 2 ) • Standard Form ax 2 +bx+c • Vertex Form a(x-h) 2 +k
Each form gives you different information!
• • • Factored form a(x-r 1 )(x-r 2 ) – Tells you direction of opening – Tells you location of x-intercepts (roots) Standard Form ax 2 +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
• x 2 opens up
Direction of opening
• ax 2 stretches x vertically by a – Here a is 1.5
Direction of opening
• ax 2 stretches x vertically by a – Here a is 0.5
– Stretching by a fraction is a squish
Direction of opening
• ax 2 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 x 2 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-r 1 )(x-r 2 ) ax 2 +bx+c – a(x-h) 2 +k
Factored form a(x-r
1
)(x-r
2
)
• • • a is the direction of opening r 1 – and r 2 are the x-intercepts Or roots, or zeros Example: -2(x-2)(x+0.5) – a is negative, opens down.
– r 1 is 2, crosses the x-axis at 2.
– r 2 is -0.5, crosses the x-axis at -0.5
Factored form a(x-r
1
)(x-r
2
)
• • • a is the direction of opening r 1 – and r 2 are the x-intercepts Or roots, or zeros Example: -2(x-2)(x+0.5) – a is negative, opens down.
– r 1 is 2, crosses the x-axis at 2.
– r 2 is -0.5, crosses the x-axis at -0.5
Standard form ax
2
+bx+c
• • • a is the direction of opening c is the y-intercept – ƒ(0)=a0 2 +b0+c=c Example: -2x 2 +3x+2 – Opens down – Crosses through the point (0,2)
Standard form ax
2
+bx+c
• • • a is the direction of opening c is the y-intercept – ƒ(0)=a0 2 +b0+c=c Example: -2x 2 +3x+2 – Opens down – Crosses through the point (0,2)
Vertex form
• Start with f(x)=x 2
Vertex form
• Stretch/Flip if you want – aƒ(x)=ax 2
• Shift right by h – aƒ(x-h)=a(x-h) 2
Vertex form
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)=-2x 2 +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)=-2x 2 +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) = -3x 2 +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) = -3 x 2 + 2 x -9 . Which of the following are true?
Standard form: a x 2 + b x+ 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
2 +
bx
+
c
=
0
x
= -
b
2
a
±
b
2 -
4
ac
2
a
What does the QF say?
+
b
2 4
ac
2
a
-
b
2 4
ac
2
a
Is the distance you have to move
b
2 4
ac
2
a
from the center left and right to get to the roots
x
= 2
b a
is the line of symmetry for the curve
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 h
= -
b
2
a
and
k
=
f
(
h
)
Example
when
f
(
x
) = 2
x
2 + 3
x
+ 2 And you want to rewrite
f
(
x
) as
f
(
x
) =
a
(
x
-
h
) 2 +
k h
= 2( 3 2) = 3 4 and
k
=
f
= 2 ( ) + ( 3 4 ) 9 4 + = 2 ( ) 2 2 = 9 8 + 18 8 + 3 ( ) + + 16 8 = 25 8 2 So
f
(
x
) = 2 (
x
3 4 ) 2 + 25 8
Given the function R(x)=(2x+6)(x-12), find an equation for its axis of symmetry.
A) x = - 9 B) x = 9 C) x = 2 D) x = 6 E) 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) (x 1 ,y 1 )=(0,1)
• How to find an equation from vertex A parabola passes has its vertex at (1,3) and passes through the point (0,1) . What is the equation of this parabola?
and point
y
=
y
=
a
(
x
-
h
) 2
a
(
x
1) 2 + + 3
k
• • (h,k)=(1,3) (x 1 ,y 1 )=(0,1) But to be finished, I need to know a!
Use: My formula is true for every x,y including x 1 ,y 1
• • • 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) (x 1 ,y 1 )=(0,1) My formula is true for every x,y; not just x 1 ,y 1
y
=
a
(
x
-
h
) 2
y
=
a
(
x
1) 2
y
1 =
a
(
x
1 1) 2 1 = 2
a
= (0
a
1) 2 (1)
a
= 2 + 3
y
= 2(
x
1) 2 +
k
+ 3 + 3 + 3
A quadratic function has vertex at (0,2) and passes through the point (1,3). Find an equation for this parabola.
A) y = (x+2) 2 B) y = x 2 +3 C) y = x 2 +1 D) y = x 2 E) 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
) 2 +
k
Plug in vertex:
y
=
a
(
x
0) 2 + 2
y
=
ax
2 + 2 Find
a
: 3 =
a
1 2 1 =
a
+ 2 Plug in
a
:
y
= 1
x
2 + 2
y
=
x
2 + 2 E