Transcript Algebra 2

Algebra 2
Quadratic Functions and
Transformations
Lesson 4-1
Goals
Goal
• To identify and graph
quadratic functions.
Rubric
Level 1 – Know the goals.
Level 2 – Fully understand the
goals.
Level 3 – Use the goals to
solve simple problems.
Level 4 – Use the goals to
solve more advanced problems.
Level 5 – Adapts and applies
the goals to different and more
complex problems.
Vocabulary
•
•
•
•
•
•
•
Parabola
Quadratic Function
Vertex Form
Axis of Symmetry
Vertex of the Parabola
Minimum Value
Maximum Value
Essential Question
Big Idea: Function and Equivalence
• What is the vertex form of a quadratic function?
Quadratic Function
• In Chapters 2 and 3, you studied linear
functions of the form f(x) = mx + b.
• A quadratic function in two variables can
be written in standard form f(x) = ax2 + bx
+ c, where a, b, and c are real numbers and
a ≠ 0.
• The simplest quadratic function f(x) = x2 is
the parent function.
• In a quadratic function, the variable is
always squared.
Parabola
• The graph of a quadratic function is a
curve called a parabola.
• A parabola is a U-shaped curve as
shown at the right.
• To graph a quadratic function, generate
enough ordered pairs to see the shape of
the parabola. Then connect the points
with a smooth curve.
Vertex
• The highest or lowest point on a parabola is the vertex.
• If a parabola opens upward, the vertex is the lowest point.
• If a parabola opens downward, the vertex is the highest
point.
Vertex is the
highest point
Vertex is the
lowest point
Axis of Symmetry
• The vertical line that divides a parabola into two symmetrical
halves is the axis of symmetry.
• The axis of symmetry always passes through the vertex of the
parabola.
Axis of
symmetry
Vertex
Vertex
Axis of
symmetry
Summary
Quadratic Function
• A second form of the
quadratic function is vertex
form.
• The vertex form of a quadratic
function is f(x) = a (x – h)2 + k
where a ≠ 0.
• It is called vertex form
because it contains the vertex
(h, k).
Quadratic Function
Transformations
• Transformations in the horizontal direction are
done to the input of a function (x).
• Transformations in the vertical direction are done
to the output of a function (f(x)).
• Parent Quadratic Function f(x) = x2.
Output
f(x) = x2
Input
Translations
Horizontal Translation of the Form
f(x) = (x + h)2
Example:
Graph the functions on one coordinate plane.
y
f(x) = x 2
8
g(x) = (x + 3)2
6
4
f(x) = x 2
g(x) = (x +
3)2
2
x
f(x)
x
g(x)
2
4
2
1
1
1
1
4
0
0
0
9
6
1
1
1
16
8
2
4
2
25
x
8
6
4
2
4
2
4
6
8
Notice that the graph
of g(x) is the graph of
f(x) shifted 3 units to
the left.
Horizontal Translation of the Form
f(x) = (x – h)2
Example:
Graph the functions on one coordinate plane.
f(x) = x 2
y
8
g(x) = (x – 1)2
6
f(x) = x 2
g(x) = (x – 1)2
4
2
x
f(x)
x
g(x)
2
4
2
9
1
1
1
4
0
0
0
1
4
1
1
1
0
6
2
4
2
1
8
x
8
6
4
2
2
4
6
8
Notice that the graph
of g(x) is the graph of
f(x) shifted 1 unit to
the right.
Vertical Translation of the Form
f(x) = x 2 + k
y
Example:
Graph the functions on one
coordinate plane.
g(x) = x 2 + 3
8
6
f(x) = x 2
g(x) = x 2 + 3
4
f(x) = x 2
2
x
f(x)
x
g(x)
2
4
2
7
1
1
1
4
0
0
0
3
1
1
1
4
6
2
4
2
7
8
x
8
6
4
2
4
2
4
6
8
Notice that the graph
of g(x) is the graph of
f(x) shifted 3 units
upward.
Vertical Translation of the Form
f(x) = x 2 – k
Example:
Graph the functions on one coordinate plane.
y
f(x) = x 2
8
6
f(x) = x 2
4
g(x) = x 2 – 2
x
f(x)
x
g(x)
2
4
2
2
1
1
1
1
0
0
0
2
1
1
1
1
2
4
2
2
g(x) = x 2 – 2
2
x
8
6
4
2
4
6
8
2
4
6
8
Notice that the graph
of g(x) is the graph of
f(x) shifted 2 units
downward.
Combining Translations
Example:
Graph the functions on one coordinate
plane.
g(x) = (x + 1)2 + 2
y
8
6
f(x) = x 2
x
f(x)
g(x) = (x +
x
1)2
+2
2
g(x)
2
4
2
3
1
1
1
2
0
0
0
3
1
1
1
6
2
4
2
11
f(x) = x 2
4
x
8
6
4
2
4
6
8
2
4
6
8
Notice that the graph
of g(x) is the graph of
f(x) shifted 1 unit to
the left and 2 units
upward.
Example:
Use the graph of f(x) = x2 as a guide, describe the transformations
and then graph each function.
g(x) = (x – 2)2 + 4
Identify h and k.
g(x) = (x – 2)2 + 4
h
k
Because h = 2, the graph is translated 2 units right. Because k = 4, the
graph is translated 4 units up. Therefore, g is f translated 2 units right
and 4 units up.
Example:
Use the graph of f(x) = x2 as a guide, describe the transformations
and then graph each function.
g(x) = (x + 2)2 – 3
Identify h and k.
g(x) = (x – (–2))2 + (–3)
h
k
Because h = –2, the graph is translated 2 units left. Because k = –3, the
graph is translated 3 units down. Therefore, g is f translated 2 units left
and 4 units down.
Your Turn:
Using the graph of f(x) = x2 as a guide, describe the
transformations and then graph each function.
g(x) = x2 – 5
Identify h and k.
g(x) = x2 – 5
k
Because h = 0, the graph is not translated horizontally.
Because k = –5, the graph is translated 5 units down. Therefore, g is f
is translated 5 units down.
Your Turn:
Use the graph of f(x) =x2 as a guide, describe the transformations
and then graph each function.
g(x) = (x + 3)2 – 2
Identify h and k.
g(x) = (x – (–3)) 2 + (–2)
h
k
Because h = –3, the graph is translated 3 units left. Because k = –2, the
graph is translated 2 units down. Therefore, g is f translated 3 units left
and 2 units down.
More Transformations
• Recall that functions can also be reflected,
stretched, or compressed.
• This includes quadratic functions.
Quadratic Function
Transformations
Vertical Stretch of the Form
f(x) = ax 2
y
Example:
Graph the functions on one coordinate
plane.
g(x) = 2x 2
8
f(x) = x 2
6
f(x) = x
x
2
f(x)
4
g(x) = 2x 2
x
2
g(x)
2
4
2
8
1
1
1
2
0
0
0
0
1
1
1
2
2
4
2
8
x
8
6
4
2
4
6
8
2
4
6
8
Notice that the graph
of g(x) is narrower
than the graph of f(x).
Vertical Compression of the Form
f(x) = ax 2
Example:
Graph the functions on one coordinate
plane.
f(x) = x
2
g(x) =
y
8
6
1 x2
2
x
f(x)
x
g(x)
2
4
2
2
1
1
1
1
2
0
0
0
0
1
1
1
1
2
2
4
2
2
f(x) = x 2
1
g(x) = x 2
2
4
2
x
8
6
4
2
4
6
8
2
4
6
8
Notice that the graph
of g(x) is wider than
the graph of f(x).
Reflection Over the x-axis of the Form
f(x) = -x 2
Example:
Graph the functions on one coordinate plane.
y
g(x) = –x 2
8
x
g(x)
6
2
4
4
f(x) = x 2
1
1
2
x
f(x)
0
0
2
4
1
1
1
1
2
4
0
0
1
1
2
4
f(x) = x 2
x
8
6
4
2
4
6
Notice that the graph of g(x) is
a reflection of the graph of f(x)
over the x-axis.
8
2
4
6
8
g(x) = – x 2
Example:
Using the graph of f(x) = x2 as a guide, describe the transformations
and then graph each function.
g (x ) =
1
x
4
2
Because a is negative, g is a
reflection of f across the x-axis.
Because |a| = , g is a vertical
compression of f by a factor of
.
Example:
Using the graph of f(x) = x2 as a guide, describe the transformations
and then graph each function.
g(x) = 3x2
Because a = 3 , g is a vertical
stretch of f by a factor of 3.
Your Turn:
Using the graph of f(x) = x2 as a guide, describe the transformations
and then graph each function.
g(x) = 2x2
Because a = 2, g is a vertical
stretch of f by a factor of 2.
Your Turn:
Using the graph of f(x) = x2 as a guide, describe the transformations
and then graph each function.
g(x) = –
x2
Because a is negative, g is a
reflection of f across the x-axis.
Because |a| = , g is a vertical
compression of f by a factor of
.
Minimums and Maximums
If a parabola opens upward, it has a lowest point, a minimum. If a parabola
opens downward, it has a highest point, a maximum. This minimum or
maximum is the vertex of the parabola.
If a > 0, the parabola opens upward.
The vertex is a minimum point.
Opens up
a>0
If a < 0, the parabola opens upward.
The vertex is a maximum point.
Vertex is the
maximum
Opens down
a<0
Vertex is the
minimum
Minimums and Maximums
More Vertex Form
• The parent function f(x) = x2 has its
vertex at the origin.
• You can identify the vertex of
other quadratic functions by
analyzing the function in vertex
form.
• The vertex form of a quadratic
function is f(x) = a(x – h)2 + k,
where a, h, and k are constants and
(h, k) is the vertex.
Because the vertex is translated h horizontal units and k
vertical from the origin, the vertex of the parabola is at (h, k).
Helpful Hint
When the quadratic parent function f(x) = x2 is written in
vertex form, y = a(x – h)2 + k,
a = 1, h = 0, and k = 0.
Summary of Transformations
f(x) = ± a(x – h)2 + k
Reflection about
the x-axis.
+ opens upward
- opens downward
Horizontal Translation
Opposite direction of sign
+ to the left
- to the right
Vertical stretch/compress
a>1 narrower
0<a<1 wider
Vertical Translation
Same direction of sign
+ up
- down
Example:
Use the description to write the quadratic function in vertex
form.
The parent function f(x) = x2 is vertically stretched by a factor
of
and then translated 2 units left and 5 units down to create
g.
Step 1 Identify how each transformation affects the constant
in vertex form.
4
4
=
a
Vertical stretch by :
3
3
Translation 2 units left: h = –2
Translation 5 units down: k = –5
Example: continued
Step 2 Write the transformed function.
g(x) = a(x – h)2 + k
= (x – (–2))2 + (–5)
= (x + 2)2 – 5
g(x) =
(x + 2)2 – 5
Vertex form of a quadratic function
Substitute
k.
Simplify.
for a, –2 for h, and –5 for
Your Turn:
Use the description to write the quadratic function in vertex
form.
The parent function f(x) = x2 is vertically
compressed by a factor of and then translated
2 units right and 4 units down to create g.
Step 1 Identify how each transformation affects the constant
in vertex form.
Vertical compression by
:a=
Translation 2 units right: h = 2
Translation 4 units down: k = –4
Continued
Step 2 Write the transformed function.
g(x) = a(x – h)2 + k
Vertex form of a quadratic function
= (x – 2)2 + (–4)
Substitute
= (x – 2)2 – 4
Simplify.
g(x) = (x – 2)2 – 4
for a, 2 for h, and –4 for k.
Your Turn:
Use the description to write the quadratic function in vertex
form.
The parent function f(x) = x2 is reflected across the x-axis and
translated 5 units left and 1 unit up to create g.
Step 1 Identify how each transformation affects the constant
in vertex form.
Reflected across the x-axis: a is negative
Translation 5 units left: h = –5
Translation 1 unit up: k = 1
Continued
Step 2 Write the transformed function.
g(x) = a(x – h)2 + k
Vertex form of a quadratic function
= –(x –(–5))2 + (1)
Substitute –1 for a, –5 for h,
and 1 for k.
= –(x +5)2 + 1
Simplify.
g(x) = –(x +5)2 + 1
Example:
What is an equation of the function
shown in the graph?
Solution:
The vertex of the parabola is at (2, –3), so h = 2 and
k = –3. Since the graph passes through (0, –1), let x = 0 and
y = –1. Substitute these values into the vertex form of the
equation and solve for a.
Solution:
Vertex form
Substitute –1 for y, 0 for x,
2 for h, and –3 for k.
Simplify.
Add 3 to each side.
Divide each side by 4.
Solution:
The equation of the parabola in vertex form is
Your Turn:
What is an equation of the function
shown in the graph?
𝑓 𝑥 = −3 𝑥 + 1
2
+2
A.
B.
C.
D.
A
B
C
D
Example: Appliication
On Earth, the distance d in meters that a dropped object falls
in t seconds is approximated by d(t)= 4.9t2. On the moon, the
corresponding function is dm(t)= 0.8t2. What kind of
transformation describes this change from d(t)= 4.9t2, and
what does the transformation mean?
Examine both functions in vertex form.
d(t)= 4.9(t – 0)2 + 0
dm(t)= 0.8(t – 0)2 + 0
Example: continued
The value of a has decreased from 4.9 to 0.8. The decrease
indicates a vertical compression.
Find the compression factor by comparing the new a-value to the
old a-value.
a from dm(t)
a from d(t)
0.8
=
4.9

0.16
The function dm represents a vertical compression of d by a factor of
approximately 0.16. Because the value of each function approximates
the time it takes an object to fall, an object dropped from the moon falls
about 0.16 times as fast as an object dropped on Earth.
Your Turn:
The minimum braking distance d in feet for a vehicle on dry
concrete is approximated by the function (v) = 0.045v2, where v is
the vehicle’s speed in miles per hour.
The minimum braking distance dn in feet for a
vehicle with new tires at optimal inflation is dn(v) = 0.039v2, where
v is the vehicle’s speed in miles per hour. What kind of
transformation describes this change from d(v) = 0.045v2, and what
does this transformation mean?
Continued
Examine both functions in vertex form.
d(v)= 0.045(t – 0)2 + 0
dn(t)= 0.039(t – 0)2 + 0
The value of a has decreased from 0.045 to 0.039. The decrease
indicates a vertical compression.
Find the compression factor by comparing the new a-value to the
old a-value.
a from dn(t)
a from d(v)
=
0.039
0.045
=
13
15
Continued
The function dn represents a vertical compression of d by a factor
of
. The braking distance will be less with optimally inflated
new tires than with tires having more wear.
Check the graph. The graph of dn appears to be vertically compressed
compared with the graph of d.
15
0
15
0
Graphing Using
Transformations
f(x) = 2(x - 1)2 - 4
1) Stretch the graph of y = x2.
Graphing Using
Transformations
f(x) = 2(x - 1)2 - 4
2) Vertically stretch the curve in #1 by a
factor of 2, to sketch y = 2x2.
Graphing Using
Transformations
f(x) = 2(x - 1)2 - 4
3) Move the curve in #2 horizontally one
unit to the right to sketch y = 2(x – 1)2.
Graphing Using
Transformations
f(x) = 2(x - 1)2 - 4
4) Move the curve in #3 vertically down
4 units to sketch y = 2(x – 1)2 -4.
Graphing Using
Transformations
f(x) = 2(x - 1)2 - 4
Graphing a Quadratic
Function in Vertex Form
• Procedure: quadratic function in vertex form
1. Identify and plot the vertex (h, k). Find the axis of
symmetry, x = h.
2. Find and plot two points on one side of the axis of
symmetry.
3. Plot the corresponding points on the other side of the
axis of symmetry.
4. Sketch the curve.
Vertex Form
f ( x ) = ( x  1)  4
2
f ( x ) = ( x  1)  4
2
(1, 4 )
1. Identify and
plot the vertex
(h, k). Find the
axis of
symmetry, x = h
f ( x ) = ( x  1)  4
2
2. Find and plot 2
points on one side of
the axis of
symmetry;
f(2)=(2-1)2+4=5
f(3)=(3-1)2+4=8,
and plot.
Axis of
Symmetry
x=1
3. Plot the
corresponding
points on the other
side of the axis of
symmetry.
4. Sketch the curve.
Vertex Form
f ( x ) = 2 ( x  1)  5
2
f ( x ) = 2 ( x  1)  5
2
1. Identify and
plot the vertex
(h, k). Find the
axis of
symmetry, x = h
Axis of
Symmetry
x=1
f ( x ) = 2 ( x  1)  5
2
2. Find and plot 2
points on one side of
the axis of
symmetry;
f(2)=-2(2-1)2+5=3
f(3)=-2(3-1)2+5=-3,
and plot.
3. Plot the
corresponding
points on the other
side of the axis of
symmetry.
f ( x ) = 2 ( x  1)  5
2
4. Sketch the curve.
Your Turn:
Graph y = –2(x – 1)2 + 3.
Solution:
Your Turn:
Graph y = 3(x + 2) 2 - 4.
Solution:
Essential Question
Big Idea: Function and Equivalence
• What is the vertex form of a quadratic function?
• The vertex form of a quadratic function is
f(x) = a(x – h)2 + k, where a ). All quadratic
functions are transformations of the parent
function, f(x) = x2. Use vertex form to identify the
transformations and graph a quadratic function.
Assignment
• Section 4-1, Pg. 209 – 211; #1 – 6 all, 8 –
26 even, 30 – 54 even.