Download Chapter 2 Study Guide

Chapter 2 Study Guide
2.1 Using Transformations to Graph Quadratic Functions
The graph of a quadratic function is a parabola. A parabola is a curve shaped like the letter U.
f(x)  a(x  h)2  k(a  0)
Quadratic function
You can make a table to graph a quadratic function.
Graph f(x)  x2   3.
x
f (x)  x2  4x  3
(x, f (x))
0
f(0)  02  4(0)  3  3
(0, 3)
1
f(1)  12  4(1)  3  0
(1, 0)
2
f(2)  22  4(2)  3  1
(2, 1)
3
f(3)  32  4(3)  3  0
(3, 0)
4
f(4)  42  4(4)  3  3
(4, 3)
Vertex Form of Quadratic Function
Horizontal and vertical translations change the vertex of f(x)  x2.
Parent Function
Transformation
f(x)  x2
g(x)  (x  h)2  k
Vertex: (0, 0)
Vertex: (h, k)
The vertex of g(x)  (x  4)2  2
is (4, 2).
The graph of f(x)  x2 is shifted
4 units right and 2 units down.
2.2 Properties of Quadratic Functions in Standard Form
You can use the properties of a parabola to graph a quadratic function in standard form:
f(x)  ax2  bx  c, a  0.
Property
Example: f(x)  x2  2x  2
a  0: opens upward
a  0: opens downward
a  1, b  2, c  2
a  0, so parabola opens downward.
Axis of symmetry: x  
 b
,
 2a
Vertex:  
b
2a
 b 
f  
 2a  
Axis of symmetry: x  
b
( 2)

 1
2a
2( 1)
 b
f     f ( 1)  1( 1)2  2( 1)  2  3
 2a 
Vertex: (1, 3)
y-intercept: c
To graph f(x)  x2  2x  2:
y-intercept is 2, so (0, 2) is a point on the graph.
2.3 Solving Quadratic Equations by Graphing and Factoring
Solve the equation ax2  bx  c  0 to find the roots of the equation.
Find the roots of x2  2x  15  0 to find the zeros of f x  x2  2x  15.
x2  2x  15  0
Factor, then multiply
x  5x  3  0
to check.
x  5  0 or x  3  0
Solve each
equation for x.
x  5 or x  3
Set each factor
equal to 0.
To check the roots, substitute each root into the original equation:
Equation:
x2  2x  15  0
x2  2x  15  0
Root:
x  5
x3
Check:
52  25  15
32  23  15
25  10  15  0 
9  6  15  0 
The roots of x2  2x  15  0 are 5 and 3.
The roots of the equation
The zeros of f x  x2  2x  15 are 5 and 3.
are the zeros of the function.
Some quadratic equations have special factors.
Difference of Two Squares:
a2  b2  a  b a  b
Perfect Square Trinomials:
a2  2ab  b2  a  b2
a2  2ab  b2  a  b2
Always write a quadratic equation in standard form before factoring.
16x2  25
16x2  25  0
16 and 25 are perfect squares.
Use the difference of two
squares to factor.
4x2  52  0
4x  54x  5  0
4x  5  0 or 4x  5  0
x
5
5
or x  
4
4
Try to factor a perfect square trinomial if the coefficient of x and the
constant term are perfect squares.
4x2  12x  9  0
2x2  22x3  32  0
4x2 and 9 are
perfect squares.
2x  32x  3  2x  32  0
2x  3  0
x
3
2
The factors are
the same.
2.4 Completing the Square
You can use the square root property to solve some quadratic equations.
Square Root Property
To solve x2  a,
take the square root
of both sides of the
equation.
Solve
Remember:
22  4, and 22  4.
x2  a
x2   a
x a
4x  5  43.
4x2  48
Add 5 to both sides.
x2  12
Divide both sides by 4.
x 2   12
Take the square root of both sides.
x   12
Simplify.
Think:
x  2 3
Solve
The coefficient of x2 should
be 1 to use the square root
property.
2
x2  12x  36  50.
x  62  50
Factor the perfect square trinomial.
 x  6 2
Take the square root of both sides.
  50
x  6   50
Subtract 6 from both sides.
x  6  50
Simplify.
Think:
x  6  5 2
You can use a process called completing the square to rewrite
a quadratic of the form x2  bx as a perfect square trinomial.
To complete the
square of x2  bx,
2
b
add   .
2
Think: Multiply the
coefficient of x by
1
. Then square it.
2
2
b
b 
x  bx      x  
2
2 
2
2
Complete the square: x2  8x  ?.
Step 1
Identify b, the coefficient of x: b  8.
Step 2
b
Find   :
2
Step 3
b
Add   :
2
Step 4
Factor:
2
2
2
2
 b   8 
  
   4   16
2  2 
2
x 2  8 x  16
x2  8x  16  x  42
Check: x  42  x  4x  4
x2  8x  16 
Use
as a factor.
2.5 Complex Numbers and Roots
An imaginary number is the square root of a negative number.
Use the definition 1  i to simplify square roots.
Simplify.
25
 25  1
Factor out 1.
 25 
Separate roots.
1
5 1
Simplify.
5i
Express in terms of i.
Imaginary
Real
Complex numbers are numbers that can be written in the form a  bi.
The complex conjugate of a  bi is a  bi.
Write as a  bi
Find 0  5i  5i
The complex conjugate of 5i is 5i.
You can use the square root property and
imaginary solutions.
1  i to solve quadratic equations with
Solve x2  64.
x 2   64
x  8i
Take the square root of both sides.
Express in terms of i.
Check each root: 8i2  64i 2  641  64
8i2  64i 2  641  64
Remember:
Solve 5x2  80  0.
5x2  80
Subtract 80 from both sides.
x  16
Divide both sides by 5.
x 2   16
Take the square root of both sides.
2
x  4i
Express in terms of i.
Check each root:
54i 2  80
54i 2  80
516i 2  80
516i 2  80
801  80
801  80
0
0
2.6 The Quadratic Formula
The Quadratic Formula is another way to find the roots of a quadratic
equation or the zeros of a quadratic function.
Find the zeros of f x  x2  6x  11.
Step 1
Set f x  0.
Step 2
Write the Quadratic Formula. x 
Step 3
Substitute values for a, b, and c into the Quadratic Formula.
x2  6x  11  0
b  b 2  4ac
2a
a  1, b  6, c  11
b  b2  4ac   6  
x

2a
Step 4
2
Simplify.
x
Step 5
 6   4 1 11
2 1
  6  
 6  4 1 11 6 

2 1
2
36  44 6  80

2
2
Write in simplest form.
x
6  80
3
2
80
3
2
16  5 
2
3
4 5
32 5
2
Remember to divide both terms of
the numerator by 2 to simplify.
The discriminant of ax2  bx  c  0 a  0 is b2  4ac.
Use the discriminant to determine the number of roots of a quadratic equation. A quadratic equation can have
2 real solutions, 1 real solution, or 2 complex solutions.
Find the type and number of solutions.
x2  10x  25
2x2  5x  3
3x2  4x  2
Write the equation in
standard form:
Write the equation in
standard form:
Write the equation in
standard form:
2x2  5x  3  0
x2  10x  25  0
3x2  4x  2  0
a  2, b  5, c  3
a  1, b  10, c  25
a  3, b  4, c  2
Evaluate the discriminant:
Evaluate the discriminant:
Evaluate the discriminant:
b2  4ac
b2  4ac
b2  4ac
52 423
102  4125
42  432
25  24
100  100
16  24
49
0
8
When b2 4ac  0,
the equation has 2 real
solutions.
When b2  4ac  0, the
equation has 1 real solution.
When b2  4ac 0, the
equation has 2 complex
solutions.
2.7 Solving Quadratic Inequalities
Graphing quadratic inequalities is similar to graphing linear inequalities.
Graph y  x2  2x  3.
Draw the graph of y  x2  2x  3.
• a  1, so the parabola opens downward.
Step 1
• vertex at (1, 4)

b
2

 1 , and f 1  4
2a
2  1
• y-intercept is 3, so the curve also passes
through (2, 3)
Draw a solid boundary line for  or .
(Draw a dashed boundary line for or .)
Step 2
Shade below the boundary of the parabola
for  or . (Shade above the boundary for  or .)
Step 3
Check using a test point in the shaded region. Use 0, 0.
y  x2  2x  3
?: 0  02  20  3
:03
2.8 Curve Fitting with Quadratic Models
When the second differences are constant in a pattern of data, the data could represent a quadratic function.
x
2
3
4
5
6
y
6
12
20
30
42
Check that the x-values are
equally spaced.
Find the first differences. This means the differences between successive y-values.
12–6
20–12
30–20
42–30
6
8
10
12
Find the second differences. This means the differences between successive first differences.
8–6
10–8
12–10
2
2
2
Data Set 1 is a quadratic function.
Second differences are
constant. They are all 2.
2.9 Operations with Complex Numbers
Graphing complex numbers is like graphing real numbers. The real axis corresponds to the
x-axis and the imaginary axis corresponds to the y-axis.
To find the absolute value of a complex number, use a  bi  a2  b2 .
7i 

3  i 
0
2
 7
2
 49
Think:
7i  0  7i;
so a  0 and
b  7.

3
2
  1
2
 9 1
Think:
3  i  3  1i;
so a  3 and
b  1.
 10
7
To add or subtract complex numbers, add the real parts and then add the
imaginary parts.
4  i   2  6i 
Remember to distribute when
subtracting. Then group to add the
real parts and the imaginary parts.
4  i   2  6i
4  2  i  6i 
6  7i
Use the Distributive Property to multiply complex numbers.
Remember that i 2  1.
3i 2  i 
6i  3i 2
Distribute.
6i  31
Use i 2  1.
3  6i
Write in the form a  bi.
4  2i 5  i 
20  4i  10i  2i 2
Multiply.
20  6i  21
Combine imaginary parts and use i 2  1.
22  6i
Combine real parts.