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.
Quadratic function f(x) = a(x - h)2 + k(a ¹ 0)
You can make a table to graph a quadratic function.
Graph f(x) = x2 - 4x + 3.
Vertex Form of Quadratic Function
Horizontal and vertical translations change the vertex of f(x) = x2.
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 ¹ 0.
To graph f(x) = -x2 - 2x + 2:
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
(x + 5)(x - 3) = 0
(x + 5) = 0 or (x - 3) = 0
x = -5 or x = 3
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 ü 9 + 6 - 15 = 0 ü
The roots of x2 + 2x - 15 = 0 are -5 and 3.
The zeros of f (x) = x2 + 2x - 15 are -5 and 3.
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
(4x)2 - (5)2 = 0
(4x - 5)(4x + 5) = 0
(4x - 5) = 0 or (4x + 5) = 0
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
(2x - 3)(2x - 3) = (2x - 3)2 = 0
(2x - 3) = 0
2.4 Completing the Square
You can use the square root property to solve some quadratic equations.
Solve 4x2 - 5 = 43.
4x2 = 48 Add 5 to both sides.
x2 = 12 Divide both sides by 4.
Take the square root of both sides.
Simplify.
Solve x2 + 12x + 36 = 50.
(x + 6)2 = 50 Factor the perfect square trinomial.
Take the square root of both sides.
x + 6 = Subtract 6 from both sides.
x = -6 Simplify.
x = -6
You can use a process called completing the square to rewrite
a quadratic of the form x2 + bx as a perfect square trinomial.
Complete the square: x2 - 8x + ?.
Step 1 Identify b, the coefficient of x: b = -8.
Step 2 Find
Step 3 Add
Step 4 Factor: x2 - 8x + 16 = (x - 4)2
Check: (x - 4)2 = (x - 4)(x - 4)
= x2 - 8x + 16 ü
2.5 Complex Numbers and Roots
An imaginary number is the square root of a negative number.
Use the definition to simplify square roots.
Simplify.
Factor out -1.
Separate roots.
Simplify.
5i Express in terms of i.
Complex numbers are numbers that can be written in the form a + bi.
The complex conjugate of a + bi is a - bi.
The complex conjugate of 5i is -5i.
You can use the square root property and to solve quadratic equations with
imaginary solutions.
Solve x2 = -64.
Take the square root of both sides.
x = ±8i Express in terms of i.
Check each root: (8i)2 = 64i 2 = 64(-1) = -64
(-8i)2 = 64i 2 = 64(-1) = -64
Solve 5x2 + 80 = 0.
5x2 = -80 Subtract 80 from both sides.
x2 = -16 Divide both sides by 5.
Take the square root of both sides.
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. x2 - 6x - 11 = 0
Step 2 Write the Quadratic Formula.
Step 3 Substitute values for a, b, and c into the Quadratic Formula.
a = 1, b = -6, c = -11
Step 4 Simplify.
Step 5 Write in simplest form.
The discriminant of ax2 + bx + c = 0 (a ¹ 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.
2.7 Solving Quadratic Inequalities
Graphing quadratic inequalities is similar to graphing linear inequalities.
Graph y £ -x2 + 2x + 3.
Step 1 Draw the graph of y = -x2 + 2x + 3.
• a = -1, so the parabola opens downward.
• vertex at (1, 4)
, and f (1) = 4
• y-intercept is 3, so the curve also passes
through (2, 3)
Draw a solid boundary line for £ or ³.
(Draw a dashed boundary line for or .)
Step 2 Shade below the boundary of the parabola
for or £. (Shade above the boundary for or ³.)
Step 3 Check using a test point in the shaded region. Use (0, 0).
y £ -x2 + 2x + 3
?: 0 £ -(0)2 + 2(0) + 3
ü : 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.
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.
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
|7i | |3 - i |
= 7
To add or subtract complex numbers, add the real parts and then add the
imaginary parts.
(4 - i ) - (-2 + 6i )
(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.