Chapter 4 Notes
4.1A Graph Quadratic Functions in Form f(x) = ax2 and
f(x) = ax2 + c
Quadratic Functions - graph is a parabola
Parent Quadratic Function: f(x) = x2
Vertex: (0, 0)
lowest point of the graph.
axis of symmetry: x = 0, passes through the
vertex and divides the parabola into mirror images
f(x) = ax2 + c
"a" positive - graph opens up
"a" negative - graph opens down
| a | > 1 graph is narrower than parent graph
| a | < 1 graph is wider than parent graph
Graph shifts up "c" units
4.1B Graph Quadratic Functions in the Form
y = ax2 + bx + c
Characteristics:
a > 0 graph opens up
a < 0 graph opens down
|a| > 1 graph is more narrow than parent graph
|a| < 1 graph is wider than parent graph
x = -b axis of symmetry
2a
-b x - coordinate of the vertex, vertex is maximum
2a
when graph opens down, minimum
when graph opens up.
c is the y-intercept so point (0,c) is on the graph
Graph:
1. Plot the vertex
2. Graph the line of symmetry
3. Make a table and plot and label 4 points
4. Draw the parabola (arrows)
4.2
Graph Quadratic Function in Vertex Form
Vertex Form: y = a(x - h)2 + k
Characteristics:
Parabola y = ax2 translated horizontally h units
and vertically k units
Vertex (h, k)
Axis of symmetry is x = h
a > 0 graph opens up
a < 0 graph opens down
Graph: Vertex, 4 other points, axis of symmetry
Graph Quadratic Function in Intercept Form
Intercept Form:
y = a(x - p)(x - q)
Characteristics:
x-intercepts are (p, 0) and (q, 0)
Axis of symmetry is half way between x-intercepts
x=p+q
2
a > 0 graph opens up
a < 0 graph opens down
Graph: Vertex, line of symmetry, and x-intercepts
4.3/4.4 Solve Quadratic Equations by Factoring
Standard Quadratic Equation: ax2 + bx + c = 0
1. Set equation equal to zero
2. Factor
3. Use the Zero Product Property ZPP
If the product of two expressions is zero, then one
or both
of the expressions equal zero. AB = 0, then A=0
or B=0
Zeros of the function: Solutions are the x-intercepts
because the function equals zero.
4.5 Solve Quadratic Equations by Finding Square Roots
Properties of Square Roots ( a> 0, b > 0 )
Product Property
Quotient Property
Rationalizing the denominator
Multiply by the radical or the conjugate
Solve:
1. Isolate the variable squared term or quantity
2. Take the square root of each side
3. Simplify
4. Check
4.6 Perform Operations with Complex Numbers
Complex Numbers
Imaginary
i = √-1
Real
irrational
rational
integers
whole
counting
i= √-1
=i
i2 = (√-1)2 = -1
i3 =-1(i )
= -i
i4 = -1(-1)
=1
complex number a + bi
Add, subtract, or multiply using i as a variable
Divide complex - multiply by the conjugate
Complex plane - horizontal is the real axis
vertical is the imaginary axis
Absolute value of a complex number z = a + bi, denoted
|z|,
is a nonnegative real number defined as distance
between z and the origin in the complex plane.
|z| = √a2 + b2
4.7 Complete the Square
1. Complete the square to write perfect square equation
a. divide each term by the x2 coeffient
b. Isolate the quadratic and linear term on 1 side
c. Add half of b squared to both sides
d Write as binomial squared
2. Solve using the square root method
3. Simplify
4. Check
Write quadratic functions in vertex form by completing the
square.
y = a(x - h)2 + k
4.8 Use the Quadratic Formula to Solve Quadratic
Equations
ax2 + bx + c = 0
x = (-b ± √(b2 - 4ac))/2a
Discriminant b2 - 4ac > 0 2 real solutions
b2 - 4ac = 0 1 real solution
b2 - 4ac < 1 no real solutions
Motion Problems:
object is dropped h = -16t2 + ho
object is thrown
h = -16t2 + vot + ho
4.9 A Graph and solve Quadratic Inequalities in 2 Variables
quadratic inequality in two variables:
y < ax2 + bx + c
y > ax2 + bx + c
y < ax2 + bx + c
y > ax2 + bx + c
Graph:
* Graph the parabola (vertex and 4points)
* Test a point inside or outside the parabola
* Shade the true area
Systems of quadratic inequalities:
*Graph, Test, & Shade each inequality
*The solution is where the shadings overlap - shade
darker
4.9B Graph and solve Quadratic Inequalities in 1 Variable
quadratic inequalities in one variable:
ax2 + bx + c < 0
ax2 + bx + c > 0
ax2 + bx + c < 0
ax2 + bx + c > 0
Solve by using a table:
*Rewrite the inequality in standard form
*Make a table of values
*Write an inequality statement where for the x values
where it fits the inequality statement
Solve by graphing:
*Find the graph's x-intercepts by setting equation = 0
*Graph the equation y = ax2 + bx + c using the vertex
and x-intercepts
*The solution consists of the x-values for which the
graph of y = ax2 + bx + c lies above x-axis for >,
below x-axis for <.
4.9B Graph and solve Quadratic Inequalities in 1 Variable
quadratic inequalities in one variable:
ax2 + bx + c < 0
ax2 + bx + c > 0
ax2 + bx + c < 0
ax2 + bx + c > 0
Solve by using a table:
*Rewrite the inequality in standard form
*Make a table of values
*Write an inequality statement where for the x values
where it fits the inequality statement
Solve by graphing:
*Find the graph's x-intercepts by setting equation = 0
*Graph the equation y = ax2 + bx + c using the vertex
And x-intercepts
*The solution consists of the x-values for which the
graph of y = ax2 + bx + c lies above x-axis for >,
below x-axis for <.
Solve algebraically:
*Write an equation in standard form
*Solve by factoring (ZPP)
*Plot the critical x-values on a number line
*Test an x value in each of the intervals to see which
Intervals satisfy the inequality statement
*Write the inequality solution
4.10 Write Quadratic Functions
Vertex Form: y = a(x - h)2 + k
Given: vertex
1 point
*substitute the point and vertex to solve for "a"
Intercept Form: y = a(x - p)(x - q)
Give: x-intercepts
1 point
*Substitute the point and intercepts to solve for "a"
Standard Form: y = ax2 + bx + c
Given: 3 points
*Write a system of three equations, three variables
*Use matrix equation or row operations to solve for
a,b,c