Download Sections R.5, 1.2 and 4.3

Sections 1.2 and 4.3
Dr. Doug Ensley
February 18, 2015
Do all terms have a common factor?
Examples. Factor each polynomial
I
x 3 − 7x
I
x(x 2 − 7)
I
2x 2 − 6x
I
2x(x − 6)
I
6x 2 − 18
I
6(x 2 − 3)
I
5x 3 + 10x 2
I
5x 2 (x + 2)
I
4x 3 + 2x 2 + 12x
I
2x(2x 2 + x + 6)
Quadratics of the form x 2 + bx + c
The goal here is to find two numbers whose product is c and
whose sum is b.
Example. Factor x 2 + 10x + 16
List all pairs of numbers whose product is 16, until you find the
pair which sum to 10.
Factor pairs:
I
1 & 16
I
2&8
I
4&4
Answer. x 2 + 10x + 16 = (x + 2)(x + 8)
Quadratics of the form x 2 + bx + c
The goal here is to find two numbers whose product is c and
whose sum is b.
Example. Factor x 2 − 3x − 18
List all pairs of numbers whose product is −18, until you find the
pair which sum to −3.
Factor pairs:
I
−1 & 18
1 & −18
I
−2 & 9
2 & −9
I
−3 & 6
3 & −6
Answer. x 2 − 3x − 18 = (x − 6)(x + 3)
Practice
Examples. Factor each polynomial
I
x 2 + 7x + 12
I
(x + 3)(x + 4)
I
x 2 − 6x − 16
I
(x − 8)(x + 2)
I
x 2 − 25
I
(x − 5)(x + 5)
I
x 2 + 10x + 25
I
(x + 5)(x + 5)
I
x 2 + 9x − 36
I
(x + 12)(x − 3)
Quadratics of the form ax 2 + bx + c
The goal here is to find two numbers whose product is ac and
whose sum is b. The two numbers will correspond to the OUTER
and INNER terms of FOIL.
Example. Factor 2x 2 + 11x + 15
List all pairs of numbers whose product is 30, until you find the
pair which sum to 11.
Factor pairs:
I
1 & 30
I
2 & 15
I
3 & 10
I
5&6
Make a table:
2x 2
6x
5x
15
Answer. 2x 2 + 11x + 15 = (2x + 5)(x + 3)
Quadratics of the form ax 2 + bx + c
The goal here is to find two numbers whose product is ac and
whose sum is b. The two numbers will correspond to the OUTER
and INNER terms of FOIL.
Example. Factor 6x 2 − 11x − 10
List all pairs of numbers whose product is −60, until you find the
pair which sum to −11.
Factor pairs:
I
1 & −60 OR −1 & 60
I
2 & −30 OR −2 & 30
I
3 & −20 OR −3 & 20
I
4 & −15 OR −4 & 15
..
.
I
Make a table:
6x 2
4x
Answer. 6x 2 − 11x − 10 = (2x − 5)(3x + 2)
−15x
−10
Practice
Examples. Factor each polynomial
I
5x 2 + 23x + 12
I
(5x + 3)(x + 4)
I
2x 2 + 3x − 20
I
(2x − 5)(x + 4)
I
4x 2 − 17x − 15
I
(4x + 3)(x − 5)
I
4x 2 − 25
I
(2x − 5)(2x + 5)
I
9x 2 − 12x + 4
I
(3x − 2)(3x − 2)
Quadratic Equations
There are three primary methods for solving quadratic equations.
Here they are in the order they should be considered:
I
The square root method is useful if you can easily find an
)2 =
.
equivalent equation of the form (
I
The factoring method involves first rewriting the equation of
= 0, and then working backwards with
the form
FOIL. If we can show the equation can be equivalently written
as Factor #1 · Factor #2 = 0, then we can conclude that
either Factor #1 is 0 or Factor #2 is 0.
I
The quadratic formula always works once your equation is
written in the form ax 2 + bx + c = 0. In this case, the
solutions are
√
−b ± b 2 − 4ac
x=
2a
Examples
Solve each equation, if possible.
I
(x − 8)2 = 25
I
x 2 + 3x = 10
I
7x 2 − 34x − 5 = 0
I
4x 2 − 196 = 0
I
9y 2 − y + 5 = 0
Examples
Solve each equation, if possible.
I
6(z 2 − 1) = 5z
I
4y 2 + 25 = 20y
I
x(x − 6) + 8 = 0
I
x 2 − 2x − 5 = 0
I
5x 2 − 12x + 7 = 0
Quadratic Functions
Example. Consider the quadratic function f (x) = x 2 + 8x + 15.
1. What is the domain of f ?
2. Is the graph of f concave up or concave down?
3. What is the vertex (h, k) of f ? Is this a local max or local
min?
4. What is the axis of symmetry of f ?
5. Find all x- and y -intercepts of f .
6. Use your answers to draw a sketch of the graph of f .
7. What is the range of f ?
Sketch
Graph of y = x 2 + 8x + 15
y
20
16
12
8
4
−9
−7
−5
−3
−1
−4
1 x
Quadratic Functions – Rules
If a quadratic function is given in general form f (x) = ax 2 + bx + c,
then we can answer these questions generically . . .
1. The graph of f (x) is concave up if a is positive; the graph of
f (x) is concave up if a is negative.
b
2. The x-coordinate of the vertex is h = − . The y -coordinate
2a
(k) is found by computing f (h). This point is a local min if
f (x) is concave up, and a local max if f (x) is concave down.
b
3. The equation of the axis of symmetry is x = − .
2a
4. The y -intercept occurs at the point (0, f (0)). The
x-intercepts are found by solving ax 2 + bx + c = 0.
5. If the graph is concave up, then the range is, “all y greater
than or equal to k, the y -coordinate of the vertex.” If the
graph is concave down, then change ”greater than” to “less
than.”
Practice
Example. Consider the quadratic function f (x) = 2x 2 − 9x + 10.
1. What is the domain of f ?
2. Is the graph of f concave up or concave down?
3. What is the vertex (h, k) of f ? Is this a local max or local
min?
4. What is the axis of symmetry of f ?
5. Find all x- and y -intercepts of f .
6. Use your answers to draw a sketch of the graph of f .
7. What is the range of f ?
Sketch of the graph
y
14
10
6
2
−1
1
−2
y
14
10
2
3
4
5
x
Quadratic Functions – Standard Form
Every quadratic function can be written in the form
f (x) = a(x − h)2 + k
where (h, k) is the vertex of the parabola. This is helpful for
finding function formulas from given data.
Example. Find a quadratic function with vertex (4, −5) and
y -intercept (0, 3).
Quadratic Functions – Applications
If a ball is thrown straight up with initial velocity 80 feet per
second, its height (in feet) after t seconds will be given by the
function
f (t) = 80t − 16t 2
1. Find the period of time for which the ball is more than 96 feet
in the air.
2. What is the maximum height reached by the ball?
3. How long is the ball in the air before it hits the ground?