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Anthony Poole & Keaton Mashtare
2nd Period
X and Y intercepts

The points at which the graph crosses or touches the
coordinate axes are called intercepts. The x-coordinate of a
point at which the graph crosses or touches the x-axis is the
x-int. The y-coordinate of a point at which the graph crosses
or touches the y-axis is the y-int.
Finding X and Y intercepts
To find the x-intercepts, let y=0 in equation and solve for x.
To find the y-intercepts, let x=0 in equation and solve for y.

1.
2.

Find the X and Y intercepts of the graph
y=x²-4
y=x²-4
=0²-4
=-4
y-intercept = -4

0=x²-4
=(x+2) (x-2)
x+2=0 and x-2=0
x=-2
and x=2
x-intercepts=-2 and 2
Find the X and Y intercepts of the graph
y=x²+9
y=x²+9
=0²+9
=9
y-intercept=9
0=x²+9
-9=x²
√9=x
±3=x
x-intercepts=-3 and 3
Try Me!!

Find the X and Y intercept(s) of the
equation y=4x²-8
y=4(0)²+8
=8
y-intercept=8
0=4x²-8
8=4x²
2=x²
√2=x
x-intercept=±2
Try Me!!

Find the X and Y intercepts of the
equation y=4x²-16
y=4(0)²-16
=-16
y-intercept=-16
0=4x²-16
=(2x-4) (2x+4)
2x-4=0 and 2x+4=0
x=2 and x=-2
x-intercept=2 and -2
Slope/Point-Slope/Slope-Intercept
The slope of a line is a measurement of
the steepness and direction ofx a nonvertical line.
 In order to determine the slope of a line,
y 2  y1
use the formula m=

m
52 3
2 
73 4
x 2  x1

If x2  x1 , L is a vertical line and the
slope m of L is undefined (since this
results in division by 0)
Slope Cont.


A line can have a positive slope, a negative slope,
a slope of 0, and an undefined slope.
If the line is declining from right to left the slope is
positive.

If the line is declining from left to right the slope is
negative.

If the line is horizontal the slope is 0.

If the line is vertical the slope is undefined.
Slope Cont.

Find slope of the line that contains the
points (7,5) and (3,2)
y 2  y1
 m
x 2  x1
52 3
m 

73 4

The slope of the line is
3
4
Try Me!!

Find the slope of the line containing the
point (4,8) and (7,2).
82 6
 m

 2
4  7 3

The slope of the line is -2
Function, Domain, Range
 A function
from set D to a set R is a
rate that assigns to every element in
D a unique element in R. The set D
of all input values is the domain of the
function, and the set R of all output
values is the range of the function.
Function
To determine whether a graph is a
function, use the Vertical Line Test.
 A graph (set of points (x,y)) in the xyplane defines y as a function of x if and
only if no vertical line intersects the
graph in more than one point.


The vertical line test states, if you draw a vertical line anywhere on the
graph and it hits the graph in only one place then the graph is a function. If
the line hits the graph in two or more places then the graph is not a function.
Function Cont.

Determine whether the following graphs
are functions.
yes
no
yes
Domain and Range

Often the domain of a function f is not
specified; instead, only the equation
defining the function is given. In even
cases, the domain of f is the largest set
of real numbers for which the value of
f(x) is a real number. The domain of f is
the same as the domain of the variable x
in the expression f(x).
Example #1
Find the domain of each of the following
functions.
a) f(x)=x²+5x
The function f tells us to square the
number and then add 5 times the
number. Since the operations can be
performed on any real number, we
conclude that the domain of f is all real
numbers.

Example #2

Find the domain of the following function
3x
a) g ( x ) 2
x 4
The function tells us to divide the 3x by x²4. Since the division by 0 is not
defined, the denominator x²-4 can
never be equal to 0, so x can never be
equal to -2 or 2. The domain function g
is {x|x≠-2, x≠2}
Try Me!!
Find the domain of the following function
a) h(t )  4  3t
The function h tells us to take the square
root of 4-3t. But only non-negative
numbers have real square roots, so the
expression under the square root must
be greater than or equal to 0. This
requires that 4-3t≥0. Therefore the
4
4
domain of h is {t|t≤ 3 } or interval (-∞, 3 ]

The Unit Circle

The unit circle is a circle whose radius 1
and whose center is at the origin of a
rectangular coordinate system.
Half-Angle Formulas
x 1  cos x

 sin
2
2
2
2
 cos

x 1  cos x

2
2
The purpose of the half angle formula is
to determine the exact values of trig and
Testing for Symmetry
Symmetry with respect to the x-axis
means that if the cartesian plane were
folded along the x-axis, the portion of
the graph above the x-axis would
coincide with the portion below the xaxis .
 Symmetry with respect to the y-axis and
the origin can be similarly explained.

Symmetry Cont.
A graph is symmetric with respect
to the x-axis if wherever (x,y) is on
the graph (x,-y) is also on the graph.
A graph is symmetric with respect
to the y-axis if whenever (x,y) is on
the graph, (-x,y) is also on the graph
A graph is symmetric with
the origin if whenever (x,y)
is on the graph, (-x,-y) is
also on the graph
Example
Is the equation y=x²-2 symmetric with
respect to the y-axis?
Solution: Yes, because the point (-x,y)
satisfies the equation.
y=x²-2
y=(-x)²-2
y=x²-2

Try Me!!
Is the equation x-y²=1 symmetric with
respect to the x-axis?
Solution: Yes, because when you replace y
with (-y) it yields an equivalent equation.
x-y²=1
x-(-y)²=1
x-y²=1

Try Me!!
x
y

 Is the equation
x2 1 symmetric with
respect to the origin?
Solution: Yes, because if you replace x
with (-x) and y with (-y) it yields an
equivalent equation.
y  2x
x 1
( x)
y 
( x)2 1
y  2x
x 1
Volume Formulas

Volume of a cylinder – V   r h
2
• Note: Think area of circular base times height

Volume of a cone
–V 
1
 r 2h
3
• Note: Think one-third the volume of the corresponding
cylinder
4 3
 Volume of a sphere – V   r
3
 Volume of rectangular prism – V  lwh
○ In order to find the volume, just simply plug in
the information into the correct place.
Example

Find the volume of a cylinder with a
height of 3 and a radius of 2.
V r h
2
V   (2) (3)
V  12
2
Try Me!!

Find the volume of a cone with height of
5 and radius of 3.
1 2
V  r h
3
1
V   (3)2 (5)
3
45
V 
3
Recognizing Graphs and their
respective equations
h
c
y
i
p
e
r
b
c
o
l
l
e
a
f ( x)  1
x
( x  h)2  ( y  k )2  r 2
p
p
n
p
o
a
e
a
s
r
g
r
i
a
a
a
t
b
t
b
i
o
i
o
v
l
v
l
e
a
e
a
f ( x)  x 2
f ( x)   x 2
Graphs and their respective
equations
y  x
( x  h) 2 ( y  k ) 2

1
b2
a2
( x  h) 2 ( y  k ) 2

1
a2
b2
Natural Log

The natural logarithm function ln(x) is the
inverse function of the exponential function
 Product Rule- ln(xy)= ln(x) + ln(y)
 Example: ln(3*7)= ln(3) + ln(7)
 Quotient Rule- ln(x/y)= ln(x) – ln(y)
y
 Power Rule- ln(x )= yln(x)
8
 Example: ln(2 )= 8ln(2)
 Derivative Rule- f(x)=ln(x)→f’(x)=1/x
 Natural log of a negative number- ln(x) is
undefined when x≤0
 Natural log of 1= 0
 Natural log of e= 1
e
x
Example
Solve log3(4x-7)=2
We can obtain an exact solution by
changing the logarithm to exponential
form.
log3(4x-7)=2
4x-7=3²
4x-7=9
4x=16
x=4

Try Me!!
Solve logx64=2
We can obtain an exact solution by
changing the logarithm to exponential
form.
log x 64=2
x²=64
x=√64=8

Number e (Euler’s Number)

The number e is defined as the base of
the natural logarithm
▪ it is an irrational number
2.7182818284590452353602874…
http://www.mathexpression.com/find-the-x-and-y-intercept-of-a-linearequation.html
https://algebra1b.wikispaces.com/Linear+Equations+1B-1
http://www.sparknotes.com/math/algebra1/graphingequations/section4.r
html
http://www.mathsisfun.com/sets/domain-range-codomain.html
http://www.s-cool.co.uk/category/subjects/gcse/maths/graphs
http://everobotics.org/projects/robo-magellan/robo-magellan.html
http://homepage.mac.com/shelleywalsh/MathArt/Symmetry.html
http://www.squarecirclez.com/blog/how-to-draw-y2-x-2/2301