PRE-CAL

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

1. To find the x-intercepts, let y=0 in equation and solve for x.

2. To find the y-intercepts, let x=0 in equation and solve for y.

Find the X and Y intercepts of the graph y=x²-4

y=x²-4 0=x²-4

=0²-4 =(x+2) (x-2)

=-4 x+2=0 and x-2=0

y-intercept = -4 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 0=x²+9

=0²+9 -9=x²

=9 √9=x

y-intercept=9 ±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 0=4x²-8

=8 8=4x²

y-intercept=8 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 0=4x²-16

=-16 =(2x-4) (2x+4)

y-intercept=-16 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 of a non-vertical line.

In order to determine the slope of a line, use the formula m=

If , L is a vertical line and the slope m of L is undefined (since this results in division by 0)

2 1

2 1

y y

x x

2x

2 1x x

5 2 3

7 3 4m

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)

The slope of the line is

2 1

2 1

y ym

x x

5 2 3

7 3 4m

3

4

Try Me!!

Find the slope of the line containing the point (4,8) and (7,2).

The slope of the line is -2

8 2 62

4 7 3m

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 xy-plane 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 noyes

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

a)

The function tells us to divide the 3x by x²-4. 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}

2

3( )

4

xg x

x

Try Me!!

Find the domain of the following function

a)

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 domain of h is {t|t≤ } or interval (-∞, ]

( ) 4 3h t t

4

3

4

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

The purpose of the half angle formula is

to determine the exact values of trig and

2 1 cossin

2 2

x x

2 1 coscos

2 2

x x

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 x-axis .

Symmetry with respect to the y-axis and the origin can be similarly explained.

Symmetry Cont. A graph is symmetric with respect A graph is symmetric with respect A graph is

symmetric with

to the x-axis if wherever (x,y) is on to the y-axis if whenever (x,y) is on the origin if whenever (x,y)

the graph (x,-y) is also on the graph. the graph, (-x,y) is also on the graph 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!!

Is the equation symmetric with respect to the origin?

Solution: Yes, because if you replace x with (-x) and y with (-y) it yields an equivalent equation.

2 1xy x

2 1xy x

2

( )( ) 1

xy x

2 1xy x

Volume Formulas

Volume of a cylinder –• Note: Think area of circular base times height

Volume of a cone –• Note: Think one-third the volume of the corresponding

cylinder

Volume of a sphere – Volume of rectangular prism –

○ In order to find the volume, just simply plug in the information into the correct place.

2V r h

21

3V r h

34

3V r

V lwh

Example

Find the volume of a cylinder with a height of 3 and a radius of 2.

2V r h2(2) (3)V

12V

Try Me!!

Find the volume of a cone with height of 5 and radius of 3.

21

3V r h

21(3) (5)

3V

45

3V

Recognizing Graphs and their respective equations

1( )f x x

2( )f x x 2( )f x x

2 2 2( ) ( )x h y k r

h

y

p

e

b

o

l

a

c

i

r

c

l

e

p

a

r

a

b

o

l

a

p

a

r

a

b

o

l

a

p

o

s

i

t

i

v

e

n

e

g

a

t

i

v

e

Graphs and their respective equations

y x 2 2

2 2

( ) ( )1

x h y k

b a

2 2

2 2

( ) ( )1

x h y k

a b

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)Power Rule- ln(x )= yln(x)

Example: ln(2 )= 8ln(2)

Derivative Rule- f(x)=ln(x)→f’(x)=1/xNatural log of a negative number- ln(x) is undefined

when x≤0Natural log of 1= 0Natural log of e= 1

xe

y

8

Example

Solve log (4x-7)=2

We can obtain an exact solution by changing the logarithm to exponential form.

log (4x-7)=2

4x-7=3²

4x-7=9

4x=16

x=4

3

3

Try Me!!

Solve log 64=2

We can obtain an exact solution by changing the logarithm to exponential form.

log 64=2

x²=64

x=√64=8

x

x

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-linear-equation.html

https://algebra1b.wikispaces.com/Linear+Equations+1B-1

http://www.sparknotes.com/math/algebra1/graphingequations/section4.rhtml

http://www.mathsisfun.com/sets/domain-range-codomain.htmlhttp://www.s-cool.co.uk/category/subjects/gcse/maths/graphshttp://everobotics.org/projects/robo-magellan/robo-magellan.htmlhttp://homepage.mac.com/shelleywalsh/MathArt/Symmetry.htmlhttp://www.squarecirclez.com/blog/how-to-draw-y2-x-2/2301