Anthony Poole & Keaton Mashtare 2 nd 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=x²-40=x²-4 =0²-4 =(x+2) (x-2) =-4x+2=0 and x-2=0 y-intercept = -4x=-2 andx=2 x-intercepts=-2 and 2  Find the X and Y intercepts of the graph y=x²+9 y=x²+90=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+80=4x²-8 =88=4x² y-intercept=82=x² √2=x x-intercept=±2

Try Me!!  Find the X and Y intercepts of the equation y=4x²-16 y=4(0)²-160=4x²-16 =-16 =(2x-4) (2x+4) y-intercept=-162x-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)

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

Try Me!!  Find the slope of the line containing the point (4,8) and (7,2).   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 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. yesnoyes

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}

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 (-∞, ]

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

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 respectA graph is symmetric with to the x-axis if wherever (x,y) is on to the y-axis if whenever (x,y) is onthe 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.

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.

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

Try Me!!  Find the volume of a cone with height of 5 and radius of 3.

Recognizing Graphs and their respective equations

Graphs and their respective equations

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/x Natural log of a negative number- ln(x) is undefined when x≤0 Natural log of 1= 0 Natural log of e= 1 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 …

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