Pre-Cal

1. Anthony Poole & Keaton Mashtare 2nd Period Pre-Cal

2. 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.

3. 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

4. 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

5. 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

6. 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)

7. 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.

8. Slope Cont. • Find slope of the line that contains the points (7,5) and (3,2) • The slope of the line is

9. Try Me!! • Find the slope of the line containing the point (4,8) and (7,2). • The slope of the line is -2

10. 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.

11. 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.

12. Function Cont. • Determine whether the following graphs are functions. yes no yes

13. 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).

14. Example #1 • Find the domain of each of the following functions. • 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.

15. Example #2 • Find the domain of the following function 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}

16. Try Me!! • Find the domain of the following function 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 (-∞, ]

17. The Unit Circle • The unit circle is a circle whose radius 1 and whose center is at the origin of a rectangular coordinate system.

18. Half-Angle Formulas • The purpose of the half angle formula is to determine the exact values of trig and

19. 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.

20. 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

21. 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

22. 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

23. 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.

24. 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.

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

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

27. Recognizing Graphs and their respective equations

28. Graphs and their respective equations

29. 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

30. 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

31. 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

32. Number e (Euler’s Number) • The number e is defined as the base of the natural logarithm • it is an irrational number 2.7182818284590452353602874…

33. http://www.mathexpression.com/find-the-x-and-y-intercept-of-a-linear-equation.htmlhttp://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.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