Course Outline. Linear Equations Linear Inequalities (1.6)

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1 Course Outline Functions/Graphing Solving Equations Applications Definitions of function, graph, domain, range, x- and y- intercepts, vertical line test(.-.) Linear functions (.-.) -Parallel and perpendicular lines Piecewise Functions: Absolute value functions (.) Quadratic Functions: Completing the square, vertex, increasing/decreasing, maximum/minimum, graph movements (.,.,.-.) Polynomial functions: Relative/Absolute Maximum/Minimum (.-.) Radical Functions: Graphing with movements Rational Functions: Horizontal/Vertical asymptotes (.) Algebra of Functions: Sums and Di erences, Products and Quotients, Compositions, Di erence Quotients (.-.) Inverse Functions (.) Linear Equations Linear Inequalities (.) Absolute Value Equations and Inequalities (.) Quadratic Equations: Complex numbers, Quadratic formula (.-.) Polynomial Equations: Long/Synthetic Division (.) Radical Equations (.) Rational Equations (.) Fahrenheit to Celsius conversion, Cost per hour, Average rate of change Maximum/minimum problems Finding Inverses Exponential Functions (.) Exponential Equations (.) Interest, Growth/Decay (.) Logarithmic Functions (.) Logarithmic Equations: Logarithm rules (.) Solving more complicated logarithmic/exponential equations (.)

2 Introduction to Functions and Graphing (Sections.-.) Definition. A function is a rule that assigns to each member of the first set (called the domain) exactly one member of a second set (called the range). x (input) f f(x) (output) Example. Determine whether or not each diagram represents a function. a a b b c c d d

3 Notation. Functions are usually given by formulas, such as: y = f(x) = g(x) = Example. Let f(x) =x +andg(x) =x +x +. Determine the outputs below. f( ) f() g( ) g() Example. Now let s try some that are a little harder. f(x) =x and g(x) = x +x. Determine the outputs below. f(t +) g(t +)

4 Definition. The graph of a function, f(x), is the picture obtained by plotting all ordered pairs of the form (x, f(x)), i.e. (input, output). Example. Determine if the following graphs correspond to functions Theorem (Vertical Line Test). If it is possible for a vertical line to cross a graph more than once, then the graph is...

5 Definition. Given a function y = f(x): x-intercept(s) of the function y-intercept of the function domain of the function range of the function Interval Notation:

6 Example. Graph the following functions using a table. Then use the graph to estimate the x- and y-intercepts, as well as the domain and range. f(x) =x +x h(x) = p x

7 Example. You can also use a table to draw graphs of equations that are not functions. y =± x x + y =

8 Example. Graph the following on your graphing calculator. Using the graph, estimate the x- and y-intercepts, as well as the domain and range. y = p x y = x x x-intercept(s) x-intercept(s) y-intercept y-intercept Domain Domain Range Range y = x + y = p x x-intercept(s) x-intercept(s) y-intercept y-intercept Domain Domain Range Range Since we can t always rely on the graph to be able to determine the x-/y-intercepts and the domain/range exactly, we will have to learn algebraic methods as well.

9 Linear Functions and Linear Equations (Sections.-.) Definition. Afunctionf is called a linear function if it can be written in the form where m and b are constants. Example. Determine whether each of the following is a linear function. y x = y x +=0 y x =0 (x y +)=0

10 0 Remark. The two constants m and b represent important parts of the graph. The constant b is The constant m is called the slope of the line and is a measure of the steepness of the line where (x,y )and(x,y )aretwopointsontheline. (x,y ) (x,y ) y y (rise) x x (run)

11 Example. Graph the following linear functions without using a table. f(x) =x y =x x + y = g(x) =

12 Example. Find the slope of the line through the two given points. (, ) and (, ) (, ) and (, ) (, ) and (, ) (, 0) and (, ) (, ) and (, ) (0, ) and (, )

13 Remark. Horizontal lines: y = b Vertical lines: x = a Example. Graph the following lines. Are linear functions Are NOT linear functions y = y = x = x = Note: The slope of a horizontal line is. The slope of a vertical line is.

14 Remark. To find the equation of a line, you can use one of the following forms of a line. However, your answer should always be given in slope-intercept form. Slope-Intercept Form Point-Slope Form y = mx + b y y = m(x x ) Example. Find the equation of the line through (, ) having slope m =. Example. Find the equation of the line given a point on the line and the slope. (, ) and m = (0, ) and m =

15 Example. Find the equation of the line through (, ) and (, ). Example. Find the equation of the line through the given points. (, ) and (0, ) (, ) and (, ) (, ) and (, ) (, ) and (, )

16 Remark. To find x- andy-intercepts algebraically: Set x =0fory-intercept Note: For linear equations in the form y = mx + b, the y-intercept is just (0,b) Set y =0forx-intercept(s) Example. Find the x- and y-intercepts for the graph of each equation. y =x + y = x + x-intercept(s) y-intercept x-intercept(s) y-intercept x = y = x-intercept(s) y-intercept x-intercept(s) y-intercept

17 Applications of Linear Functions Average Rate of Change (Section.) Remark. The average rate of change between any two data points on a graph is the slope of the line passing through the two points. Example. Kevin s savings account balance changed from $0 in January to $0 in April. Find the average rate of change per month. Example. The table below indicates the number of cases of West Nile Virus over a period of four months. What is the average rate of change, in cases per month, between months and? Month #ofcases 0 Example. Find the average rate of change for each function on the given interval. f(x) =x +x x on [, ] f(x) =x on[, ]

18 Applications of Linear Functions Modeling (Section.-.) Example. Given the following information, determine a formula to convert degrees Celsius to degrees Fahrenheit. Water freezes at 0 Celsius and Fahrenheit. Water boils at 00 Celsius and Fahrenheit. Example. Ataxiservicechargesa$.pickupfeeandanadditional$.permile. Ifthe cab fare was $.0, how many miles was the cab ride?

19 Example. A lake near the Arctic Circle is covered by a meter thick sheet of ice during the winter months. When spring arrives, the warm air gradually melts the ice, causing its thickness to decrease at a constant rate. After weeks, the sheet is only. meters thick. Determine a formula to represent the thickness of the ice sheet x weeks after the start of spring. Example. In a certain College Algebra class at IUP, final grades are computed according to the following chart: Exams: % Final Exam: 0% Homework: % Quizzes: 0% Astudentscores%onExam,%onExam,and%onExam. OnMyMathLab,their homework average is %; and their quiz average is %. Determine a linear function that will compute the students final grade based on their score on the final exam. Use this function to determine what the student needs on the final exam to get an A, to get a B, and to get a C.

20 0 Parallel and Perpendicular Lines (Section.) Definition. Two distinct lines are called parallel i () they are both vertical lines, or () they have the same slope Example. Determine if the two lines are parallel. x y = y =x y =x y +x = Example. Find the equation of the line through the given point and parallel to the given line. (0, ), y = x + (, ), y =

21 (, ), y = x + (, ), x = Definition. Two distinct lines are called perpendicular i () one is vertical and one is horizontal, or () their two slopes, m and m,satisfym m = Example. Determine if the two lines are perpendicular. y = x + y = x y +x +=0 x y =

22 Example. Find the equation of the line through the given point and perpendicular to the given line. (0, ), y = x + (, ), y = (, ), y = x + (, ), x =

23 Intersection Points and Zeros (Section.) Definition. An intersection point of two functions is a point (x, y) where the graphs of each function cross (i.e. it is a point that is in common to both graphs). Example. Find the intersection point (if any) between the two given lines. y =x + and y =x y =x andy =x + y = x +andy = x y =andx = Example. A private plane leaves Midway Airport and flies due east at a speed of 0 km/h. Two hours later, a jet leaves Midway and flies due east at a speed of 00 km/h. After how many hours will the jet overtake the private plane?

24 Definition. Zeros of a function are the values of x that make the function equal 0 (also the x-coordinate of the x-intercept). Example. Find the zero(s) of each function (if any). f(x) =x + g(x) = g(x) = x f(x) =x + Example. Aplaneisdescendingfromaheightof,000feetataconstantrateof00feetper minute. After how long will the plane have landed on the ground?

25 Linear Inequalities (Section.) Rules: Thesignstaysthesameunlessyoumutliply/dividebyanegative,thenthesignflips. Example. Solve each inequality. Then graph the solution set, and then write the solution set in interval notation. x +apple x + x >x + apple x +< 0 < x +apple Example. Macho Movers charges $00 plus $ per hour to move a household across town. Brute Force Movers charges a flat fee of $ per hour for the same move. For what lengths of time does it cost less to hire Brute Force Movers?

26 Piecewise Defined Functions (Section.) These are just functions whose rules are defined in di erent parts/pieces Example. Determine the function values. x +,x> (a) f(x) = x,xapple f() < p x +,xapple 0 (b) g(x) = x +, 0 <xapple :,x> g( ) g(0) f() g() f( ) g()

27 Absolute Value Function (Sections. and.) x,x < 0 Definition. h(x) = is the absolute value function, denotedbyh(x) = x. x,x 0 Remark. To solve absolute value equations: = a No solution if a<0 Equivalent to = a or = a if a 0 Example. Solve each of the following. x = x + = x += x + =

28 Remark. To solve absolute value inequalities when a>0: <a means a< <a (Also true for apple) >a means >aor < a (Also true for ) Example. Solve each of the following. Write your answer in interval notation. x < x 0. apple0. x x >

Course Outline. Linear Equations Linear Inequalities (1.6)

Course Outline. Linear Equations Linear Inequalities (1.6) Course Outline Functions/Graphing Solving Equations Applications Definitions of function, graph, domain, range, x- and y- intercepts, vertical line test(.-.) Linear functions (.-.5) -Parallel and perpendicular