Advanced Algebra/Trig notes Chapter 1

1-3- Linear Functions Linear Function- a function f defined by f(x) = ax + b, where a and b are real numbers y = 3x + 6 is a “linear equation in two variables” Linear functions are straight lines, so if we know the coordinates of two points, we can graph the line. x-intercept is the point where the graph crosses t x-axis. It is where y = 0 y-intercept is the point where the graph crosses t y-axis. It is where x = 0 zero of a function- let f be a function. Then any number c for which f(c) = 0 is called a zero of the function f. If c is a real zero of f, then f(c) = 0 and c is an x-intercept of the graph of f

1-3- Linear Functions

A vertical line has an undefined slope Slope-intercept form Y = mx + bwhere m is the slope, and b is the y-intercept

1-3- Linear Functions

Find the slope between the points: (3, 9), and (-1, 0) (3, -2), and (3, 8) (-2, 3), and (12, 3)

1-4: Equations of Lines and Linear Models

From the information given, Find the equation of the line in slope-intercept form, Point-Slope form, and Standard form a)Slope = 4, through point (1, 0) b)Through (-1, 4), parallel to x + 3y = 5 c)Through (-5, 7), perpendicular to y = -2 d)Through (1, 6), perpendicular to 3x + 5y = 1 e)X-intercept (3, 0), and y-intercept (0, -1)

1-5: Solving Linear Equations in One Variable Linear Equation in one variable- an equation that can be written: ax + b = 0, where a ≠ 0 To solve: 1)Analytic approach- use pencil and paper to continually transform the equation into simpler equations (algebraically) 2)Graphical approach- use graphs/tables Addition Property of Equality a = b and a + c = b + c are equivalent Multiplication Property of Equality a = b and ac = bc are equivalent if c ≠ 0

1-5: Solving Linear Equations in One Variable Solving Graphically Think of the equation as being f(x) = g(x) Graph each separately, find where they cross X-coordinate is the answer If f and g are linear functions, then their graphs are lines that intersect at a single point, no point, or infinitely many points Example Solve graphically: 4x – 3 = 2x + 3  treat each side of the equation as a separate function graph: f(x) = 4x – 3 and g(x) = 2x + 3 The x-coordinate of the intersection is the solution

1-5: Solving Linear Equations in One Variable X-intercept method Graph y = f(x) – g(x) X-intercept (or zero of function) is the solution Example Solve graphically using the x-intercept method: 4x – 3 = 2x + 3  this time we graph y = (4x – 3) – (2x + 3) the x-intercept is the solution Contradiction- an equation with no solution solution set is the empty set of null set, denoted by ø Identity- an equation that is true for all values in t domain of its variables

1-5: Solving Linear Equations in One Variable Solving linear inequalities of one variable ax + b > 0, ax + b < 0, ax + b ≥ 0, ax + b ≤ 0 Use interval notation for the answers Solve linear inequalities graphically Graph f(x), g(x) Solutions are the x-values that correspond with the right direction Example Solve graphically: 3x – 4 < 0  graph f(x) = 3x – 4 and g(x) = 0 solutions are the x-coordinates where f(x) < g(x)

1-5: Solving Linear Equations in One Variable

For quiz Thursday: Use Pythagorean Theorem, Distance Formula, and Midpoint Formula Determine if a relation is a function Find the domain and range of a function- write in interval notation From a linear equation or graph, find x-intercept y-intercept slope zeros From information given, find the equation to a linear function -including finding the slope from parallel or perpendicular lines Graph linear functions From two points, find the slope Evaluate functions at given values (from equation or graph) Transform point-slope and standard form equations into slope-intercept form Graphically solve linear equations and inequalities in one variable

1-5: Solving Linear Equations in One Variable Also for the chapter 1 test on Aug 31: Word problems similar to those found in 1-6 Solve direct variation problems Rearrange formulas to solve for a specified variable Classify numbers (natural, whole, integers, etc) Anything else found in chapter 1 that may not be on this list (other than linear models and regression in 1.4)

1-5: Solving Linear Equations in One Variable Also for the chapter 1 test on Aug 31: Word problems similar to those found in 1-6

1-6: Applications of Linear Functions Solving application problems: Read the problem (more than once!) Assign a variable for what you need to find Write an equation, draw pictures, list out information given Solve the equation- determine the solution Look back and check Mixture-of-concentration problems How much pure alcohol should be added to 20 liters of 40% alcohol to increase the concentration to 50% alcohol Liters: 20 + x = 20 + x Alcohol concentration: = (20) x = 0.50(20 + x)

1-6: Applications of Linear Functions Direct Variation y = kx plug the x-value into x and the y-value into y, solve for k rewrite the problem as y = kx with the k you found Perimeter of a rectangle P = 2L + 2W