MAC 1105-College Algebra LSCC, S. Nunamaker Chapter 1-Graphs, Functions, and Models 1.1 Introduction to Graphing I. Reasons for using graphs A. Visual presentations enhance understanding. B. Visual presentations enhance recall. II. Cartesian Coordinate System A.Characteristics 1. Origin, the reference point of the system, represents (0, 0).. Horizontal axis is the x-axis (unless otherwise specified). 3. Vertical axis is the y-axis (unless otherwise specified). 4. Ordered pair (x, y) represents a unique point on the graph. 5. Quadrants: I (+, +). II (-, +), III (-, -), IV (+, -) 6. Usually, the x variable is the independent and y variable is the dependent variable. x is the first coordinate or abscissa while y is the second coordinate or ordinate. 7. x-intercept is at the point (a, 0 ): let y=0 and solve for x. y-intercept is at the point (0, a): let x=0 and solve for y. III. Be able to use graphing calculator to graph various equations. A. Linear equations: straight line, ex. y=x+1. a. Is ( 1, 4 ) or ( 1, 3 ) a solution of y x 1? b. Graph y x 1 by plotting at least points. Examples: 1. 3x 5 y 15. 4x 1 y 3. 4x 6 y 1 1
B. nd degree equations: parabolic line/curve, ex. y x 1 LSCC, S. Nunamaker a. Is ( 0, 1 ) or (1, 3 ) a solution of y x 1? b. Graph y x 1using graphing calculator. C. Circle: ex. x y 4. a. Is (, 0 ) or ( 0, - ) a solution of x y 4? b. Graph x y 4 IV. The Distance Formula The distance between any two points ( x1, y1 ) is given by: d ( x x ) ( y y ) 1 1 Example: Find the distance between (-3, 7 ) and (, 10 ) Find the distance between ( 3, 5 ) and ( 6, 0 ) The point (0, 1) is on a circle that has a center (-3, 5). Find the length of the diameter of the circle. * Use the distance formula and the Pythagorean theorem (a b c s.t. a and b are two legs and c is the hypoteneuse of a right triangle) to determine whether this set of points :(-3, 1 ), (, -1 ), and ( 6, 9 ) could be vertices of a right triangle.
V. The Midpoint Formula LSCC, S. Nunamaker If the endpoints of a segment are (x x1 x y1 y are: (, ), y ) and (x, y ), then the coordinates of the midpoint 1 1 Example: 1. The diameter of a circle connects two points (, -3) and (6, 4 ) on the circle. Find the coordinates of the center of the circle.. Find the midpoint between: (, 4 ) and ( 6, 8 ). 3. Find the midpoint between: (, ) 5 1 and ( 6, 16) VI. The Equation of a Circle The equation of a circle with center ( a, b ) and radius r is: ( x a) ( y b) r Example:1. Find the equation of a circle having radius 6 and with center at (, -4 ). Find the equation of a circle having diameter 10 and center (-1, 4). 3. Find the equation of a circle having radius 8 and center (4, 6 ). 4. Find the equation of a circle having radius 4 and center (-4, -6) 5. Find the center and the radius of each circle. 3
Example: Graph each circle by hand: LSCC, S. Nunamaker a. x y 4 b. x ( y 4) 16 c. ( x 4) ( y 5) 9 d. ( x 1) ( y ) 64 e. ( x 1) ( y ) 64 f. x y 10x 6 y 30 4
LSCC, S. Nunamaker 1. Functions and Graphs I. Definition of a function A function is a correspondence between a first set, called the domain, and a second set, called the range, such that each member of the domain corresponds to exactly one member of the range. This correspondence or relationship can be expressed in ordered pairs or in equations or in graphs. A function is a set of ordered pairs in which no first coordinate is repeated. A. In ordered pairs: {(, 10), (3, 15), (4, 0)} This relation is a function. Domain: {, 3, 4 } Range: {10, 15, 0 } {(5, 0), (3, -1), (5, -1)} This relation is not a function. How about {(3, 1), (5, 1), (7, 1)}, {(5, ),(5, 3), (7, 1)}? B. In equations: y x (variable y is a function of variable x) x is the independent variable and y is the dependent variable. Inputs are values of x substituted into the equation. Outputs are resulting values of y. So, if we use the function of y x, if we call the function f, we can use x to represent an arbitrary input, and notation of a function, f(x), read "f of x", or "the value of f at x to represent the corresponding output. So, for f() = = 4, f(3) = 3 = 9, f(4)= 4 = 16 Example: 1. Given f ( x) x x 3 Find: f ( 0 ) = f ( 5a) f ( 7) 5
f ( a 4 ) LSCC, S. Nunamaker. Given g( x) x 3 Find: g( ) = g( ) g( 3 y) g( h) g( a h) g( a) = h C. In graphs, use the vertical-line test to determine whether the graph is of a function. If it is possible for a vertical line to cross a graph more than once, then the graph is not a graph of a function. Example: Is this a function? A. y x B. Is this a function? y 3x 5 C. Is this a function? y x D. Is this a function? y 1 x 4 E. Is this a function? y x *At different intervals, a function may be increasing, decreasing, or is constant. * f( x) x, the Greatest Integer Function, the greatest integer less than or equal to x. Example: 35. 3, 36. 4, 3. 4, 78. 7, 1. 1 II. Domain of a Rational Expression Recall that a rational expression is the quotient of two polynomials. For example, 3,, 5 x 3 y y 3 4x 5 are rational expressions Any number that makes the denominator zero is not in the domain of a rational expression. 6
LSCC, S. Nunamaker Examples: A. Find the domain of the following functions ( a-h): 5 a. f ( x) x 5 x 4 3 b. f ( x) c. f ( x) x 4x 5 t 4 d. f ( x) 4 x *For a, use set-builder notation : {x x is a real number and x 5} e. f ( x) 8 x f. f ( x) 3 5x x 5x 6 g. f ( x) x 1 h. f ( x) x 1 5x 3x 9 B. Sketch/graph and determine the domain and range of each function: a. f ( x) 7x 4 b. f ( x) 3x c. f ( x) 4 x d. f x x ( ) 5 x e. f ( x) x 1 4x 5 1.3 Linear Functions, Slope, and Applications 1.4 Equations of Lines and Modeling I. Linear equation form (straight line based on variables, x and y) A. Ax +By = C B. Slope-Intercept Equation: y = mx + b (such that m = slope or steepness of a line or m y y x x 1 1 y x y 1 x 1 and (0, b) is the y-intercept.) II. A horizontal line (constant function) is represented by: y = constant, m=0, y b or f ( x) b. III. A vertical line is represented by x= constant, m=undefined IV. A straight line bisecting first and third quadrant, with a positive slope, is an identity function, s.t. m = 1, b = 0, y = x represents an identity function. V. If given points: P1 ( x1, y1 ) and P ( x, y ), in order to obtain slope (m), use the equation 7
m= rise / run (such that rise = change of y = y y1, run = change of x = x x LSCC, S. Nunamaker 1 ).Then to obtain the equation defined by these two points, simply use y = mx + b. Since m is obtained, use either P 1 ( x y ) or P ( x y ) to plug into the equation y = mx + b and solve for b. 1, 1, VI. If given 1 point and the value of slope (m) of the line, use ( y y1 ) / ( x x ) m, the point-slope equation, to obtain y y1 m( x x1 ). VII. A line with positive slope slants upward to the right. VIII. A line with negative slope slants downward to the right. IX. Two quantities are directly proportional if and only if their graph is a straight line through the origin. X. Two lines are parallel if they have the same slope (m) but different y-intercept (b) value.. XI. Two lines are perpendicular if their slopes are negative reciprocals of one another. ( m 1 1 or m m m ) 1 1 XII. x-intercept and y-intercept: where the line crosses the x-axis or y-axis. Example: 1. 3x 5 y 15 x-intercept: ( 5, 0 ), y-intercept: ( 0, 3 ). 6x 3 y 18 3. y 5x 0 Examples: Which of the following is(are) linear? 1. 5a b 1. 3x y 3. xy 0 4. y 10 5. 4x 6 y 0 6. b 6a 7. b 5 a 8. x 5 6 y 9. Find the slope of the straight line passing through given pair of points: a.(4, 6) and (3, 0) b.(6, 10) and (-4, 8) c.(, -1) and (, 4) 8
LSCC, S. Nunamaker 10. Find the slopes (m) and the y-intercept or vertical intercepts for the straight lines defined by each equation: a. x 3 y 6 b. 10x 5 y 50 c. p q 10 (p on the horizontal axis) d. y 7 11. Find the equation of the straight line that passes through the given point and has the given slope: a. ( 1, 0 ), m 5 b. ( -1, ), m c. ( 0, 1 ), m 3 d. ( 8, 8 ), m 40 1. Find the equation of the straight line that passes through the given pairs of points: a. ( 8, 8 ) and ( 8, 0 ) b. ( -1, ) and ( 3, ) c. (700, 1 ) and ( 700, 10 ) d. ( 4, -7 ) and (-8, ) 13. Find the equation of the line that passes through ( 10, 0 ) and is parallel to the line defined by the equation x 4 y 900 14. Find the equation of the line that passes through ( -1, 50 ) and is perpendicular to the line defined by the equation y 4x 1 15. Determine which of the following equations have graphs that are parallel to the graph of y 3x 7, which have graphs that are perpendicular to it, and which have graphs that are neither perpendicular nor parallel. a. y 3x 8 d. 6x y 8 b. 3 y x 8 e. x 6 y 1 c. y 6x 8 f. x 6 y 1 9
1.6 The Algebra of Functions LSCC, S. Nunamaker I. Compute the functions of the sum, the difference, the product, and the quotient of two functions, and determine the domains. A. Function values of the sum: ( f + g) (x) = f(x)+g(x) Example: Given: f(x) = x + g(x) = x 1 Find (f + g ) (5): Determine if 5 is in the domain of each function. Solution: (f + g) (5) = f (5) + g (5)=(5 +) +(5 1) =7 +6 = 33 Given: f ( x ) = 3x+1 g ( x ) = 4x-3 Find: ( f+g ) (3): Given: f (x ) = x 3 5 g (x ) = x Find: ( f + g ) (- ): B. Function values of the difference: ( f - g ) (x ) = f ( x ) - g (x ) Example: Given: f (x ) = x + g ( x ) = x 1 Find ( f - g ) ( 5 ): Determine if 5 is in the domain of each function. Solution: ( f - g ) ( 5 ) = f (5 ) - g (5 ) = (5+)-(5 1) = 7-6 = -19 10
Given: f (x ) = 3x + 1 g (x ) = 4x - 3 LSCC, S. Nunamaker Find (f - g ) (3 ): Given: f ( x ) = x 3 5 g ( x ) = x Find ( f - g ) (- ): C. Function values of the products ( f g ) ( x ) = f ( x ) g (x ) = f (x ) g ( x ) Example: Given: f (x ) = x + g (x ) = x 1 Find ( fg ) ( 5 ): Determine if 5 is in the domain of each function. Solution: ( fg ) ( 5 ) = f ( 5 ) g ( 5 )= (5+) (5 1) =76= 18 Given: f ( x ) = 3x + 1 g (x ) = 4x - 3 Find: ( fg ) ( 3 ): Given: f (x ) = x 3 5 g (x ) = x Find (fg ) ( - ): 11
D. Function values of the quotients LSCC, S. Nunamaker ( f / g ) ( x ) = f ( x ) / g ( x ) = f ( x ) g( x) Examples: Given: f (x ) = x + g (x ) = x 1 Find ( f / g ) ( 5 ): Determine if 5 is in the domain of each function. Solution: ( f / g ) ( 5 ) = f ( 5 ) / g ( 5 )= ( 5 + ) / (5 1) = 7 / 6 Given: f ( x ) = 3x + 1 g ( x ) = 4x - 3 Find ( f / g ) ( 3 ): Given: f ( x ) = x 3 5 g ( x ) = x Find ( f / g ) ( - ): II. Composition of Functions The composite function f is defined as ( f Example: g, the composition of f and g, g ) ( x ) = f ( g( x)), such that x is in the domain of f. Given: f ( x) x 5 g( x) x 3x 8 Find ( f g)( 5 ) and ( g f )( 5 ): Solution: ( f g)( 5) f ( g( 5)) f ( 5 35 8) = f ( 18 ) ---> f ( 18 ) = 18 5 = 31 ( g f )( 5) g( f ( 5)) g( 5 5) g( 5) ---> g( 5) 5 35 8 18 1
Given: f ( x) 3x 1 g( x) 4x 3 LSCC, S. Nunamaker Find ( f g)( 3 ) and ( g f )( 3 ): 3 Given: f ( x) x 5 g( x) x Find ( f g)( ) and ( g f )( ): Given: f ( x) x 3 g( x) x 1 Find ( f g)( 3 ), ( f g)( 3 ), ( f g)( 1 ), ( fg)( 1 ), ( f / g)( ), ( g f )( 3) III. Decomposing a Function as a Composition If h( x) ( x 4) 3, find f ( x), g( x) such that h( x) ( f g)( x) Solution: f ( x) x 3 and g( x) x 4 Example: 1. If f ( x) ( x 7) 5, g( x) x 10, find h( x) such that h( x) ( f g)( x). If h( x) x 5, find f ( x ), g ( x ), such that h ( x ) ( f g )( x ) 3. If h( x) x 4, find f ( x), g( x), such that h( x) ( f g)( x) 13
IV. In graphs, use the vertical-line test to determine whether the graph is of a function. If it is possible for a vertical line to cross a graph more than once, then the graph does not represent a function. A. Is this a function? y x B. Is this a function? y 3x 5 C. Is this a function? y x D. Is this a function? y 1 x E. Is this a function? y x 4 *At different intervals, a function may be increasing, decreasing, or is constant. V. Intercepts of a Graph/Equation A. x-intercept is the point ( a, 0 ), the point at which the graph intersects x-axis is. Let the y be zero and solve the equation for x. Ex. x-intercept of y 3x 3 is ( -1, 0 ) B. y-intercept is the point ( 0, b ), the point at which the graph intersects y-axis. Let the x be zero and solve the equation for y. Ex. y-intercept of y 3x 3 is ( 0, 3 ) C. Try to find x- and y-intercepts of: 1. x y y 6x 4 1. y 5x 1 3. VI. Symmetry of a Graph A. A graph is symmetric with respect to y-axis: if whenever ( x, y )is a point on a graph, ( -x, y) is also on the graph. ( If replacing x by -x, graph yields an equivalent equation.) 4 (Ex. y x x, y x ) 14
B. A graph is symmetric with respect to x-axis: if whenever (x, y ) is a point on a graph, ( x, -y ) is also on the graph.( If replacing y by -y, graph yields an equivalent equation.) ( Ex. x y, x y 1 ) C. A graph is symmetric with respect to origin: if whenever (x, y) is a point on a graph, ( -x, -y ) is also on the graph. ( If replacing x by -x, and y by -y, graph yields an equivalent equation.) (Ex. y x 3 x) VII. Points of Intersection A. A solution to a system of linear equations is a set of numbers, one for each variable, that satisfies all the equations in the system. VIII. Transformations of Functions Some family of graphs have the same basic shape. (Ex.: y x, y x 6, y ( x 4), y x, y 1 ( x 4) ) Basic Types of Transformations ( c>0 ) Original graph y f ( x) Horizontal shift c units to the right y f ( x c) Horizontal shift c units to the left y f ( x c) Vertical shift c units downward y f ( x) c Vertical shift c units upward y f ( x) c Reflection (about x-axis) y f ( x) Reflection (about y-axis) y f ( x) Reflection (about origin) y f( x) Stretched vertically by a factor of a y a f ( x) if a 1, shrunken if 0 a 1 15
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