Welcome to MS 101 Intermediate Algebra

Chapter 1 Linear Equations and Linear Functions Qualitative Graphs Graphing Linear Equations Slope Finding Linear Equations Functions

1.1 Qualitative Graphs Definition: Graph without scaling (tick marks and their numbers) on the axis. Medication Costs p Linear Curve price t Time

Independent/Dependent Variables Price p is dependent on the year t. (p is dependent of t) p is a dependent variable Year t does not depend on the price p. (t is independent of p) t is an independent variable

Examples You are taking a patient’s pulse counting the beats b per minute m. The number of people n that can run m miles. Waiting inline in the cafeteria for lunch. Use n for number of people in front and w for the wait time. Answers: Independent - m, m, n Dependent - b, n, w

Independent/Dependent Variable (time) Independent most the time. Something happens over time temperature over time, growth, population, etc. Dependent Time it takes for something to happen wait time, time taken to cook, time taken to move a distance, etc

Dependent Variable Independent Variable

Intercepts Average Age of Nurses m A m-intercept A-intercept Average Age n-intercept parabola n t Years Since 1900 An intercept of a curve is the point where the curve intersects an axis (or axes).

Graph Characteristics Increasing Curve upward from left to right Decreasing Curve downward from left to right Quadrants

Examples Height of a ball thrown straight up Heart rate of a person on a treadmill as the pace is steadily increased The temperature in Celsius of boiled water placed in a freezer The value of a car after a fixed amount of years

1.2 Graphing Linear Equations Solution, Satisfy, and Solution Set An ordered pair (a,b) is a solution of an equation in x and y if the equation becomes a true statement when a substituted for x and b is substituted for y. We say (a,b) satisfies the equation. Solution set is the set of all solutions of the equation.

Example y = 4x – 2 Find y if x = 2 y = 4(2) – 2 = 8 – 2 = 6 Thus the ordered pair (2,6) satisfies the equation y = 4x – 2 and (2,6) is a solution of the equation y = 4x – 2

Continued Solution set of the equation y = 4x – 2 includes (2,6) x y -2 -10 -1 -6 0 -2 1 2 2 6 Ordered pair (a,b) the independent variable is First (left) followed by the dependent variable in the second position (right).

Graphing Equations y = 5x + 2 Chose values, ex. 0, 1 Organize in table = 0 + 2 = 2 Solution: (0,2) y = 5(1) + 2 = 5 + 2 = 7 Solution: (1,7) Chose values, ex. 0, 1 Organize in table x y -2 -8 -1 -3 0 2 7 12 17

(3, 17) (2, 12) (1, 7) (0, 2) (-1,-3) (-2,-8)

(3, 17) A graph of an equation in two variables is a visual representation of the solutions of an equation. (2, 12) (1, 7) (0, 2) (-1,-3) (-2,-8)

(1, 9) does not satisfy the equation (3, 17) (2, 12) (1, 9) (0, 2) (1, 9) does not satisfy the equation (-1,-3) (-2,-8)

Linear Equation Forms Slope-Intercept Form Standard Form General Form y = mx + b m and b are both constants (m is the slope, b is the y-intercept) Standard Form Ax + By = C A, B, C are integers (A and B are not both equal to zero) General Form Ax + By + C = 0 Point-Slope Form (y – y ) = m (x – x ) Points (x, y) & (x ,y ), slope m 1 1 1 1

General to Slope-Intercept Form 3y - 12x + 9 = 0 3y - 12x + 9 -9 = 0 -9 3y – 12x = -9 3y - 12x + 12x = -9 + 12x 3y = -9 + 12x 3 3 3 y = -3 + 4x y = 4x - 3 General to Standard Try to get y by itself Divide Simplify

Distributive Law First Distribute 2(2y-3) = 2(3x-4) + 2y Get y by itself Divide Simplify 2(2y-3) = 2(3x-4) + 2y 4y – 6 = 6x – 8 +2y 4y – 6 -2y = 6x – 8 + 2y -2y 2y – 6 +6 = 6x – 8 +6 2y = 6x – 2 2 2 2 y = 3x - 1

Graphing Equations with Fraction Slopes y =¾x - 4 Chose x values that are multiples of the denominator. Ex. 0, 4, 8, 12 Make a chart and graph x y 0 y = ¾ (0) – 4 = -4 4 y = ¾ (4) – 4 = -1 8 y = ¾ (8) – 4 = 2

Intercepts of Graphs To find the x-intercept, substitute 0 for y and solve for x. It will always be in the form (x,0). To find the y-intercept, substitute 0 for x and solve for y. It will always be in the form (0,y). y = - ½x +3 0 = - ½x + 3 0 -3 = - ½x -3 -3 *(-2) = - ½x *(-2) 6 = x ~x-intercept (6,0) y = - ½ (0) +3 y = 3 ~y-intercept (0,3) y-intercept (0,3) x-intercept (6,0)

Vertical/Horizontal Lines If a and b are constants, Vertical lines are in the form x=a Horizontal lines are in the form y=b y=6 x=8

How do we measure steepness? 1.3 How do we measure steepness?

Comparing Steepness Calculate the ratio of vertical to horizontal distance. Patient 1 has his foot elevated 2 feet above the bed over a vertical distance of 2.5 feet. Patient 2 has his foot elevated 1.5 feet above the bed over a vertical distance of 2 feet. Which leg is elevated steeper? Patient 1’s foot is steeper because the ratio is slightly greater. Vertical Distance Horizontal Distance 2 1.5 2.5 2 2 1.5 = .8 = .75 2.5 2

Slope Let (x , y ) and (x , y ) be two points on a line. The slope of the line is: vertical change rise y - y horizontal change run x - x 1 1 2 2 2 1 m = = = 2 1 Run is positive to the right Run is negative to the left Rise is positive going up Rise is negative going down

Increasing vs Decreasing positive rise positive run or negative rise negative run positive slope positive rise negative run or negative rise positive run negative slope

Finding Slope of a Graph m = rise run m = 2 1 m = 2 m = y - y x - x m = 3 – 1 2 – 1 m = 2 = 2 1 2 1 (2,3) 2 1 2 (1,1) 1

Investigating Slope of Horizontal/Vertical Lines 2 – 2 0 Slope of a horizontal line 4 – 2 2 is always 0. 3 – 1 2 Slope of a vertical line 1 – 1 0 is always undefined. m = = = 0 m = = = Undefined y = 2 (2,2) (4,2) x = 1 (1,1) (1,3)

Parallel/Perpendicular Lines Parallel lines have the same slope m = m Slopes of perpendicular lines are opposite reciprocals 1 1 m 2 l 2 Parallel lines never intersect 2 1 l 1 l 2 m = 2 = 2 l 1 1 Perpendicular lines form 90 degree (right) angles

Finding Slope of Linear Equations 1.4 Finding Slope of Linear Equations y = -3x - 2 Graph rise -3 run 1 If in slope-intercept form: m = slope, m = -3 b = y-intercept, b = -2 1 m = = = -3 -3 1 -3

Using Slope and y-intercept to graph an equation Slope = 2 = rise 2 run 1 y-intercept (0, -2) Plot y-intercept (0, b) Use m = to plot 2nd point. Sketch line passing through the 2 points. rise run (0, -2)

Vertical Change Property For a line y = mx + b, if the run in 1, then the rise is the slope m. m 1 1 m Slope Addition Property For a line y = mx + b, if the independent variable increases by 1, then the dependent variable changes by slope m. y = -6x + 3 , x increases by 1, y changes by -6

Identifying Parallel/Perpendicular Lines Are the lines 3y = 2x – 6 and 6y – 4x = 24 parallel, perpendicular or neither? 3y = 2x – 6 3 3 3 y = 2/3x - 3 6y – 4x + 4x = 18 +4x 6y = 4x + 24 6 6 6 y = 4/6x + 24/6 y = 2/3x + 4 Get y by itself Divide Simplify Get the equations in slope-intercept form Slope in both equations is 2/3, therefore the lines are parallel.

Indentifying Linear Equations from points Set 1 Set 2 Set 3 Set 4 X Y 10 24 6 7 33 -2 4 11 20 12 34 -3 16 8 17 35 -4 13 9 23 36 -5 14 27 37 -6

Answers Set 1 – slope -4 Set 2 – not a line Set 3 – slope 0

Linear Equation Forms Slope-Intercept Form Standard Form General Form 1.5 Slope-Intercept Form y = mx + b m and b are both constants (m is the slope, b is the y-intercept) Standard Form Ax + By = C A, B, C are integers (A and B are not both equal to zero) General Form Ax + By + C = 0 Point-Slope Form (y – y ) = m (x – x ) Points (x, y) & (x ,y ), slope m 1 1 1 1

Find an equation given slope and a point Slope 4, point (6,2) Start with putting the slope into y=mx + b y = 4x + b Plug the coordinates of the point in for x and y (2) = 4(6) + b Solve for b 2 -24 = 24 + b -24 -22 = b y = 4x - 22

Find an equation of the line using two points (3, 5) (4, -2) Find the slope y - y -2 – 5 -7 x - x 4 – 3 1 Place slope into y = mx + b y = -7x + b Then chose either point to plug in for x and y… (3, 5) (5) = -7(3) + b Solve for b 5 +21 = -21 + b +21 26 = b… y = -7x + 26 m = = = 2 1 2 1

Finding the approximate equation of a line (-4.56,-5.24) (7.72, -4.93) Find the slope y - y -4.93 – (-5.24) .31 x - x 7.72 – (-4.56) 12.28 Place slope into y = mx + b y = .03x + b Then chose either point to plug in for x and y… (7.72, -4.93) (-4.93) = .03(7.72) + b Solve for b -4.93 -.2316 = .2316 + b -.2316 -5.16 = b… y = .03x – 5.16 m = = = = .02524 = .03 2 1 2 1

Finding an equation of a line parallel to a given line y = 3x – 2 , point (1,6) m = 3, so slope is 3 Parallel lines have similar slopes, so: y = 3x + b Plug in the coordinates for x and y (6) = 3(1) + b 6 -3 = 3 + b -3 b = 3 y = 3x + 3 is parallel to y = 3x -2

Finding an equation of a line perpendicular to a given line y = ½ x – 6, point (2,-2) Slopes of perpendicular lines are opposite reciprocals: flip the numerator and denominator and change the sign or -2 y = -2x + b (-2) = -2(2) + b -2 +4 = -4 + b +4 2 = b, y = -2x + 2 is perpendicular to y = ½ x - 6 12 21

Point-slope Form If a non-vertical line has a slope m and contains the point (x ,y ), then the equation of the line is: (y - y ) = m (x - x ) Given a point and the slope: m = 4 point (6,-3) y – (-3) = 4 (x - 6 ) y + 3 -3 = 4x – 24 -3 y = 4x – 27 1 1 1 1

Finding an equation of a line using point-slope form Given two points (2,14) and (-8,-6) Find slope -6 – 14 -20 -8 – 2 -10 (y - y ) = 2(x - x ) Substitute in one of the points y – (14) = 2(x – (2)) y – 14 +14 = 2x -4 +14 y = 2x + 10 m = = = 2 1 1

1.6 Functions Relation – set of ordered pairs Domain – set of all values of the independent variable (x-values) Range – set of all values of the dependent variable (y-values) Function – a relation in which each input leads to exactly one output Domain = input, Range = output

Equations by input/output x 5 6 7 y 3 1 4 & 3 x 2 4 y 7 8 9 RELATIONS RELATIONS input (domain) output (range) input (domain) output (range)

Is it a function? y = ± x y² = x y = x² x = 5 , y = ± 5 or y = 5 and y = -5 input has 2 outputs, not a function y² = x x = 16, y = ± 4 or y = 4 and y = -4 y = x² x = 3, y = 3² = 9 input has 1 output, function

Is it a function? Set 1 Set 2 Set 3 Set 4 x y 4 1 5 6 10 9 -1 7 11 2 input y output 4 1 5 6 10 9 -1 7 11 2 12 8 13 -2 14

Answers Set 1 – not function, slope undefined, vertical line Set 2 – not a function Set 3 – function, slope 0, horizontal line Set 4 – function, slope -3, decreasing line

Is it a function? When x =2 y = 3 and y = -3, not a function

Vertical line test A relation is a function if and only if every vertical line intersects the graph of the relation at no more than one point.

Functions from equations Is y =-3x -2 a function? Graph it! Any vertical line would intersect only once….Yes.

Linear Function All non-vertical lines are functions. Linear function: a relation whose equation can be put in y=mx + b form (m and b are constants)

Rule of Four Four ways to describe functions x y -2 -8 -1 -3 0 2 7 12 y = 5x + 2 Four ways to describe functions Equation Graph Table Verbally (input-output) x y -2 -8 -1 -3 0 2 7 12 17 For each input-output pair, the output is two more than 5 times the input.

Finding Domain and Range Domain = width = input = independent variable = x-values Range = height = output = dependent variable = y-values Leftmost point (-3,1) Rightmost point (4, 1) Domain -3 ≤ x ≤ 4 Highest point (2, 3) Lowest point (-1,-2) Range -2 ≤ y ≤ 3

Domain/Range Continued D = all real numbers R = y ≥ -1 D = x ≥ 0 R = y ≥ -1 D = all real numbers R = all real numbers