Merced College; Don Power

 

INTERMEDIATE ALGEBRA - CH 3, LECTURE

 

3.1  Rectangular Coordinate System - Equations in two variables

 

Ordered pairs, Coordinates, ordered pairs

 

Plotting points

Finding coordinates of points from a graph

 

Presenting tables of data as bar graphs and as line graphs

 

Ex:  Data set;  Several different types of graphs can be done based on this.  These were done with Excel.

x

y

15

23

20

34

25

56

30

79

35

67

40

56

 

 

Show the same data on a scatter plot

 

We'll need to extend to negative values

Definitions
      ordered pair

      x-coordinate and y-coordinate

      rectangular coord system:  two real number lines at right angles

      origin

      x-axis, y-axis

      quadrants:  Example

 

      Ex: plotting (graphing) ordered pairs (in the different quadrants)

 

 

3 approaches to problem solving

      Algebraic - use a formula

      Numerical - build a table; guess and check

      Graphing - Draw a graph

 

Verifying solutions to equations in two variables

      Substitute the solution (x,y) into the equation

      Simplify each side independently

      If both sides simplify the same, the ordered pair is a solution

            (the nature of the equation should tell you whether other solutions can be expected)

 

Distance Formula - From the Pythagorean Theorem

 

      Example

 

      Applications

            Showing that points are vertices of a right triangle:  d12 + d22 = d32

            Collinear points:  d1 + d2 = d3

 

Midpoint Formula

 

      Example

 

 

3.2  Graphs of Equations

 

Point-plotting

      Isolate one variable, if possible

      Make a table of values

            Your choice, but for each type of equations there are usually special points that you should get

                  Any equations:

                        y-intercept(s):  set x = 0 and solve for y.  You get where the graph crosses the y-axis.

                        x-intercept(s):  set y = 0 and solve for x.  You get where the graph crosses the x-axis.

                  Absolute value:  Find the center:

                        Set the expression inside the brackets to 0, solve for x.

                        Then back-substitute and solve for y

      Plot your ordered pairs

      Connect the points

            Go from left to right  (valid whenever y = expression in x)

            Use as smooth a curve as you can, but

                  Linear equations graph as straight lines, and

                  Absolute value equations have V-shaped graphs

 

Application:  straight line depreciation

      Model:  Final value = Initial value – (depreciation per year) * t      (The * indicates multiplication)

                  Usually we add the inequality 0 ≤ t ≤ service life.  (Otherwise we could get negative values)

            The final value will be y (vertical axis)

            t is the time in years (horizontal axis)

            Depreciation per year is how much value is lost each year. 

                        This is the "slope" or "rate of change"

            Since it's a "linear equation," it graphs as a straight line.

 

3.3       Slope

 

Definition / formula

 

Formula for slope:  change in y / change in x

Different forms of formula:

      rise/run

      (y2 - y1)  /  (x2 - x1)

      delta y / delta x

      change in y when x changes by exactly 1  (rate of change in application problems)

      - (y-intercept / x-intercept)

 

Concept:

      Horiz:  slope = 0  (note that Δy is 0)

      Rising line:  Slope is positive  (When Δx is positive, Δy is also positive)

            The steeper the line, the higher the slope

            As the line approaches the vertical, the slope approaches ∞

      Vertical:  Slope is undefined (Δx is 0)

      Dropping line:  Slope is negative  (When Δx is positive, Δy is negative)

     

     

 

Finding slope of a line

      From 2 points

            #33:  Find y so that the line through the points has the given slope

                  (sub into formula, solve the proportion)

      From a table of values

            Related task:  Given one point and slope, find additional points

      From a graph

      From an equation

            In slope-intercept form:  y = mx + b

             In the form  or ax + by = c (or the form ax + by + c = 0)

                  First, solve for y (converting to slope-intercept form)

      From intercepts

 

Using slope and y-intercept to graph a line

Use x- and y-intercepts to graph a line

Parallel and perpendicular lines

      Parallel:  slopes are the same

      Perpendicular:  m1 = 1/m2 or m1*m2 = -1

Applications:

      Road grade or pitch of a roof

       Slope as a rate of change  (or a "unit change") 87c or 88c

      From previous text:  Find 2 points P on curve y=x2+2 such that slope of line thru (1,4) and P is 2.

 

3.4  Equations of Lines   

 

Slope-intercept form  y = mx + b

Derive from slope formula with points (x,y), (0,b), known m, b

Observation:  x and y are variables (letters), and m and b are constants (numbers)

      Ex:  Write an equation of the line with slope -2/5 and y-intercept 4/5

Ex:  Graph 2x + 3y = 9 using the y-intercept and slope

      [first, solve for y to get slope-intercept form; read m and b from there]

 

Point-slope form  y - y1 = m(x - x1)

Derivation:  from slope formula with points (x,y) and (x1,y1)

Ex:  Write an equation of the line through (4,-3) and (-1,2); convert to point-slope form

      First, find slope, then use point-slope formula with either point

      Resulting equation is not unique (you could use either point)

General form:  ax + by + c = 0

 

Converting one form to another

      To get slope-intercept form, solve for y

      To get general form, clear fractions and set everything equal to 0.

      Ex:  Convert previous results to general form

 

Vertical and Horizontal lines:  x=h or y=k.

      Ex:  Find eqn of line that passes through (–5,2) that is

            Horizontal

            Vertical

 

Parallel and perpendicular lines:  use previous concept of slope

 

Find slope, take negative reciprocal to get slope of perpendicular line.

 

Ex:  Write the equation (in standard form) of line through (-4,1)
      1.   That is parallel to the graph of  4x-y = 2.

      2.   That is perpendicular to the graph of  4x-y = 2.

     

Applications

      Write an equation of a line based on info that in a word picture

            First, organize the information in a table

                  If two points are given, use them to calculate the slope

                  If a rate of change is given, that rate is the slope.

            If time is involved, use t instead of x

            In general, make the dependent variable y and the independent variable x or t.

                  (the dependent variable should be the quantity that depends on the other)

            To simplify equations, indexing or scaling is often used

                  Examples: "let t=5 represent 1995,"  "the cost, in thousands of dollars, is given by ..."

      Use the equation for prediction

            Interpolation:  Calculating a value in the middle of the existing data points.

            Extrapolation:  Calculating a value that is outside the range of the existing data points.

 

3.5  Graphing Linear Inequalities in Two Variables

 

Ex:  2x - 3y > 6

      Solution set is every ordered pair (x,y) that satisfies the statement -- infinite solutions

      e.g. (4,1), (0,-3), (40,3), ...  It's impossible to list them all or even describe them conveniently.

The best way to represent the solution is with a shaded graph:

      Every point on the shaded side of the line 2x-3y=6 is included in the solution set.

Every point on the other (unshaded) side of the line 2x-3y=6 is not included in the solution set.

Points on the line itself may be included or excluded from the solution set,
      depending on the symbol used <, >, £, or ³.

Steps:

      Graph the equation.  Use solid line for £ or ³  [points on line are included in solution set]

            and broken line for < or >  [points on line are not included in solution set].

      Pick a test point that is not on the line (I suggest the origin or a point on the x-axis or y-axis)

            Substitute the test point into the inequality statement.

                  If the result is true, shade the side of the graph that includes the point.

                  If the result is false, shade the other side of the graph, the side that does not include the point.

Ex:  Graph the solution set for x>-3

Ex:  Graph the solution set for  2y £ 3x

 

3.6  Relations and Functions

 

Informal look:

Wage table related to a line graph (for $7.50/hr):  each time results in unique pay.

            Terms "input" and "output"

            Terms "domain" and "range"

Note:  these are relations between variables, and they can be pictures/described:

      as formulas

      as tables of data

      as graphs

      as verbal statements ("I earn $7.50 an hour").

Formal look

Def: rule that pairs/links each element in one set (domain) with exactly one element in second set (range)

Illustrate with examples:  y = abs(x) [yes] versus x=y2 [no]

      From equation:  given x, can you get more than one y?         What are domain and range?

      From table of values, given x, can you get more than one y?       What are domain and range?

      From graph, given x, can you get more than one y?        What are domain and range?

Functions as ordered pairs

Alternate definition:  a set of ordered pairs in which no two ordered pairs have the same 1st coord

Illustrate with orig example

Example that is not a function:

Auto prices from newsp for one make/model are related to year of manuf,
            but you can get multiple prices for the same age vehicle

Def of relation:  Rule that pairs/links each element of domain with one or more elements of the range

      i.e.  any collection of ordered pairs.

 

Function Notation           

Wage example

      Let  f = {(x,y)|y=7.5x, 0£x£40}

            Alternate designation for "y" is f(x):  identify y in terms of the x and the rule that calculates y

      For a second employee, let g = {(x,y)|y=6.5x, 0£x£40}

      Ordered pairs for f are

            (0,f(0)), i.e.(0,0)

            (1,f(1)), i.e. (1,7.50) etc

            in general (x,f(x))

      What are

            f

            f(x)       The value of the function when we have an input of x

                        or, the second member of the ordered pair whose first member is x

            f(2)

            g(4)

 

 

 

3.7       Graphing relations and functions

 

Point plotting -- the basic technique

Function notation and graphs:  plot f(x) on y-axis

 

Graphs of Basic Functions --

 

Constant     y = c

Identity y = x

Line through origin  y = mx

Absolute Value y = abs(x)     Table:  Use x = 0, ±1, ±2

Square root      y = sqrt(x)        Table:  Use x = 0, 1, 4, 9

Squaring     y = x2                     Table:  Use x = 0, ±1, ±2

Cubing       y = x3                     Table:  Use x = 0, ±1, ±2

 

Graphing piecewise-defined functions

 

Vertical line test

Examples:  y = abs(x) [yes] versus x=y2 [no]

 

Transformations (Shifts)

 

Remember y – y1 = m (x – x1)            Same as y = m (x – x1) + y1

      Three interpretations:

      Line with slope m that passes through (x1,y1)

      Takes the line y = mx and moves the origin to the point (x1,y1)

      Takes the line y = mx and shifts x1 units to the right, and y1 units up

 

y = f(x)±c

      + shifts up

      – shifts down

 

y=f(x±c)

      + shifts left

      – shifts right

 

y = -f(x)     Flips graph upside down

 

In general, if the modification is inside the function argument, it affects x, i.e. a horizontal effect

      and, if the modification is outside the function argument, it affects y, i.e. a vertical effect

 

y=cf(x) and -cf(x)     Not in this lesson

y=f(cx) and f(x/c)     Not in this lesson

y=f(–x)                       Not in this lesson

 

 

 

 

 

Return to:  Merced College; Don Power               Updated 09/17/07 by Don Power