"The only way to be totally free is through education" - Jose Marti

Study Journal Notes

Relations and Functions:

Definitions you should know:

• relation

• linear relation

• non-linear relation

• independent variable

• dependent variable

• discrete data

• continuous data

• domain

• range

• function

• non-function

• slope

Lesson 1:

• Be able to identify the relationship between the independent variable (x-axis) and the dependent variable (y-axis) in order to describe the "story-line" of the graph

• Graph titles are often given as "dependent variable" versus "independent variable"

  • for example, distance versus time, or speed versus time

• In general, time is an independent variable and we show change over time on the x-axis

• Make sure you know the general trends of the three graphs on page 5 of your notes

  • a straight line is a linear graph, it shows constant change​

  • a curved line is non-linear, the change is not constant

  • a horizontal line is a linear graph, it shows no change

Lesson 2:

• Know the definitions of relation, linear relation, non-linear relation, independent variable, dependent variable, discrete data, and continuous data

• The independent variable is always given on the x-axis

• the dependent variable is always given on the y-axis

• There are several indicators of a linear relationship:

  • in a table of values, the x-values change at a constant rate, and the y-values change at a constant rate​

  • on a graph, the data makes a straight line

  • in an equation, the variables are to degree 1 or 0, and the coefficients are real numbers

  • in a set of ordered pairs, there is a constant change in the x-values and in the y-values

Lesson 3:

• The domain tells us what x-values exist on the graph and the range tells us what y-values exist on the graph

• Domain and range can be expressed in different ways:

  • as words​

  • as a number line

    • open dot = up to but not including​

    • closed dot = up to and including

  • as interval notation (only for continuous data)

    • ) means up to but not including​

    • ] means up to and including

  • as set notation (use greater than and less than signs, say what number set (integers or real numbers) the values belong to)

  • as a list (only for discrete data)

• You must be able to read the domain and range from a graph and from a table of values

Lesson 4:

• in a function there is only one "y-value" for each "x-value"

  • on a graph we can test this using the vertical line test​ - if any vertical line passes through the line of a graph more than once the graph is a non-function

• function notation tells us what kind of variable we are putting into the function, or what kind of number we are substituting for the variable in a function

  • for example f(x) = 2x +3 means that the function 2x + 3 is written in terms of x, x is the input for the function, and f(x) is the output for the function.​

  • for example, if we have f(4) for the function 2x + 3 then we have 2(4) + 3 = 11, so f(4) = 11. 4 is the input for the function, and 11 is the output for the function.

Lesson 5:

• the slope of a line describes numerically how the line changes vertically as it changes horizontally

  • calculated as rise over run (vertical change over horizontal change)​

• Know how to calculate slope if you know two points of data using the slope formula: m = (y2-y1)/(x2-x1)

• There are four conditions for slope:

  • positive - goes up as we travel from left to right​

  • negative - goes down as we travel from left to right

  • zero - no change up or down as we travel from left to right

  • undefined - straight up and down (vertical) with no change in horizontal distance

• If you know the slope and you are given the graph, you can find other points on the graph using the staircase method

​

Linear Equations

Lesson 1:

• Know the slope formula if you are given two points on a line or can identify two points on a graph

  • m = (y2-y1)/(x2-x1)

• Know how to draw a line if you are given a point and the slope (use the staircase method)

• Know the slope-intercept formula: y = mx + b

  • m = slope​

  • b = the y-intercept

• Know how to use the slope-intercept formula to solve problems by substituting values for the variables.

Lesson 2:

• Recall: slope-intercept form is y = mx + b

• If you have slope-intercept form, then you have enough information to graph the line:

  • use the value of "b" to identify the y-intercept​

  • use the slope to identify another point on the graph by following the rules of ries and run from the y-intercept (use the staircase method)

  • Once you have at least two points, but preferably three points, then you can connect the line - make sure you include arrows on the ends of the line since you don't know where the line begins or ends without the domain and range limitations.

• Know how to rearrange an equation using algebraic rules in order to get it in to the form y = mx + b

• Know how to use y = mx + b to solve a problem (identify what the y-intercept represents and what the slope represents if you are given a context or story)

Lesson 3:

• General form for a linear equation is Ax+By+C = 0

  • A must be a positive whole number​

  • B and C must be integer whole numbers (no fractions allowed)

• This form is purely algebraic (you can make it by rearranging other forms, or you can rearrange it to make other forms)

Lesson 4:

• x-intercepts are where the graph of the line crosses the x-axis and y = 0

• the y-intercept is where the graph of the line crosses the y-axis and x=0

• any equation in the form y=k where k is a number, has a slope of zero and therefore is a horizontal line with a y-intercept of k

• any equation in the form x = k where k is a number, has an undefined slope and therefore is a vertical line with an x-intercept of k

Lesson 5:

• Slope point form is y-y1 = m(x-x1)

  • use this when you have the slope and a point​

  • use this when you have two points and can solve for the slope

  • use this when you have a graph of a line but you cannot easily determine the y-intercept or the slope, but you can identify points on the line

Lesson 6:

• when you have two points, calculate the slope first (see lesson 1), then substitute the slope and one of the points into the slope-point form

• you can rearrange the slope-point form to create any other form

Lesson 7:

• parallel lines are at an equal distance from each other at all points

• parallel lines have the same slope (you can test this with congruent triangles)

• perpendicular lines intersect at right angles

• perpendicular lines have slopes that are negative reciprocals of each other (flip the fraction, change the sign)

• When you have to find a mystery part of a slope given a slope that is parallel, cross multiply!

• When you have to find a mystery part of a slope given a slope that is perpendicular, flip the faction, change the sign, and cross multiply!

Lesson 8:

• a line that is perpendicular to the y-axis has a slope of zero

• a line that is perpendicular to the x-axis has an undefined slope

• a line that is parallel to the y-axis has an undefined slope

• a line that is parallel to the x-axis has a slope of zero

• Let what you have define the form of the equation that you will use:

  • if you have a point and the slope, use point-slope form​

  • if you have the slope and the y-intercept, use slope-intercept form

  • if you have two points, solve for the slope and then use slope-point form

• To solve for the x-intercept, make y = 0 and solve for x

• To solve for the y-intercept, make x = 0 and solve for y