Section 6 – 1 Rate of Change and Slope Rate of change = change in the dependent variable change in the independent variable Ex1. Susie was 48’’ tall at age 10 and 54’’ at age 13. Find the rate of change. Dep. variable = height and indep. = age Slope = vertical change = rise = y 2 – y 1 horizontal change run x 2 – x 1

Ex2. Find the slope between (5,7) & (-3,4) Horizontal lines have the equation y = h and they have a slope = 0 Vertical lines have the equation x = k and they have an undefined slope If you are given a graph, find 2 points and then determine the slope from them (either by using the formula or counting rise over run) Leave slope in fraction form (not mixed number or decimal)

Section 6 – 2 Slope-Intercept Form Linear equations → graphs are lines y = mx + b is slope-intercept form for a linear equation m = slope and b = y-intercept (where the line crosses the y-axis) A negative slope makes one number in the fraction negative NOT both To graph: 1) place the y-intercept on the y- axis 2) place another point by using rise over run for slope 3) connect

Ex1. Graph y = -⅔x + 5 Ex2. Write an equation for a line that has a slope of 4 and a y-intercept of -6 To find an equation of a line given two points: 1) find slope 2) pick one point to use the x & y-coordinates with m to determine b 3) write m and b into slope-intercept form

Ex3. Write an equation of a line that passes through (2,7) & (-3,-5) To test if a point is on a given line: 1) plug given coordinates into equation for x and y 2a) If both sides of the equation are equal, then the point is on the line 2b) If the two sides of the equation do not come out equal, then the point is NOT on the line

Section 6 – 3 Standard Form Ax + By = C is a linear equation in standard form A, B, and C are integers (not decimals or fractions) In most books (not this one), the etiquette is that A is not negative To graph in standard form: 1) plug 0 in for x to find the y-intercept 2) plug 0 in for y to find the x-intercept 3) plot these two points and connect

Graph in standard form Ex1. 3x + 2y = 12Ex2. 4x – 2y = 8 Ex3. Change y = -⅔x + 4 to standard form Ex4. Change y = ¾x – 5 to standard form

Section 6 – 4 Point-Slope Form & Writing Linear Equations Point-slope form y – y 1 = m(x – x 1 ) m, x 1, and y 1 will be numbers Ex1. Give the equation of a line that passes through (-2, 6) and has a slope of 5 (in point- slope form) To graph a line in point-slope form: 1) determine a point (x,y) from the equation 2) plot that point on the graph 3) determine the slope and find another point (rise over run) 4) connect

Ex2. Graph y – 5 = ⅔(x + 4) Ex3. Write an equation in point-slope form of a line that passes through (-2, 8) and (3, 4) Ex4. Change y – 3 = 4 / 3 (x + 5) to slope- intercept form

If data points are given (even in table form): to determine if points are linear, there must be a constant slope (see examples 4 and 5 on page 306) Ex5. Change y + 6 = ¾(x – 5) to standard form

Section 6 – 5 Parallel & Perpendicular Lines Lines are parallel if they have the same slope but different y-intercepts Ex1. Are the graphs of y = ½x – 5 and 2x + 4y = 8 parallel? Ex2. Write an equation for the line that passes through (2, 6) and is parallel to y = 3 / 2 x – 8 Symbol for parallel = ║ or

The slopes of perpendicular lines are negative reciprocals (when you multiply them, the product is -1) Vertical and horizontal lines are perpendicular (they are the only exception to the negative reciprocals rule) y = ¾x + 2 and y = - 4 / 3 x + 8 are perpendicular (symbol is and up-side-down capital T) Ex3. Write the equation of a line that passes through (6,8) & is perpendicular to y = 3x – 9

Section 6 – 6 Scatter Plots & Equations of Lines A scatterplot is a collection of points on a graph Often times the points follow a trend (they generally go in the same direction) If the points follow a trend, you can draw a trend line that goes through the center of the points to the best of your ability The trend line that most accurately shows the trend is called the line of best fit

The correlation coefficient r, tells how closely the equation actually models the data Graphing calculators can find the line of best fit and the correlation coefficient 1) 2 nd – catalog (formerly 0) – scroll down to DiagnosticOn – Enter twice 2) Stat – Edit – in L 1 put the x values and in L 2 put the corresponding y values 2 nd – Quit (formerly Mode) to exit Stat – Calc – LinReg – Enter = linear equation

Ex1. Find the line of best fit and the correlation coefficient for If a value is given in years, don’t enter the year as it appears, but as the number of years after 1900 (1994 becomes 94) Without a table or a graphing calculator, draw your own line of best fit, pick 2 points, find the slope, and write the answer in point- slope form

Section 6 – 7 Graphing Absolute Value Equations All absolute value equations are v-shaped y = │x│ is a v-shaped graph with its vertex at (0, 0) and slopes of 1 and -1 If │x│ is added or subtracted by a number, the original graph is being translated (scooted) – same shape, just moved y = │x│ + k shifts k units UP y = │x│ – k shifts k units DOWN

Ex1. Graph a) y = │x│ – 3 b) y = │x│ + 5 If the number being added or subtracted is inside the absolute value symbols with the x, it translates left or right y = │x – h│ is h units to the RIGHT y = │x + h│ is h units to the LEFT

Ex2. Graph a) y = │x – 3│ b) y = │x + 4│ The graph of y = -│x│ is y = │x│ reflected over the x-axis (flipped up-side-down) Ex3. Describe the translation of y = │x – 3│ – 5 as compared to y = │x│