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Transcript of Ch2: Graphs
Ch2: GraphsQuadrant I (+, +)Quadrant II (-, +)Quadrant III (-, -)Quadrant IV (+, -)Origin (0, 0)24 6-6 -2(-6,-3)(5,-2)When distinct points are plotted as abovethe graph is called a scatter plot points that are scattered aboutGraphs represent trends in data.For example:x number of years in businessy thousands of dollars of profitEquation : y = x 3(0,-3)(6,0)y interceptxinterceptA point in the x/y coordinate plane is described by an ordered pair of coordinates (x, y)
2.1 Distance & MidpointOrigin (0, 0)24 6-6 -2(-6,-3)The Distance FormulaTo find the distance between 2 points (x1, y1) and (x2, y2)
d = (x2 x1)2 + (y2 y1)2
The Midpoint FormulaTo find the coordinates of the midpoint (M)of a segment given segment endpoints of (x1, y1) and (x2, y2)
x1 + x2, y1 + y2 2 2MA point in the x/y coordinate plane is described by an ordered pair of coordinates (x, y) (5,-2)Things to know:Find distance or midpoint given 2 pointsGiven midpoint and 1 point, find the other pointGiven distance and 1 point, find the other point
2.2 & 2.3 Linear EquationsThe graph of a linear equation is a line.A linear function is of the form y = ax + b, where a and b are constants.
y = 3x + 2
y = 3x + 5x
y = -2x 3
y = (2/3)x -1
y = 4
6x + 3y = 12
x y=3x+2 x y=2/3x 1
0 2 0 -1
1 5 3 1All of these equations are linear.Three of them are graphed above.
X and Y intercepts(0,-3)(6,0)y interceptxinterceptEquation: y = x 3
The y intercept happens where y is something & x = 0: (0, ____)
Let x = 0 and solve for y: y = (0) 3 = -3
The x intercept happens where x is something & y = 0: (____, 0)
Let y = 0 and solve for x: 0 = x 3 => 3 = x => x = 6-36
Things to know:Find slope from graphFind a point using slopeFind slope using 2 pointsUnderstand slope between 2 points is always the same on the same lineSlopeSlope = 5 2 = 3 1 - 0
Slope = 1 (-1) = 2 3 0 3
y = mx + b m = slope b = y interceptx y=3x+2 x y=2/3x 1
0 2 0 -1
1 5 3 1
Slope is the ratio of RISE (How High) y2 y1 y (Change in y)
RUN (How Far) x2 x1 x (Change in x)
The Possibilities for a Lines Slope (m)Example: y = 2Example: x = 3Example: y = x + 2Example: y = - x + 1Things to know:Identify the type of slope given a graph.Given a slope, understand what the graph would look like and draw it.Find the equation of a horizontal or vertical line given a graph.Graph a horizontal or vertical line given an equationEstimate the point of the y-intercept or x-intercept from a graph.Question: If 2 lines are parallel do you know anything about their slopes?
Linear Equation Forms (2 Vars)Standard FormAx + By = CA, B, C are real numbers.A & B are not both 0.
Example: 6x + 3y = 12
Slope Intercept Formy = mx + bm is the slopeb is the y intercept
Example: y = - x - 2
Point Slope Formy y1 = m(x x1)
Example: Write the linear equation through point P(-1, 4) with slope 3 y y1 = m(x x1) y 4 = 3(x - - 1) y 4 = 3(x + 1)Things to know:Find Slope & y-interceptGraph using slope & y-interceptThings to know:Change from point slope to/from other forms.Find the x or y-intercept of any linear equationThings to know:Graph using x/y chartKnow this makes a line graph.
Parallel and PerpendicularLines & SlopesPARALLEL Vertical lines are parallel Non-vertical lines are parallel if and only if they have the same slopey = x + 2
y = x -8Same SlopePERPENDICULARAny horizontal line and vertical line are perpendicular If the slopes of 2 lines have a product of 1 and/or are negative reciprocals of each other then the lines are perpendicular.y = x + 2
y = - 4/3 x - 5Negative reciprocal slopes3 -4 = -12 = -14 3 12Productis -1Things to know:Identify parallel/non-parallel lines.Things to know:Identify (non) perpendicular lines.Find the equation of a line parallel or perpendicular to another line through a point or through a y-intercept.
Practice Problems Find the slope of a line passing through (-1, 2) and (3, 8)
Graph the line passing through (1, 2) with slope of -
Is the point (2, -1) on the line specified by: y = -2(x-1) + 3 ?
Parallel, Perpendicular or Neither? 3y = 9x + 3 and 6y + 2x = 6
Find the equation of a line parallel to y = 4x + 2 through the point (-1,5)
Find the equation of a line perpendicular to y = - x 8 through point (2, 7)
Find a line parallel to x = 7; Find a line perpendicular to x = 7Find a line parallel to y = 2; Find a line perpendicular to y = 2
8. Graph (using an x/y chart plotting points) and find intercepts of any equation such as: y = 2x + 5 or y = x2 4
Symmetry in GraphingY-Axis Symmetry even functions f (-x) = f (x) For every point (x,y), the point (-x, y) is also on the graph.Test for symmetry: Replace x by x in equation. Check for equivalent equation.
Origin Symmetry odd functions f (-x) = -f (x)For every point (x, y), the point (-x, -y) is also on the graph.Test for symmetry: Replace x by x , y by y in equation. Check for equivalent equation.
X-Axis Symmetry(For every point (x, y), the point (x, -y) is also on the graph.)Test for symmetry: Replace y by y in equation. Check for equivalent equation.Symmetry Test-y = (-x)3-y = -x3y = x3Symmetry Testy = (-x)2y = x2Symmetry Testx = (-y)2X = y2
A Rational Function Graph & Symmetryy = 1 x
x y-2 -1/2-1 -1-1/2 -20 Undefined 2 1
Intercepts:No intercepts existIf y = 0, there is no solution for x.If x = 0, y is undefinedThe line x = 0 is called a vertical asymptote.The line y = 0 is called a horizontal asymptote.Symmetry:y = 1/-x => No y-axis symmetry-y = 1/-x => y = 1/x => origin symmetry-y = 1/x => y = -1/x => no x-axis symmetry
Application: Solar EnergyThe solar electric generating systems at Kramer Junction, California, use parabolic troughs to heat a heat-transfer fluid to a high temperature. This fluid is used to generate steam that drives a power conversion system to produce electricity. For troughs, 7.5 feet wide, an equation for the cross-section is: 16y2 = 120x 225
(a) Find the intercepts of the graph of the equation(b) Test for symmetry with respect to the x-axis, y-axis, and origin
Application: Cigarette UseA study by the Partnership for a Drug-Free America indicated that, in 1998, 42% of teens in grades 7 through 12 had recently used cigarettes. A similar study in 2005 indicated that 22% of such teens had recently used cigarettes.
Write a linear equation that relates the percent of teens y to the number of years after 1998, x.
Find the intercepts of the graph of your equation.
Do the intercepts have any meaningful interpretation?
Use your equation to predict the percent for the year 2019. Is this result reasonable?
2.5 Proportions & VariationProportion equality of 2 ratios. Proportions are used to solve problems in everyday life.
If someone earns $100 per day, then how many dollars can the person earn in 5 days? 100 x (x)(1) = (100)(5) 1 5 x = 500
2.If a car goes 210 miles on 10 gallons of gas, the car can go 420 miles on X gallons
210 420 (210)(x) = (420)(10) 10 x (210)(x) = 4200 x = 4200 / 210 = 20 gallons
If a person walks a mile in 16 min., that person can walk a half mile in x min.
16 x (x)(1) = (16) 1 x = 8 minutes===
Direct Variationy = kxy is directly proportional to x.y varies directly with xk is the constant of proportionality
Example: y = 9x (9 is the constant of proportionality)Let y = Your payLet x Number of Hours workedYour pay is directly proportional to the number of hours worked.Example1: Salary (L) varies directly as the number of hours worked (H). Write an equation that expresses this relationship. Salary = k(Hours) L = kH
Example 2: Aaron earns $200 after working 15 hours.Find the constant of proportionality using your equation in example1.. 200 = k(15) So, k = 200/15 = 13.33
Inverse Variationy = k y is inversely proportional to x x y varies inversely as xExample: y varies inversely with x.If y = 5 when x = 4, find the constant of proportionality (k)
5 = k So, k = 20 4Example: P. 199 #7 : F varies inversely with d2; F = 10 when d = 5F = k/ d2 10 = k/52 10 = k/25K = 250 => F = 250/ d2
Another Example (Inverse Variation)y = k y is inversely proportional to x x y varies inversely as xP. 197: Example 2: The maximum weight W that can be safely supported by a 2-inch by 4-inch piece of point varies inversely with its length L. Experimenters indicate that the maximum weight that a 10 foot long 2 x 4 Piece of pine can sup