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Ch2: Graphs y axis x axis Quadrant I (+, +) Quadrant II (-, +) Quadrant III (-, -) Quadrant IV (+, -) Origin (0, 0) 2 4 6 -6 -2 (-6,-3) (5,-2) n distinct points are plotted as above graph is called a scatter plot – ‘points t are scattered about y - $$ in thousand x Yrs Graphs represent trends in data For example: x – number of years in business y – thousands of dollars of pro Equation : y = ½ x – 3 (0,-3) (6,0) y intercept x intercept point in the x/y coordinate plane is escribed by an ordered pair of oordinates (x, y)

Ch2: Graphs

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y axis. y - $$ in thousands. x axis. x Yrs. Ch2: Graphs. Quadrant II (-, +). Quadrant I (+, +). Origin (0, 0). (6,0). -6 -2. 4 6. 2. y intercept. x intercept. (5,-2). (-6,-3). (0,-3). Quadrant III (-, -). Quadrant IV (+, -). Graphs represent trends in data. - PowerPoint PPT Presentation

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Page 1: Ch2: Graphs

Ch2: Graphsy axis

x axis

Quadrant I (+, +)Quadrant II (-, +)

Quadrant III (-, -) Quadrant IV (+, -)

Origin (0, 0)

2 4 6-6 -2

(-6,-3) (5,-2)

When distinct points are plotted as abovethe graph is called a scatter plot – ‘points that are scattered about’

y - $$ in thousands

xYrs

Graphs represent trends in data.For example:x – number of years in businessy – thousands of dollars of profitEquation : y = ½ x – 3

(0,-3)

(6,0)

y intercept x

intercept

A point in the x/y coordinate plane is described by an ordered pair of coordinates (x, y)

Page 2: Ch2: Graphs

2.1 Distance & Midpointy

xOrigin (0, 0)

2 4 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 2M

A point in the x/y coordinate plane is described by an ordered pair of coordinates (x, y)

(5,-2)

Things to know:1. Find distance or midpoint given 2 points2. Given midpoint and 1 point, find the other point3. Given distance and 1 point, find the other point

Page 3: Ch2: Graphs

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

y

x

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.

Page 4: Ch2: Graphs

X and Y interceptsy

x

(0,-3)

(6,0)

y intercept x

intercept

Equation: y = ½ x – 3

The y intercept happens where y is something & x = 0: (0, ____)

Let x = 0 and solve for y: y = ½ (0) – 3 = -3

The x intercept happens where x is something & y = 0: (____, 0)

Let y = 0 and solve for x: 0 = ½ x – 3 => 3 = ½ x => x = 6

-3

6

Page 5: Ch2: Graphs

Things to know:1. Find slope from graph2. Find a point using slope3. Find slope using 2 points4. Understand slope between 2 points is always the same on the same line

Slope

Slope = 5 – 2 = 3 1 - 0

Slope = 1 – (-1) = 2 3 – 0 3

y = mx + b m = slope b = y intercept

x 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)

y

x

=

Page 6: Ch2: Graphs

The Possibilities for a Line’s Slope (m)Positive Slope

x

y

m > 0

Line rises from left to right.

Zero Slope

x

y

m = 0

Line is horizontal.

m is undefined

Undefined Slope

x

y

Line is vertical.

Negative Slope

x

y

m < 0

Line falls from left to right.

Example: y = 2

Example: x = 3

Example: y = ½ x + 2

Example: y = -½ x + 1

Things to know:1. Identify the type of slope given a graph.2. Given a slope, understand what the graph would look like and draw it.3. Find the equation of a horizontal or vertical line given a graph.4. Graph a horizontal or vertical line given an equation5. Estimate 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?

Page 7: Ch2: Graphs

Linear Equation Forms (2 Vars)Standard Form Ax + By = C A, B, C are real numbers.

A & B are not both 0.

Example: 6x + 3y = 12

Slope Intercept Form y = mx + b m is the slopeb is the y intercept

Example: y = - ½ x - 2

Point Slope Form y – 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:1. Find Slope & y-intercept2. Graph using slope & y-intercept

Things to know:1. Change from point slope to/from other forms.2. Find the x or y-intercept of any linear equation

Things to know:1. Graph using x/y chart2. Know this makes a line graph.

Page 8: Ch2: Graphs

Parallel and PerpendicularLines & Slopes

PARALLEL• Vertical lines are parallel• Non-vertical lines are parallel if and only if they have the same slope

y = ¾ x + 2

y = ¾ x -8Same Slope

PERPENDICULAR

•Any 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 = ¾ x + 2

y = - 4/3 x - 5

Negative reciprocal slopes

3 • -4 = -12 = -14 3 12

Productis -1

Things to know:1. Identify parallel/non-parallel lines.

Things to know:1. Identify (non) perpendicular lines.2. Find the equation of a line parallel or perpendicular to another line through a point or through a y-intercept.

Page 9: Ch2: Graphs

Practice Problems

1. Find the slope of a line passing through (-1, 2) and (3, 8)

2. Graph the line passing through (1, 2) with slope of - ½

3. Is the point (2, -1) on the line specified by: y = -2(x-1) + 3 ?

4. Parallel, Perpendicular or Neither? 3y = 9x + 3 and 6y + 2x = 6

5. Find the equation of a line parallel to y = 4x + 2 through the point (-1,5)

6. Find the equation of a line perpendicular to y = - ¾ x –8 through point (2, 7)

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

Page 10: Ch2: Graphs

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.

y = x3

Origin Symmetry

y = x2 Y-axisSymmetry

x = y2

X-axis Symmetry

Symmetry Test-y = (-x)3

-y = -x3

y = x3

Symmetry Testy = (-x)2

y = x2

Symmetry Testx = (-y)2

X = y2

Page 11: Ch2: Graphs

A Rational Function Graph & Symmetryy = 1 x

x y-2 -1/2-1 -1-1/2 -20 Undefined½ 21 12 ½

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

Page 12: Ch2: Graphs

Application: Solar Energy

• The 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

Page 13: Ch2: Graphs

Application: Cigarette Use

A 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.

(a)Write a linear equation that relates the percent of teens y to the number of years after 1998, x.

(a)Find the intercepts of the graph of your equation.

(b)Do the intercepts have any meaningful interpretation?

(c)Use your equation to predict the percent for the year 2019. Is this result reasonable?

Page 14: Ch2: Graphs

2.5 Proportions & VariationProportion equality of 2 ratios. Proportions are used to solve problems in everyday life.

1. 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

3. If a person walks a mile in 16 min., that person can walk a half mile in x min.

16 x (x)(1) = ½(16) 1 ½ x = 8 minutes

=

=

=

Page 15: Ch2: Graphs

Direct Variationy = kx y 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

Page 16: Ch2: Graphs

Inverse Variation

y = k y is inversely proportional to x x y varies inversely as x

Example: y varies inversely with x.If y = 5 when x = 4, find the constant of proportionality (k)

5 = k So, k = 20 4

Example: 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

Page 17: Ch2: Graphs

Another Example (Inverse Variation)y = k y is inversely proportional to x x y varies inversely as x

P. 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 support is 500 lbs. Write a general formula relating the Maximum weight W that can be safely supported by a length of 25 feet.

W = k/L So, 500 = k/10 => k = 5000 Thus, W = 5000/L

So, Maximum weight safely supported for Length of 25ft is: W = 5000/25 = 200 lbs.

Page 18: Ch2: Graphs

Direct Variation with Power

y = kxn

y is directly proportional to the nth power of x

Example: Distance varies directly as the square of the time (t)

Distance = kt2

D = kt2

Joint Variation

y = kxp • y varies jointly as x and p

Page 19: Ch2: Graphs

Example: (Joint Variation)

P. 198 Example4:The force F of the wind on a flat surface positioned at a right angleTo the direction of the wind varies jointly with the area A of theSurface and the square of the speed v of the wind. A wind of 30 miles per hour blowing on a window measuring 4 ftBy 5 ft has a force of 150 lbs. What is the force on a window measuring 3 ft by 4 ft caused by a wind of 50 mph?

y = kxp • y varies jointly as x and p

F = kAv2 A = 4 x 5 = 20 v = 30

150 = k (20)(900)

150 = 18000kK = 150/18000 = 1/120

F = (1/120)Av2 , so, when wind = 50mph on 3 x 4 (A = 12), F = (1/120)(12)(2500) = 250lbs

Page 20: Ch2: Graphs

Example: Combined VariationP. 199 S#13The square of T varies directly with the cube of A and Inversely with the square of D

T = 2 when A = 2 and D = 4.

Write a general formula to describe each variation

T2 = kA3 4 = k(8) 8k = 64 k = 8 D2 16

Thus, T2 = 8A3

D2