Ch+2+0910

= Chapter 2 Wiki= Editor: Kirsten Lyons Definitions: Sarah Lehman Notes Sections 1-4: Erica Little Notes Sections 5-8: Carly McGlade Ex. Problems: Emily Gross Multimedia: Dylan Horne

 = CHAPTER 2 = = = Definitions: Solution of an inequality in two variables- an ordered pair (x,y) that, when x and y are substituted in the inequality gives a true statement.
 * Domain**- set of input values.
 * Range**- set of output values.
 * Function**- a relation with exactly one output for each input.
 * Relation**- a mapping, or pairing of input values with output values.
 * Solution of an Equation in Two Variables**- an ordered pair (x,y) that makes the equation a true statement when the values o x and y are substituted in the equation.
 * Independent Variable**- the input variable in an equation.
 * Dependent Variable**- the output variable in an equation, which depends on the value of the input variable.
 * Linear Function**- a function in the form of y = mx + b where m and b are constants. The graph of a linear function is a line.
 * Function Notation**- use of the symbol f(x) for the dependent variable of a function.
 * Slope**- the ratio of vertical change (rise) to horizontal change (run) for a nonvertical line.
 * Standard Form of a Linear Equation**- a linear equation written in the form Ax + By = C where A and B are not both zero.
 * X-Intercept**- the x-coordinate of the point where a line intersects the x-axis. Given the equation of a line, it is the value of x when y = 0.
 * Y-Intercept**- if the graph of an equation intersects the y-axis at the point (0,b), then the number b is the y-intercept of the graph. Given the equation of the graph, it is the value of y when x = 0.
 * Direct Variation**- two variables x and y show direct variation provided by y = kx where k is a nonzero constant.
 * Constant of Variation**- the nonzero constant (k) in a direct variation equation (y = kx).
 * Scatter Plot**- a graph of ordered pairs used to determine whether there is a relationship between paired data.
 * Correlation**- positive, negative, or relatively none.
 * Linear Inequality**- an inequality that can be written as Ax + By (>/</ equal to).
 * Half Planes**- the two regions of a coordinate plane that are separated by the boundary of an inequality. One region contains the points that are the solutions of the inequality and the other contains points that are not.
 * Piecewise Functions**- a function represented by a combination of equations, each corresponding to a part of the domain.
 * Step Function**- a piecewise function whose graph resembles a set of stair steps.
 * Vertex of an Absolute Value Graph**- the corner point of the graph of an absolute value function.

 __2.1__  __Function and__ __Their__  __Graphs:__

Relation: A set of ordered pairs on a graph This is not a function because for the input of 1, there are 2 different outputs.
 * || 0 || 1 || -2 || 3 || 1 ||
 * y || 1 || -1 || 0 || 0 || 2 ||

__Input Output__ -3 3 1 5 4 8 Domain: -3,1,4 Range: 3, 5,8

A relation is a function if and only if no vertical line intersects the graph of the relation at more than one point.

Graphing equations in 2 variables 1. Construct table of vaules 2. Graph enough points to reecognize a pattern. 3. Connect points with a line or a curve.

__2.2__ __Slope and Rate of Change:__

The slope of a nonvertical line passing through the points (x1,y1) and (x2,y2) is:

M= y 2 -y 1 ———​ x 2 -x 1 (Think, rise over run)

REMEMBER – When calculating slope, subtract the coordinates in the correct order.

Postive slope m>0 Negative slope m<0 Zero slope (horizontal line) m=0 Undefined slope (vertical line) m=undefined

Determining steepness -> Find slope of both lines. Whichever has higher absolute value is steeper.

Parallel lines have same slope. Perpendicular lines have opposite reciprocal slopes.

Slope is often used to describe rate of change.

__2.3__  __Quick Graphs of Linear Equations;__

Slope intercept form -> y=mx+b m = slope b = y-intercept 1. Write equation in slope-intercept by solving for y. 2. Find y-intercept and plot point. 3. Find slope, and use to plot other points. 4. Connect points

Standard form --> Ax+By=C A must not be a fraction or negetive. 1. Write equation in standard form 2. Find x-intercept by plugging 0 in for y. (, 0) 3. Find y-intercept by plugging 0 in for x. (0, ) 4. Plot points and connect them

__2.4__ __Writing Equations of Lines:__

Given slope m and y-intercept b use: slope intercept y=mx+b Given slope m and a point use: point-slope y-y1=m(x-x1) Given two points use: difference of y's over difference of x's.

Direct variation y=kx

​ Practice Problems: ​​ Lesson 2.1  Evaluate the function when x= -2. 1. f(x)= x+17 2.f(x)= -x+3 3.f(x)= -5+8x 4.f(x)= -x-48  5.f(x)=|x+3|-9

Lesson 2.2 <span style="color: #000000; font-family: 'Times New Roman',Times,serif; font-size: 12pt; line-height: 115%;"> Tell whether the lines are parallel, perpendicular, or neither. 1. Line 1: through (4,4) and (1,6) Line 2: through (-1,0) and (3,5)

2.Line 1: through (-1,-9) and (1,-3) Line 2: through (-6,7) and (-3,6)

<span style="color: #800080; font-family: 'Times New Roman',Times,serif; font-size: 12pt; line-height: 115%;">Lesson 2.3 <span style="color: #000000; font-family: 'Times New Roman',Times,serif; font-size: 12pt; line-height: 115%;"> Find the slope and y intercept of the line. 1. y=2 2. y=5 3. y=2x-5 4. Y=3x+7 5. -2x+y=10 6. 5x-y=12 <span style="color: #00ffff; font-family: 'Times New Roman',Times,serif; font-size: 12pt; line-height: 115%;">

<span style="color: #008080; font-family: 'Times New Roman',Times,serif; font-size: 12pt;">Lesson 2.4 <span style="color: #000000; font-family: 'Times New Roman',Times,serif; font-size: 12pt;">Write an equation of a line that has the given properties. 1. slope: 2, y intercept:-4 2. Slope :0, y intercept: 2 3. slope : 4/5, y intercept: 5 4. Slope: 2, passes through (1, -3)

<span style="color: #00ff00; font-family: 'Comic Sans MS',cursive; font-size: 200%;">ANOTHER SOURCE FOR HELP: =Holt Student text – user name: alogin5 Password: n5p6= <span style="color: #000000; font-family: 'Times New Roman',Times,serif; font-size: 12pt; line-height: 115%;">

Lesson 2.5 <span style="font-family: 'Calibri','sans-serif'; font-size: 12pt; line-height: 115%;">What type of correlation do these graphs have? <span style="color: #000000; font-family: 'Times New Roman',Times,serif;">1.

2. Lesson 2.6 Check whether the ordered pairs are solutions of the inequality 1. x__<__ -5, (0,2), (-5,1) 2. 2y__>__ 7, (1,-6), (0,4) 3. 19x + y__>__ -0.5, (2,3), (-1,0) Lesson 2.7 Evaluate the function with the given x values. F(x)={3x+2, if x__<__ 1 {x+4, if x>1 1. F(-2) 2.f(0) 3. f(5) Lesson 2.8 Identify the Vertex and tell whether the graph opens up or down, or if it is narrower or wider, or the same width as y=|x|. 1. y= |x+5| 2. y= |x|+5 3. y= -|x- 1/2| -14

<span style="color: #00eeff; font-family: 'Comic Sans MS',cursive; font-size: 200%;">__2.5__ <span style="color: #00ff00; font-family: 'Comic Sans MS',cursive; font-size: 200%;"> __Correlation and Best – Fitting Lines:__ <span style="font-family: 'Times New Roman',Times,serif;">Correlation – When deciding on a correlation (positive, negative, relatively none) <span style="color: #00ffff; font-family: 'Times New Roman',Times,serif;">: Best Fitting Lines – When trying to create a best fitting line: <span style="color: #800000; font-family: 'Comic Sans MS',cursive; font-size: 200%;">__2.6__ <span style="color: #00ff00; font-family: 'Comic Sans MS',cursive; font-size: 200%;">__Linear Inequalities in Two Variables:__ Graphing- 1. Graph the boundary line of the inequality (dashed for > or <, solid for >/=or </=.) 2. Then test to see if it’s a solution by choosing any point (try picking a point clearly on one side or the other). The easiest point to use being the origin (0,0). 3. If the point that you chose satisfies the equation then shade that side of the boundary, if it doesn’t, shade the other side. And when checking your solutions for any inequality, just graph. <span style="color: #0000ff; font-family: 'Comic Sans MS',cursive; font-size: 200%;">__2.7__ <span style="color: #00ff00; font-family: 'Comic Sans MS',cursive; font-size: 200%;"> __Piecewise Functions:__ Evaluating: Example: Evaluate f (x) when (**a**) x=3 (**b**) x=0 and (**c**) x=6  F(x)={x + 2, if x</=3 ; 2x + 1, if x>3} f (3) = x (3) + 2 = 5. [Solution after substituting 3 for x.] f (0) = x (0) + 2 =2. [Solution after substituting 0 for x.] f (6) = 2(6) + 1=13. [Solution after substituting 6 for x.]
 * 1) Look at the graph
 * 2) If the y tends to increase as the x increases: it’s positive
 * 3) If the y tends to decrease as the x increases: it’s negative
 * 4) If there is no linear pattern: there’s relatively no correlation[[image:http://t1.gstatic.com/images?q=tbn:dTIh-QlvoFP89M:http://www.s-cool.co.uk/a-/psy-/resmet-/a-psy-resmet-ql4_files/a-psy-resmet-dia06.gif width="120" height="93" link="http://images.google.com/imgres?imgurl=http://www.s-cool.co.uk/a-/psy-/resmet-/a-psy-resmet-ql4_files/a-psy-resmet-dia06.gif&imgrefurl=http://www.kgv.net/psychology/year%252012/Directed%2520Study.htm&usg=__69AljFARNZ-8KFmUFOeCyaXplw4=&h=271&w=349&sz=4&hl=en&start=71&um=1&tbnid=dTIh-QlvoFP89M:&tbnh=93&tbnw=120&prev=/images%3Fq%3Dpictures%2Bof%2Bcorrelation%2Band%2Bbest%2Bfitting%2Blines%26ndsp%3D20%26hl%3Den%26rls%3Dcom.microsoft:en-us%26rlz%3D1I7ADSA_en%26sa%3DN%26start%3D60%26um%3D1"]][[image:http://t3.gstatic.com/images?q=tbn:1Vl_5WC7u_1w-M:http://www.regentsprep.org/Regents/math/ALGEBRA/MultipleChoiceReview/prac1.gif width="112" height="113" link="http://images.google.com/imgres?imgurl=http://www.regentsprep.org/Regents/math/ALGEBRA/MultipleChoiceReview/prac1.gif&imgrefurl=http://www.regentsprep.org/Regents/math/ALGEBRA/MultipleChoiceReview/WorkingData.htm&usg=__MSwgfwtNuH7pMydeMlrxzsHQxG4=&h=271&w=269&sz=5&hl=en&start=68&um=1&tbnid=1Vl_5WC7u_1w-M:&tbnh=113&tbnw=112&prev=/images%3Fq%3Dbest%2Bfitting%2Bline%2Bwith%2Bpositive%2Bcorrelation%26ndsp%3D20%26hl%3Den%26rls%3Dcom.microsoft:en-us%26rlz%3D1I7ADSA_en%26sa%3DN%26start%3D60%26um%3D1"]]
 * 1) Carefully draw a scatter plot of the data.
 * 2) Sketch the line that seems to follow the pattern of points the best. A good rule of thumb is to have as many points below the line as above.
 * 3) Choose two points on the line and estimate the coordinates of these points. (The two points you choose do not have to be original data points.)
 * 4) Find the equation of the line that passes through the two points you chose. This equation will model the data in the problem. [[image:http://t1.gstatic.com/images?q=tbn:J6QNFeczLTDhsM:http://www.bbc.co.uk/schools/gcsebitesize/maths/images/graph_55.gif width="127" height="97" link="http://images.google.com/imgres?imgurl=http://www.bbc.co.uk/schools/gcsebitesize/maths/images/graph_55.gif&imgrefurl=http://www.bbc.co.uk/schools/gcsebitesize/maths/data/scatterdiagramsrev_print.shtml&usg=__snuOk6wzj7nfhbQv8iajzmOIDik=&h=343&w=451&sz=22&hl=en&start=8&um=1&tbnid=J6QNFeczLTDhsM:&tbnh=97&tbnw=127&prev=/images%3Fq%3Dpictures%2Bof%2Bcorrelation%2Band%2Bbest%2Bfitting%2Blines%26hl%3Den%26rls%3Dcom.microsoft:en-us%26rlz%3D1I7ADSA_en%26sa%3DN%26um%3D1"]]
 * 1) f (x) = x +2 [Because x (3) is less than or equal to 3.]
 * 1) f (x) = x + 2 [Because x (0) is less than or equal to 3]
 * 1) f (x) = 2x + 1 [Because x (6) is greater than 3.]

Graphing a Piecewise Function: If you were to use the same function as above you would: Graphing a Step Function: Example: f(x) = {4, if 0</= x < 1 ; 3, if 1</= x < 2 ; 2, if 2 </= x < 3 ; 1, if 3</= x < 4} This is called a step function because all the line segments together look like a set of stairs. To start graphing this you would have all x’s from 0 to 1 have a y of 4, and the segment would have a closed point at 0 and an open one at 1. Then all the x’s from 1 to 2 would have a y of 3 and a closed point at 1 and a closed point at 2.
 * 1) Graph y = x+2 to the left of 3, and on the solution to 3 place an open point.
 * 2) Graph y = 2x +1 to the right of 3, and on the solution to 3 place a closed point.
 * 3) Some graphs will have to rays with a common initial point, although this one will not. [[image:http://t1.gstatic.com/images?q=tbn:E52pbVK6-7gS8M:http://www.analyzemath.com/Graphing/square_root_graph_6.gif width="123" height="83" link="http://images.google.com/imgres?imgurl=http://www.analyzemath.com/Graphing/square_root_graph_6.gif&imgrefurl=http://www.analyzemath.com/Graphing/graphing_square_root_func.html&usg=__qta3wjP4isyH8xQksbEVk8vTBQA=&h=263&w=389&sz=3&hl=en&start=12&um=1&tbnid=E52pbVK6-7gS8M:&tbnh=83&tbnw=123&prev=/images%3Fq%3Dpiecewise%2Bfunction%26hl%3Den%26rls%3Dcom.microsoft:en-us%26rlz%3D1I7ADSA_en%26um%3D1"]][[image:http://t3.gstatic.com/images?q=tbn:rcETyJpiaN1NtM:http://cnx.org/content/m11475/latest/pieprob.png width="100" height="110" link="http://images.google.com/imgres?imgurl=http://cnx.org/content/m11475/latest/pieprob.png&imgrefurl=http://cnx.org/content/m11475/latest/&usg=__Tl3yo4oSYBAr3MvqEgSZ0dGkirI=&h=243&w=222&sz=3&hl=en&start=15&um=1&tbnid=rcETyJpiaN1NtM:&tbnh=110&tbnw=100&prev=/images%3Fq%3Dpiecewise%2Bfunction%26hl%3Den%26rls%3Dcom.microsoft:en-us%26rlz%3D1I7ADSA_en%26um%3D1"]][[image:http://t0.gstatic.com/images?q=tbn:r-PELQSoTncgEM:http://www.icoachmath.com/Sitemap/images/Step%2520Function1.gif width="108" height="107" link="http://images.google.com/imgres?imgurl=http://www.icoachmath.com/Sitemap/images/Step%2520Function1.gif&imgrefurl=http://www.icoachmath.com/SiteMap/StepFunction.html&usg=__q6MYBRyxDOoG5OltyiZL2rH-mvg=&h=225&w=227&sz=3&hl=en&start=1&um=1&tbnid=r-PELQSoTncgEM:&tbnh=107&tbnw=108&prev=/images%3Fq%3Dpiecewise%2Bstep%2Bfunction%26hl%3Den%26rls%3Dcom.microsoft:en-us%26rlz%3D1I7ADSA_en%26um%3D1"]]

<span style="color: #00ff00; font-family: 'Comic Sans MS',cursive; font-size: 200%;"> <span style="font-family: 'Comic Sans MS',cursive; font-size: 200%;">__2.8__ <span style="color: #00ff00; font-family: 'Comic Sans MS',cursive; font-size: 200%;"> __Absolute Value Functions:__ Graphing – Before you graph you must know that the equation for an absolute value function is y = a abs(x-h) + k, you also must know that: 1. The vertex of the graph is (h, k) and is symmetric in the line x = h. 2. The graph will be V-shaped, opening up when a > 0, down when a < 0. 3. The graph is wider than the graph of y = abs(x) if abs(a) < 1, and the graph will be narrower than the graph of y = abs(x) if (a) > 1. When you do graph an absolute value function you will probably find it helpful to plot the vertex and one other point. Then find a third point by using symmetry to complete it.