# Introduction to graphing

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## The coordinates determine distance and direction

A positive number means a direction to the right or up . A negative number means a direction to the left or down .

## Plotting points

Since points and ordered pairs are so closely related, the two terms are sometimes used interchangeably. The following two phrases have the same meaning:

1. Plot the point $\left(a,\text{\hspace{0.17em}}b\right)$ .
2. Plot the ordered pair $\left(a,\text{\hspace{0.17em}}b\right)$ .

## Plotting a point

Both phrases mean: Locate, in the plane, the point associated with the ordered pair $\left(a,\text{\hspace{0.17em}}b\right)$ and draw a mark at that position.

## Sample set a

Plot the ordered pair $\left(2,\text{\hspace{0.17em}}6\right)$ .

We begin at the origin. The first number in the ordered pair, 2, tells us we move 2 units to the right ( $+2$ means 2 units to the right) The second number in the ordered pair, 6, tells us we move 6 units up ( $+6$ means 6 units up).

## Practice set a

Plot the ordered pairs.

$\left(1,\text{\hspace{0.17em}}3\right),\text{\hspace{0.17em}}\left(4,\text{\hspace{0.17em}}-5\right),\text{\hspace{0.17em}}\left(0,\text{\hspace{0.17em}}1\right),\text{\hspace{0.17em}}\left(-4,\text{\hspace{0.17em}}0\right)$ .

(Notice that the dotted lines on the graph are only for illustration and should not be included when plotting points.)

## Exercises

Plot the following ordered pairs. (Do not draw the arrows as in Practice Set A.)
$\left(8,\text{\hspace{0.17em}}2\right),\text{\hspace{0.17em}}\left(10,\text{\hspace{0.17em}}-3\right),\text{\hspace{0.17em}}\left(-3,\text{\hspace{0.17em}}10\right),\text{\hspace{0.17em}}\left(0,\text{\hspace{0.17em}}5\right),\text{\hspace{0.17em}}\left(5,\text{\hspace{0.17em}}0\right),\text{\hspace{0.17em}}\left(0,\text{\hspace{0.17em}}0\right),\text{\hspace{0.17em}}\left(-7,\text{\hspace{0.17em}}-\frac{3}{2}\right)$ .

As accurately as possible, state the coordinates of the points that have been plotted on the following graph.

Using ordered pair notation, what are the coordinates of the origin?

$\text{Coordinates\hspace{0.17em}of\hspace{0.17em}the\hspace{0.17em}origin\hspace{0.17em}are}\left(0,0\right)$ .

We know that solutions to linear equations in two variables can be expressed as ordered pairs. Hence, the solutions can be represented as points in the plane. Consider the linear equation $y=2x-1$ . Find at least ten solutions to this equation by choosing $x\text{-values}$ between $-4$ and 5 and computing the corresponding $y\text{-values}$ . Plot these solutions on the coordinate system below. Fill in the table to help you keep track of the ordered pairs.

 $x$ $y$

Keeping in mind that there are infinitely many ordered pair solutions to $y=2x-1$ , speculate on the geometric structure of the graph of all the solutions. Complete the following statement:

The name of the type of geometric structure of the graph of all the solutions to the linear equation
$y=2x-1$ seems to be __________ .

Where does this figure cross the $y\text{-axis}$ ? Does this number appear in the equation $y=2x-1$ ?

Place your pencil at any point on the figure (you may have to connect the dots to see the figure clearly). Move your pencil exactly one unit to the right (horizontally). To get back onto the figure, you must move your pencil either up or down a particular number of units. How many units must you move vertically to get back onto the figure, and do you see this number in the equation $y=2x-1$ ?

Consider the $xy\text{-plane}$ .

Complete the table by writing the appropriate inequalities.

 I II III IV $x>0$ $x<0$ $x$ $x$ $y>0$ $y$ $y$ $y$

In the following problems, the graphs of points are called scatter diagrams and are frequently used by statisticians to determine if there is a relationship between the two variables under consideration. The first component of the ordered pair is called the input variable and the second component is called the output variable . Construct the scatter diagrams. Determine if there appears to be a relationship between the two variables under consideration by making the following observations: A relationship may exist if
1. as one variable increases, the other variable increases
2. as one variable increases, the other variable decreases

 I II III IV $x>0$ $x<0$ $x<0$ $x>0$ $y>0$ $y>0$ $y<0$ $y<0$

A psychologist, studying the effects of a placebo on assembly line workers at a particular industrial site, noted the time it took to assemble a certain item before the subject was given the placebo, $x$ , and the time it took to assemble a similar item after the subject was given the placebo, $y$ . The psychologist's data are

 $x$ $y$ 10 8 12 9 11 9 10 7 14 11 15 12 13 10

The following data were obtained in an engineer’s study of the relationship between the amount of pressure used to form a piece of machinery, $x$ , and the number of defective pieces of machinery produced, $y$ .

 $x$ $y$ 50 0 60 1 65 2 70 3 80 4 70 5 90 5 100 5

Yes, there does appear to be a relation.

The following data represent the number of work days missed per year, $x$ , by the employees of an insurance company and the number of minutes they arrive late from lunch, $y$ .

 $x$ $y$ 1 3 6 4 2 2 2 3 3 1 1 4 4 4 6 3 5 2 6 1

A manufacturer of dental equipment has the following data on the unit cost (in dollars), $y$ , of a particular item and the number of units, $x$ , manufactured for each order.

 $x$ $y$ 1 85 3 92 5 99 3 91 4 100 1 87 6 105 8 111 8 114

Yes, there does appear to be a relation.

## Exercises for review

( [link] ) Simplify ${\left(\frac{18{x}^{5}{y}^{6}}{9{x}^{2}{y}^{4}}\right)}^{5}$ .

( [link] ) Supply the missing word. An is a statement that two algebraic expressions are equal.

equation

( [link] ) Simplify the expression $5xy\left(xy-2x+3y\right)-2xy\left(3xy-4x\right)-15x{y}^{2}$ .

( [link] ) Identify the equation $x+2=x+1$ as an identity, a contradiction, or a conditional equation.

( [link] ) Supply the missing phrase. A system of axes constructed for graphing an equation is called a .

how do they get the third part x = (32)5/4
can someone help me with some logarithmic and exponential equations.
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
I don't understand what the A with approx sign and the boxed x mean
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
oops. ignore that.
so you not have an equal sign anywhere in the original equation?
Commplementary angles
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Sherica
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Sherica
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Uday
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algebra 2 Inequalities:If equation 2 = 0 it is an open set?
or infinite solutions?
Kim
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Al
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if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
rolling four fair dice and getting an even number an all four dice
Kristine 2*2*2=8
Differences Between Laspeyres and Paasche Indices
No. 7x -4y is simplified from 4x + (3y + 3x) -7y
is it 3×y ?
J, combine like terms 7x-4y
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Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
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Cied
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Porter
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Porter
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Stotaw
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Azam
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At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
how did you get the value of 2000N.What calculations are needed to arrive at it
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