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Proficiency Exam

Module by: Denny Burzynski, Wade Ellis. E-mail the authors

Summary: This module is from Elementary Algebra by Denny Burzynski and Wade Ellis, Jr. In this chapter the student is shown how graphs provide information that is not always evident from the equation alone. The chapter begins by establishing the relationship between the variables in an equation, the number of coordinate axes necessary to construct its graph, and the spatial dimension of both the coordinate system and the graph. Interpretation of graphs is also emphasized throughout the chapter, beginning with the plotting of points. The slope formula is fully developed, progressing from verbal phrases to mathematical expressions. The expressions are then formed into an equation by explicitly stating that a ratio is a comparison of two quantities of the same type (e.g., distance, weight, or money). This approach benefits students who take future courses that use graphs to display information. The student is shown how to graph lines using the intercept method, the table method, and the slope-intercept method, as well as how to distinguish, by inspection, oblique and horizontal/vertical lines. This module contains the proficiency exam for the chapter "Graphing Linear Equations and Inequalities in One and Two Variables".

Proficiency Exam

For the following problems, construct a coordinate system and graph the inequality.

Exercise 1

((Reference)) −6x+4>−14 −6x+4>−14

Solution

x<3 x<3

A number line labeled x with arrows on each end, labeled from negative three to four, in increments of one. There is an open circle at three. A dark line is orginating from this circle and heading towards the left of three.

Exercise 2

((Reference)) −8<x+6≤−4 −8<x+6≤−4

Solution

−14<x≤−10 −14<x≤−10

A number line labeled x with arrows on each end, labeled at negative fourteen and negative ten. There is a closed circle at negative ten and an open circle at negative fourteen. These circles are connected by a black line

Exercise 3

((Reference)) Plot the ordered pairs (3, 1),(−2, 4),(0, 5),(−2, −2) (3, 1),(−2, 4),(0, 5),(−2, −2) .

An xy-plane with gridlines, labeled negative five and five on the both axes.

Solution

Total four points plotted in an xy-coordinate plane. The coordinates of these points are negative two, negative two; negative two, four; zero, five and three, one.

Exercise 4

((Reference)) As accurately as possible, label the coordinates of the points that have been plotted on the graph.

Total four points plotted in an xy plane. The coordinates of these points are negative three, zero; zero, one; three, three and two, negative three.

Solution

( 0,1 ),( 3,3 ),( −3,0 ),( 2,−3 ) ( 0,1 ),( 3,3 ),( −3,0 ),( 2,−3 )

Exercise 5

((Reference)) What is the geometric structure of the graph of all the solutions to the equation 2y+3x=−4 2y+3x=−4 ?

Solution

straight line

Exercise 6

((Reference)) In what form is the linear equation in two variables ax+by=c ax+by=c ?

Solution

general form

Exercise 7

((Reference)) In what form is the linear equation in two variables y=mx+b y=mx+b ?

Solution

slope-intercept

Exercise 8

((Reference)) If an ordered pair is a solution to a linear equation in two variables, where does it lie geometrically?

Solution

It lies on the line.

Exercise 9

((Reference)) Consider the graph of y= 2 7 x+16 y= 2 7 x+16 . If we were to place our pencil at any point on the line and then move it horizontally 7 units 7 units to the right, how many units and in what direction would we have to move our pencil to get back on the line?

Solution

2 units up

For the following two problems, find the slope, if it exists, of the line containing the following points.

Exercise 10

((Reference)) (−6, −1) and (0, 8) (−6, −1) and (0, 8)

Solution

3 2 3 2

Exercise 11

((Reference)) (−2, −8) and (−2, 10) (−2, −8) and (−2, 10)

Solution

no slope; vertical line at x=−2 no slope; vertical line at x=−2

Exercise 12

((Reference)) Determine the slope and y−intercept y−intercept of the line 3y+2x+1=0 3y+2x+1=0 .

Solution

slope=− 2 3 ,y-intercept is( 0,− 1 3 ) slope=− 2 3 ,y-intercept is( 0,− 1 3 )

Exercise 13

((Reference)) As we look at a graph left to right, do lines with a positive slope rise or decline?

Solution

rise

For the following problems, find the equation of the line using the information provided. Write the equation in slope-intercept form.

Exercise 14

((Reference)) Slope = 4, y-intercept = -3 Slope = 4, y-intercept = -3 .

Solution

y=4x−3 y=4x−3

Exercise 15

((Reference)) Slope = − 3 2 , y-intercept =  4 3 Slope = − 3 2 , y-intercept =  4 3 .

Solution

y=− 3 2 x+ 4 3 y=− 3 2 x+ 4 3

Exercise 16

((Reference)) slope =  2 3 , passes through (−1, 2) slope =  2 3 , passes through (−1, 2) .

Solution

y= 2 3 x+ 8 3 y= 2 3 x+ 8 3

Exercise 17

((Reference)) slope = 7, passes through (0, 0) slope = 7, passes through (0, 0) .

Solution

y=7x y=7x

Exercise 18

((Reference)) passes through the points (5,2) (5,2) and (2,1) (2,1) .

Solution

y= 1 3 x+ 1 3 y= 1 3 x+ 1 3


For the following problems, graph the equation of inequality.

Exercise 19

((Reference)-(Reference)) y =  1 3 x−2 y =  1 3 x−2

An xy-plane with gridlines, labeled negative five and five on the both axes.

Solution

y= 1 3 x−2 y= 1 3 x−2

A graph of a line passing through two points with coordinates zero, negative two and three, negative one.

Exercise 20

((Reference)-(Reference)) 5y−2x+15=0 5y−2x+15=0

An xy-plane with gridlines, labeled negative five and five on the both axes.

Solution

A graph of a line passing through two points with coordinates zero, negative three and five, negative one.

Exercise 21

((Reference)-(Reference)) 4(x+y)=8 4(x+y)=8

An xy-plane with gridlines, labeled negative five and five on the both axes.

Solution

4( x+y )=8 4( x+y )=8

A graph of a line passing through four points with coordinates negative two, four; zero, two; two, zero; and four, negative two.

Exercise 22

((Reference)—(Reference)) 3 2 y+2=0 3 2 y+2=0

An xy-plane with gridlines, labeled negative five and five on the both axes.

Solution

A graph of a line parallel to x-axis and passing through a point with coordinates zero, negative four over three.

Exercise 23

((Reference)—(Reference)) x=−2 x=−2

An xy-plane with gridlines, labeled negative five and five on the both axes.

Solution

x=−2 x=−2

A graph of a line parallel to y-axis in an xy plane. The line crosses the x-axis at x equals negative two.

Exercise 24

((Reference)) 2x+3y>6 2x+3y>6

An xy-plane with gridlines, labeled negative five and five on the both axes.

Solution

A dashed line in an xy plane passing through two points with coordinates zero, two and three, zero. The region above the line is shaded.

Exercise 25

((Reference)) Reading only from the graph, determine the equation of the line.

A graph of a line sloped up and to the left.

Solution

y=− 1 3 x+3 y=− 1 3 x+3

A graph of a line passing through two points with coordinates negative three, four and three, two.

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