Math III · A-CED.2

Creating Equations in Two or More Variables and Interpreting Graphs with Labels and Scales

Multi-variable equations let students model relationships where one quantity depends on another, which is how real costs, measurements, rates, and scientific relationships are usually described.

Concept Algebra

Domain Creating Equations

Read time 8 minutes

What this learning objective is really asking you to learn

This objective asks students to create equations in two or more variables, graph those equations, and interpret the relationships with appropriate labels and scales. Students first met this idea in Math I with simpler linear relationships. In Math III, the same modeling skill returns with a larger toolkit. The equations may now involve polynomial, rational, radical, exponential, logarithmic, or other studied function types.

A two-variable equation describes a relationship between quantities. It does not usually give one answer. Instead, it describes a set of ordered pairs that make the equation true. If $C = 25 + 8n$, then every allowed value of $n$ gives a cost $C$. If $A = πr^2$, then radius $r$ determines area $A$. If $d = 16t^2$, then time $t$ may determine falling distance $d$ under a simplified gravity model. The equation is a relationship machine.

The objective also insists on graphing and interpretation. A graph without labels and scales is not a serious model. If the horizontal axis is time, say time in what units: seconds, days, years, hours since launch? If the vertical axis is cost, say dollars, thousands of dollars, or cost per unit? Scale determines what features can be seen. A graph from 0 to 10 tells a different visual story from a graph from 0 to 10,000.

The learning target is not merely writing $y = ...$. It is choosing variables, defining units, writing a relationship, choosing a reasonable domain, graphing with a meaningful window, and explaining the graph's features in context. Math III students should be able to work with more than the straight-line models of Math I. They should be able to say, “This relationship is quadratic because area depends on a squared length,” or “This relationship is rational because average cost is total cost divided by number of units,” or “This relationship is exponential because the quantity changes by a percent each period.”

This objective is one of the main bridges between algebra and the real world. It teaches students that an equation is not just something to solve; it is a representation of how quantities move together.

Why students should learn this math

Students should learn this because real situations usually involve relationships, not isolated numbers. The cost of a project depends on hours, materials, distance, and fixed fees. The area of a design depends on dimensions. The pressure of a gas depends on volume and temperature. A company's average cost depends on total cost and number of units. A population depends on time. A screen size depends on width, height, and diagonal. A dose may depend on body weight. These are variable relationships.

If students only know how to solve equations handed to them, they are dependent on someone else to do the modeling. Creating equations gives them control. They can read a situation, decide which quantities matter, and write the relationship themselves. That is the difference between procedural algebra and mathematical modeling.

Graphing adds another layer of power. A formula gives exact calculation. A graph shows behavior. It can reveal whether a quantity increases, decreases, levels off, has a maximum or minimum, crosses a threshold, or grows too fast to remain realistic. Graphs also make communication possible. In science, business, engineering, and public policy, relationships are often communicated visually because humans read patterns faster from graphs than from equations alone.

Labels and scales are part of truthfulness. A graph can mislead if the scale is distorted or the axes are unlabeled. A cost graph that cuts off the vertical axis may exaggerate differences. A growth graph with uneven time intervals may hide acceleration. A model with no units may be impossible to interpret. This objective therefore builds quantitative literacy, not just algebra skill.

The “why” is that equations in two or more variables are the language of relationships. Students need this language to model real systems, evaluate claims, and communicate mathematical ideas clearly.

The historical machinery: equations become graphs

The ability to graph equations depends on coordinate geometry. When mathematicians began representing points with coordinates, equations and geometry became linked. A line, curve, circle, or parabola could be described algebraically. An equation became a visual object, and a graph became a set of solutions.

This was one of the great unifications in mathematics. Algebra could solve geometry problems, and geometry could visualize algebra problems. Later, calculus, physics, engineering, economics, and data science all grew from this relationship between equations and graphs.

The historical significance for students is that graphing is not a decorative add-on. It is one of the main ways mathematical models become understandable. A two-variable equation has infinitely many possible solution pairs. A graph compresses that infinite or large set into a picture.

In modern technology, graphing is everywhere. Spreadsheets, scientific software, financial dashboards, weather models, fitness trackers, and app analytics all use graphs to show relationships. But technology does not automatically choose meaningful variables, units, or scales. Humans still need to understand the model.

Where this fits in the big map of mathematics

This objective revisits Math I's equation-creation work at a Math III level. Objective 002 introduced creating two-variable equations and graphing them. Objective 141 extends that skill across more function types and more sophisticated contexts.

It connects to functions. Many two-variable equations can be written as $y = f(x)$, making one variable depend on another. But some relationships may be implicit or involve more than two variables.

It connects to domain and range. A graph should represent meaningful input values, not necessarily all algebraically possible values.

It connects to graph features: intercepts, asymptotes, maxima, minima, end behavior, and rates of change. These features gain meaning only when axes are labeled and scaled.

It connects to modeling. Students choose variables, units, and structures based on the situation.

It connects forward to systems and constraints in Objective 142. Once students can create one relationship, they can combine several relationships into a system.

The big-map role is representation. Students learn to move from context to equation to graph to interpretation.

How to execute the skill technically

A strong process begins with variables. Define each variable clearly, including units. For example: let $t$ be time in months and $V$ be value in dollars. Or let $r$ be radius in centimeters and $A$ be area in square centimeters.

Next, identify the relationship type. If there is a fixed amount plus a constant rate, use a linear model. If area depends on a squared dimension, use a quadratic relationship. If a total is divided by a variable, use a rational relationship. If a quantity changes by a percentage over equal intervals, use an exponential relationship.

Then write the equation. Suppose a device starts at $900 and loses 18% of its value each year. Let $t$ be years and $V$ be value. The equation is

\[V = 900(0.82)^t\].

Now graph with labels. The horizontal axis should be “time in years.” The vertical axis should be “value in dollars.” A reasonable domain might be $0 \le t \le 10$. A reasonable vertical scale might go from 0 to 900. The graph should decrease and level toward 0.

Another example: area of a circular garden as a function of radius.

\[A = πr^2\].

The horizontal axis is radius in meters. The vertical axis is area in square meters. The domain is $r \ge 0$. The graph increases and curves upward because area grows with the square of radius.

Students should interpret graph features. In the depreciation example, the y-intercept is 900, the initial value. In the circle-area example, the graph starts at $(0, 0)$ because radius 0 gives area 0. The slope increases because each extra meter of radius adds more area than the previous meter.

Worked example: average cost model

A company has a fixed cost of $2,000 and a variable cost of $15 per item. Create an equation for average cost per item as a function of number of items produced.

Let $x$ be the number of items produced. Let $A$ be average cost per item in dollars.

Total cost is

\[2000 + 15x\].

Average cost is total cost divided by number of items:

\[A = (2000 + 15x)/x\].

This can be rewritten as

\[A = 2000/x + 15\].

The domain is $x > 0$, and if items are whole units, $x$ should be positive integers. A graph should have horizontal axis “items produced” and vertical axis “average cost per item in dollars.” The graph decreases toward 15 as production increases. The 15 is the variable cost per item, and the fixed cost is spread over more items.

This example shows why rational relationships matter. The average cost is not linear. It decreases quickly at first and then levels off.

Upgrade example: choosing scales responsibly

Suppose a city models the cost of maintaining a road segment as

\[C = 5000 + 1200m\]

where $m$ is miles of road and $C$ is annual cost in dollars. If the graph is drawn with the vertical axis from 0 to 10,000 dollars, only a small range of miles will fit. If the graph is drawn from 0 to 100,000 dollars, the model can show a larger planning range. Neither scale is automatically correct. The scale should match the question.

If the question is about maintaining 0 to 5 miles of road, a vertical scale up to about 12,000 dollars may be useful. If the question is about a regional plan from 0 to 50 miles, the scale must be much larger. A responsible graph is designed for the decision being made.

A useful exercise is to change the graph window and see how the same equation can look steep, flat, detailed, or compressed. Then ask which graph is best for a given question. The lesson is that graphing is communication, not decoration.

Multiple variables before graphing

Some equations naturally contain more than two variables. For example, the volume of a cylinder is

\[V = πr^2h\].

This relationship includes volume, radius, and height. To graph it on a two-dimensional coordinate plane, students usually hold one variable constant. If height is fixed at 10, then

\[V = 10πr^2\].

Now volume can be graphed as a function of radius. If radius is fixed at 3, then

\[V = 9πh\].

Now volume can be graphed as a function of height.

This is an important Math III modeling idea. A formula with several variables can generate many two-variable relationships depending on what is held constant. Scientists do this constantly in experiments: hold some variables fixed, vary one input, and observe one output. The graph is not the whole formula; it is a slice of the relationship.

Model assumptions

Every equation makes assumptions. The depreciation model $V = 900(0.82)^t$ assumes a constant 18% yearly decrease. The area model $A = πr^2$ assumes a perfect circle. The average cost model assumes fixed and variable costs remain as stated. In real life, assumptions may fail.

Students should learn to write one assumption sentence after creating an equation. For example: “This model assumes the percentage loss is the same every year,” or “This model assumes the cost per item remains $15 no matter how many items are produced.” This habit makes modeling honest and prepares students for later statistics and applied work.

Common misconceptions and how to avoid them

One common mistake is writing an equation without defining variables. Without variable definitions, the equation cannot be interpreted.

Another mistake is graphing without labels or units. A graph with unlabeled axes is not a complete model.

A third mistake is choosing a scale that hides important behavior. The graph window should fit the context.

A fourth mistake is using all real numbers as the domain when the context restricts inputs.

A fifth mistake is forcing every relationship to be linear. Math III students must choose from a broader toolkit.

The big takeaway

Creating equations in two or more variables is the mathematics of relationships. Graphing those equations makes the relationship visible. Labels, scales, units, and domain restrictions make the model honest. This objective teaches students to move from real situations to symbolic equations to meaningful graphs.

Problem Library

Problems in the App From This Objective

186 problems across 15 archetypes in the app.

define variables and model relationship.

12 problems Warmup Practice Mixed Review Assessment

Problem 1

Create a polynomial equation in two variables from context rectangle area with width x and length y is 60.

Problem 2

Create a polynomial equation in two variables from context revenue equals price x times quantity y.

Problem 3

Create a polynomial equation in two variables from context projectile height y=-16t^2+64t+5.

Problem 4

Create a polynomial equation in two variables from context volume or area context with variables x and y.

Problem 5

Create a polynomial equation in two variables from context perimeter of a rectangle with length L and width W is 50.

Problem 6

Create a polynomial equation in two variables from context volume of a rectangular prism with length L, width W, and height 5.

Problem 7

Create a polynomial equation in two variables from context profit P from selling x items at price y, with a fixed cost of 200.

Problem 8

Create a polynomial equation in two variables from context distance D traveled by an object at speed S for time T.

Problem 9

Create a polynomial equation in two variables from context area A of a triangle with base b and height h.

Problem 10

Create a polynomial equation in two variables from context surface area S of a cylinder's side with radius r and height h.

Open in simulator

Problem 11

Create a polynomial equation in two variables from context the square of the hypotenuse c of a right triangle with legs a and b.

Problem 12

Create a polynomial equation in two variables from context area A of a path around a square garden with side x, and path width y.

model inverse or rate relationships.

15 problems Warmup Practice Mixed Review Assessment

Problem 13

Create a rational equation in two variables from context distance 120 miles, speed x, time y.

Problem 14

Create a rational equation in two variables from context average cost y with fixed cost 500 and quantity x plus unit cost 8.

Problem 15

Create a rational equation in two variables from context density y from mass m and volume x.

Problem 16

Create a rational equation in two variables from context inverse variation context.

Problem 17

Create a rational equation in two variables from context time y to travel distance 300 miles at speed x.

Problem 18

Create a rational equation in two variables from context average cost y with fixed cost 1000 and quantity x plus unit cost 15.

Problem 19

Create a rational equation in two variables from context current I from voltage 12V and resistance R.

Problem 20

Create a rational equation in two variables from context pressure P from force 50N and area A.

Problem 21

Create a rational equation in two variables from context y is inversely proportional to x, with constant of proportionality 25.

Problem 22

Create a rational equation in two variables from context average number of apples per basket y from total apples 150 and number of baskets x.

Problem 23

Create a rational equation in two variables from context time y to fill a 500 liter tank at a flow rate of x liters per minute.

Open in simulator

Problem 24

Create a rational equation in two variables from context average speed y for a 400 km trip taking time x.

Problem 25

Create a rational equation in two variables from context brightness B from a light source with intensity 100 at distance d.

Problem 26

Create a rational equation in two variables from context time y to complete a task if x workers work at a constant rate and the task requires 20 worker-hours.

Problem 27

Create a rational equation in two variables from context concentration C of a solution with 50 grams of solute in volume V liters.

model square-root or distance relationships.

12 problems Warmup Practice Mixed Review Assessment

Problem 28

Create a radical equation in two variables from context distance y from point (x,0) to (0,3).

Problem 29

Create a radical equation in two variables from context side length y of square with area x.

Problem 30

Create a radical equation in two variables from context speed y from height x in model y=sqrt(2gx).

Problem 31

Create a radical equation in two variables from context square-root physical or geometry relationship.

Problem 32

Create a radical equation in two variables from context distance y from point (x, 5) to origin.

Problem 33

Create a radical equation in two variables from context radius r of a circle with area A.

Problem 34

Create a radical equation in two variables from context leg x of a right triangle with hypotenuse 10 and other leg y.

Problem 35

Create a radical equation in two variables from context side length s of an equilateral triangle with area A.

Open in simulator

Problem 36

Create a radical equation in two variables from context time period T of a simple pendulum of length L.

Problem 37

Create a radical equation in two variables from context velocity v of an object with kinetic energy K and mass 2kg.

Problem 38

Create a radical equation in two variables from context diagonal d of a square with side length s.

Problem 39

Create a radical equation in two variables from context distance d from point (x,0) to (1,2).

model inverse growth/decay relationships.

15 problems Warmup Practice Mixed Review Assessment

Problem 40

Create an exponential or logarithmic equation in two variables from context amount y grows from 500 by 4% for x years.

Problem 41

Create an exponential or logarithmic equation in two variables from context remaining amount y decays by half every 6 hours x.

Problem 42

Create an exponential or logarithmic equation in two variables from context pH y from hydrogen concentration x.

Problem 43

Create an exponential or logarithmic equation in two variables from context time y needed for amount x in exponential model.

Problem 44

Create an exponential or logarithmic equation in two variables from context investment y grows from $1000 at 5% compounded annually for x years.

Problem 45

Create an exponential or logarithmic equation in two variables from context population y starts at 1 million and increases by 2.5% each year x.

Problem 46

Create an exponential or logarithmic equation in two variables from context amount y of a substance with a half-life of 10 days starting from 100g after x days.

Problem 47

Create an exponential or logarithmic equation in two variables from context value y of an asset depreciates by 15% annually from an initial value of $25,000 over x years.

Open in simulator

Problem 48

Create an exponential or logarithmic equation in two variables from context bacteria population y doubles every 3 hours starting with 100 cells after x hours.

Problem 49

Create an exponential or logarithmic equation in two variables from context Richter scale magnitude y from seismic wave amplitude x.

Problem 50

Create an exponential or logarithmic equation in two variables from context absorbance y of a solution given its transmittance x.

Problem 51

Create an exponential or logarithmic equation in two variables from context amount y from $2000 invested at 3% compounded continuously for x years.

Problem 52

Create an exponential or logarithmic equation in two variables from context time y for an investment to grow from $1000 to amount x at 6% annual interest.

Problem 53

Create an exponential or logarithmic equation in two variables from context time y for a substance to decay to amount x from initial 50g with a half-life of 20 years.

Problem 54

Create an exponential or logarithmic equation in two variables from context sound level y in decibels from intensity x.

choose window showing key features.

12 problems Warmup Practice Mixed Review Assessment

Problem 55

Graph the advanced equation with labeled axes and scale for y=500(1.2)^x for 0<=x<=10.

Problem 56

Graph the advanced equation with labeled axes and scale for y=120/x for positive x.

Problem 57

Graph the advanced equation with labeled axes and scale for y=sqrt(x-2).

Problem 58

Graph the advanced equation with labeled axes and scale for advanced model context.

Problem 59

Graph the advanced equation with labeled axes and scale for y = log_3(x-1).

Open in simulator

Problem 60

Graph the advanced equation with labeled axes and scale for y = 2cos(x/2) - 1 for 0 <= x <= 4pi.

Problem 61

Graph the advanced equation with labeled axes and scale for y = |2x - 6|.

Problem 62

Graph the advanced equation with labeled axes and scale for y = -x^3 + 4x.

Problem 63

Graph the advanced equation with labeled axes and scale for y = 25 * (0.8)^x for x >= 0.

Problem 64

Graph the advanced equation with labeled axes and scale for y = (x^2 - 1)/(x^2 - 4).

Problem 65

Graph the advanced equation with labeled axes and scale for y = -sqrt(4 - x^2).

Problem 66

Graph the advanced equation with labeled axes and scale for y = cot(x) for 0 < x < 2pi.

connect ordered pairs to quantities and units.

12 problems Warmup Practice Mixed Review Assessment

Problem 67

Interpret key points on advanced graph in context: x-intercept at t=4 in height model.

Problem 68

Interpret key points on advanced graph in context: maximum at (3,120) in revenue graph.

Problem 69

Interpret key points on advanced graph in context: vertical asymptote x=0 in average cost.

Problem 70

Interpret key points on advanced graph in context: endpoint (2,0) on radical graph.

Problem 71

Interpret key points on advanced graph in context: y-intercept at (0, 50) in population growth model.

Problem 72

Interpret key points on advanced graph in context: horizontal asymptote y=100 in drug concentration graph.

Problem 73

Interpret key points on advanced graph in context: minimum at (5, 10) in cost function.

Problem 74

Interpret key points on advanced graph in context: x-intercept at x=10 in profit function.

Problem 75

Interpret key points on advanced graph in context: point (2, 25) on radioactive decay curve.

Problem 76

Interpret key points on advanced graph in context: endpoint (0, 0) on distance vs. time graph.

Problem 77

Interpret key points on advanced graph in context: interval (0, 7) where profit function is increasing.

Open in simulator

Problem 78

Interpret key points on advanced graph in context: point of inflection at (4, 500) in sales growth model.

combine algebraic and contextual restrictions.

12 problems Warmup Practice Mixed Review Assessment

Problem 79

Identify domain and range restrictions for two-variable model y=120/x as time-speed model.

Problem 80

Identify domain and range restrictions for two-variable model y=sqrt(x-3).

Problem 81

Identify domain and range restrictions for two-variable model y=log(x+2) in context.

Open in simulator

Problem 82

Identify domain and range restrictions for two-variable model box dimensions model.

Problem 83

Identify domain and range restrictions for two-variable model y = 1/(x^2 - 9).

Problem 84

Identify domain and range restrictions for two-variable model y = sqrt(10 - 2x).

Problem 85

Identify domain and range restrictions for two-variable model y = ln(x^2 + 4).

Problem 86

Identify domain and range restrictions for two-variable model A = s^2 as area of a square.

Problem 87

Identify domain and range restrictions for two-variable model P = 2L + 2W as perimeter of a rectangle.

Problem 88

Identify domain and range restrictions for two-variable model h = -16t^2 + 64t as height of a projectile.

Problem 89

Identify domain and range restrictions for two-variable model C = 500 / n as cost per item.

Problem 90

Identify domain and range restrictions for two-variable model y = 1 / (x - 5).

set axes to reveal relevant behavior.

12 problems Warmup Practice Mixed Review Assessment

Problem 91

Choose graph scale for advanced model values exponential grows from 100 to 100000 over 10 units.

Problem 92

Choose graph scale for advanced model values rational model has key behavior near x=0 and for x up to 100.

Problem 93

Choose graph scale for advanced model values log model changes slowly from x=1 to x=1000.

Problem 94

Choose graph scale for advanced model values target values are small decimals.

Problem 95

Choose graph scale for advanced model values quadratic equation has roots at -500 and 500.

Problem 96

Choose graph scale for advanced model values trigonometric function has a period of 0.05 and an amplitude of 2000.

Problem 97

Choose graph scale for advanced model values rational function has vertical asymptotes at x=-3 and x=3, and a horizontal asymptote at y=0.

Open in simulator

Problem 98

Choose graph scale for advanced model values polynomial has local maximum at x=-10 and local minimum at x=10.

Problem 99

Choose graph scale for advanced model values data points range from 9999.5 to 10000.5.

Problem 100

Choose graph scale for advanced model values model's significant changes occur only between x=-0.5 and x=0.5.

Problem 101

Choose graph scale for advanced model values exponential model starts at 0.01 and grows to 1000 over 5 units.

Problem 102

Choose graph scale for advanced model values function has a removable discontinuity (hole) at x=2.

represent two relationships and interpret intersections.

12 problems Warmup Practice Mixed Review Assessment

Problem 103

Create equations for competing models and compare graphs: linear plan y=20x+50 versus exponential plan y=60(1.1)^x.

Problem 104

Create equations for competing models and compare graphs: polynomial revenue and fixed target line.

Problem 105

Create equations for competing models and compare graphs: rational average cost and price line.

Problem 106

Create equations for competing models and compare graphs: two model candidates f and g.

Open in simulator

Problem 107

Create equations for competing models and compare graphs: two competing taxi fare structures: y=2x+5 (Company A) and y=3x+2 (Company B).

Problem 108

Create equations for competing models and compare graphs: a quadratic profit function P(x) = -0.5x^2 + 10x - 30 and a target profit of $50.

Problem 109

Create equations for competing models and compare graphs: an exponential decay model for drug concentration C(t) = 100(0.8)^t and a linear elimination model C(t) = 100 - 5t.

Problem 110

Create equations for competing models and compare graphs: a logarithmic model for learning curve progress P(t) = 20ln(t+1) and a mastery threshold of 60 points.

Problem 111

Create equations for competing models and compare graphs: a rational function for average cost AC(q) = (1000 + 5q) / q and a market price function P(q) = 10 - 0.1q.

Problem 112

Create equations for competing models and compare graphs: a model for temperature deviation from ideal T(h) = |h-12| + 5 and a maximum acceptable deviation of 10 degrees.

Problem 113

Create equations for competing models and compare graphs: a sinusoidal model for daylight hours D(d) = 3sin(2pi/365 * d) + 12 and a linear model for energy consumption E(d) = 0.01d + 10.

Problem 114

Create equations for competing models and compare graphs: a cubic volume function V(x) = x^3 - 10x^2 + 30x and a target volume of 50 cubic units.

infer family and parameters from features.

12 problems Warmup Practice Mixed Review Assessment

Problem 115

Write an equation from advanced graph features polynomial zeros -2 and 3, crosses both, positive leading.

Problem 116

Write an equation from advanced graph features rational vertical asymptote x=4 and horizontal asymptote y=2.

Problem 117

Write an equation from advanced graph features radical endpoint (3,0) increasing.

Problem 118

Write an equation from advanced graph features exponential y-intercept 5 and growth factor 2.

Open in simulator

Problem 119

Write an equation from advanced graph features polynomial zero -1 (touches) and 2 (crosses), positive leading.

Problem 120

Write an equation from advanced graph features rational vertical asymptote x=-3 and horizontal asymptote y=-1.

Problem 121

Write an equation from advanced graph features radical endpoint (-2,0) decreasing.

Problem 122

Write an equation from advanced graph features exponential y-intercept 10 and decay factor 0.5.

Problem 123

Write an equation from advanced graph features logarithmic vertical asymptote x=0 and passes through (e,1).

Problem 124

Write an equation from advanced graph features absolute value vertex (1, -2) opening upwards.

Problem 125

Write an equation from advanced graph features polynomial zeros 0, 1, and 4, crosses all, negative leading.

Problem 126

Write an equation from advanced graph features cube root passing through (0,0) and (8,2).

identify pattern and parameters.

12 problems Warmup Practice Mixed Review Assessment

Problem 127

Write an equation from advanced-function table values values double each step and y(0)=3.

Problem 128

Write an equation from advanced-function table values xy=24 for all rows.

Problem 129

Write an equation from advanced-function table values y^2=x and y>=0 pattern.

Problem 130

Write an equation from advanced-function table values second differences constant.

Problem 131

Write an equation from advanced-function table values values halve each step and y(0)=10.

Problem 132

Write an equation from advanced-function table values y is the cube of x.

Open in simulator

Problem 133

Write an equation from advanced-function table values y is the cube root of x.

Problem 134

Write an equation from advanced-function table values x values are powers of 2 and y is the exponent.

Problem 135

Write an equation from advanced-function table values values triple each step and y(0)=2.

Problem 136

Write an equation from advanced-function table values y is the reciprocal of x squared.

Problem 137

Write an equation from advanced-function table values y is the reciprocal of x cubed.

Problem 138

Write an equation from advanced-function table values y values are 5 more than x squared.

explain limiting behavior and restrictions.

12 problems Warmup Practice Mixed Review Assessment

Problem 139

Interpret asymptote or endpoint in context graph vertical asymptote x=0 in y=120/x time model.

Problem 140

Interpret asymptote or endpoint in context graph horizontal asymptote y=8 in average cost model.

Open in simulator

Problem 141

Interpret asymptote or endpoint in context graph radical endpoint (5,0).

Problem 142

Interpret asymptote or endpoint in context graph log vertical asymptote x=2.

Problem 143

Interpret asymptote or endpoint in context graph horizontal asymptote y=0 in drug concentration C(t) = 5t/(t^2+1).

Problem 144

Interpret asymptote or endpoint in context graph vertical asymptote x=0 in sound intensity model I = 10log(x).

Problem 145

Interpret asymptote or endpoint in context graph horizontal asymptote y=5000 in logistic population model P(t).

Problem 146

Interpret asymptote or endpoint in context graph endpoint (0,0) in projectile range R(h) = sqrt(2h/g).

Problem 147

Interpret asymptote or endpoint in context graph vertical asymptote x=10 in cost per item C(x) = 1000/(x-10).

Problem 148

Interpret asymptote or endpoint in context graph horizontal asymptote y=90 in learning curve L(t) = 100 - 10e^-0.5t.

Problem 149

Interpret asymptote or endpoint in context graph endpoint (0,0) in area A(x) = x*sqrt(100-x^2).

Problem 150

Interpret asymptote or endpoint in context graph vertical asymptote x=0 in time T = 50/x for workers.

identify representation errors.

12 problems Warmup Practice Mixed Review Assessment

Problem 151

Critique graph representation for axis starts at 50 and exaggerates small differences.

Problem 152

Critique graph representation for negative time shown for physical model.

Problem 153

Critique graph representation for rational graph window hides vertical asymptote.

Problem 154

Critique graph representation for axes unlabeled in context graph.

Problem 155

Critique graph representation for logarithmic scale used on a linear axis without clear indication.

Problem 156

Critique graph representation for discrete data points connected by a continuous line.

Problem 157

Critique graph representation for parabolic graph window excludes the vertex.

Problem 158

Critique graph representation for missing units on the y-axis representing temperature.

Problem 159

Critique graph representation for graph shows a continuous line through a point of discontinuity (hole).

Problem 160

Critique graph representation for y-axis range is too broad, making significant fluctuations appear flat.

Open in simulator

Problem 161

Critique graph representation for trigonometric graph window shows less than one full period.

Problem 162

Critique graph representation for multiple data series plotted without a legend.

define variables, graph relationship, and interpret.

12 problems Warmup Practice Mixed Review Assessment

Problem 163

Build a multi-variable formula and solve one variable graphically for volume V=pi r^2 h, fixed V=500, graph h versus r.

Problem 164

Build a multi-variable formula and solve one variable graphically for distance d=rt, graph t versus r for fixed d.

Problem 165

Build a multi-variable formula and solve one variable graphically for area A=lw with fixed A.

Problem 166

Build a multi-variable formula and solve one variable graphically for science formula with target variable.

Problem 167

Build a multi-variable formula and solve one variable graphically for perimeter P=2l+2w, fixed P=100, graph w versus l.

Open in simulator

Problem 168

Build a multi-variable formula and solve one variable graphically for simple interest I=Prt, fixed I=1000, fixed r=0.05, graph t versus P.

Problem 169

Build a multi-variable formula and solve one variable graphically for volume of a cone V=(1/3)pi r^2 h, fixed V=300, graph h versus r.

Problem 170

Build a multi-variable formula and solve one variable graphically for surface area SA=2pi r h + 2pi r^2, fixed SA=600, graph h versus r.

Problem 171

Build a multi-variable formula and solve one variable graphically for Ohm's Law V=IR, fixed V=12, graph I versus R.

Problem 172

Build a multi-variable formula and solve one variable graphically for density D=m/V, fixed m=200, graph D versus V.

Problem 173

Build a multi-variable formula and solve one variable graphically for kinetic energy KE=(1/2)mv^2, fixed KE=500, graph v versus m.

Problem 174

Build a multi-variable formula and solve one variable graphically for Pythagorean theorem a^2+b^2=c^2, fixed c=10, graph b versus a.

catch variable, unit, scale, domain, and family mistakes.

12 problems Warmup Practice Mixed Review Assessment

Problem 175

Correct the advanced two-variable modeling error: A graph uses negative quantities for a physical length variable.

Problem 176

Correct the advanced two-variable modeling error: A reciprocal rate context is modeled as y=kx.

Problem 177

Correct the advanced two-variable modeling error: A graph comparison uses different x-variables for two models.

Problem 178

Correct the advanced two-variable modeling error: A table with constant ratios is modeled linearly.

Problem 179

Correct the advanced two-variable modeling error: A model for area calculation combines length in feet and width in inches directly.

Problem 180

Correct the advanced two-variable modeling error: A population growth model for a limited resource environment is unbounded exponential.

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Problem 181

Correct the advanced two-variable modeling error: The height of a ball thrown upwards is modeled by a linear function of time.

Problem 182

Correct the advanced two-variable modeling error: The oscillation of a pendulum over time is modeled by a simple linear decay function.

Problem 183

Correct the advanced two-variable modeling error: A model for the number of cars produced daily allows for fractional values.

Problem 184

Correct the advanced two-variable modeling error: A model for ice cream sales uses temperature as the dependent variable.

Problem 185

Correct the advanced two-variable modeling error: The relationship between sound intensity and perceived loudness is modeled linearly.

Problem 186

Correct the advanced two-variable modeling error: The intensity of light from a point source is modeled as decreasing linearly with distance.