Math II · A-CED.2

Creating Two-Variable Equations and Graphing Relationships with Meaningful Axes

This objective teaches students how to describe relationships, not just find isolated answers. Most real situations involve quantities changing together, and graphs make those relationships visible.

Concept Algebra

Domain Creating Equations

Read time 10 minutes

What this learning objective is really asking you to learn

This objective asks students to move from single-answer algebra to relationship algebra. A one-variable equation often asks for the value that makes a condition true. A two-variable equation describes an entire relationship between quantities. Instead of asking only “What is x?” it asks “How does y depend on x?” or “Which pairs of values can exist together?” That shift is one of the biggest ideas in high-school mathematics.

A two-variable equation might be as simple as $C = 12n + 5$, where $C$ is total cost and $n$ is the number of items. It might be quadratic, such as $h = -16t^2 + 64t + 5$, where $h$ is height and $t$ is time. It might be exponential, such as $A = 500(1.06)^t$, where $A$ is account value and $t$ is years. It might be rational, such as $t = d/r$, where time depends on distance and rate. It might involve more than two variables, such as $V = lwh$, $PV = nRT$, or $A = P(1 + r)^t$. The key idea is that the equation captures how quantities are connected.

Graphing the equation turns the relationship into a visual object. Each point on the graph represents a pair of values that satisfies the equation. In a cost model, a point might mean “10 items cost 125 dollars.” In a height model, a point might mean “after 2 seconds, the object is 69 feet high.” In a growth model, a point might mean “after 6 years, the investment is worth this amount.” A graph is not just a picture; it is a collection of possible states.

The standard specifically mentions labels and scales because a graph without labels and scales is not a mathematical model. Labels tell what the variables mean. Scales tell how much each step represents. Units tell how to interpret distance on the axes. A beautifully shaped curve can be useless or misleading if the axes are unlabeled. Is the horizontal axis seconds, days, years, meters, or dollars? Does each gridline represent 1 unit, 10 units, 100 units, or 0.1 units? Are values starting at zero or at a shifted origin? These choices affect interpretation.

Creating equations in two or more variables also requires students to recognize variable roles. Sometimes one variable is the input and the other is the output. In function language, the input is often placed on the horizontal axis and the output on the vertical axis. But not every equation starts as an explicit function. For example, $x^2 + y^2 = 25$ describes a circle. Neither variable is naturally “the output” everywhere because some x-values correspond to two y-values. The broader idea is that equations describe solution sets. A graph is a visual display of those solution sets.

Math II makes this objective richer because students now work with more function types. In Math I, linear and exponential relationships were central. In Math II, quadratics become a major focus. A linear graph shows constant rate of change. A quadratic graph shows a parabolic shape with a maximum or minimum. An exponential graph shows repeated multiplication and often grows or decays faster than a line. A rational graph may show asymptotic behavior or inverse variation. Students must learn that graph shape carries meaning.

For example, a linear cost graph with positive slope says each additional item adds the same cost. A quadratic area graph says area grows with the square of a length. A projectile graph rises, reaches a maximum, and falls. An exponential population graph may start slowly and then accelerate. A rational travel-time graph $t = d/r$ for fixed distance says that increasing speed reduces time, but the relationship is not linear: the time savings from increasing speed are large at low speeds and smaller at high speeds.

This objective is not satisfied by writing an equation and drawing any graph. The student must connect equation, graph, and context. The graph should use a domain that makes sense. If the input is number of people, negative values and fractions may not make sense. If the input is time after launch, negative time may be outside the model. If the output is physical height, values below zero may be irrelevant after the object hits the ground. A mathematically complete graph may include points that are not viable in the real situation. A modeling graph must show or explain the relevant part.

Why students should learn this math

Students should learn this objective because relationships are the basic material of modern decision-making. A single value can answer one question, but a relationship lets you answer many questions. If you know the equation connecting distance, rate, and time, you can solve for different trips. If you know the equation connecting price and revenue, you can compare choices. If you know the equation connecting time and height, you can predict when an object will land or how high it will go. If you know the equation connecting dosage and concentration, you can estimate when a medicine may fall below a useful level.

Graphs matter because humans are visual thinkers. A table can list values, and an equation can compute values, but a graph reveals structure at a glance. Is the relationship increasing or decreasing? Is it steep or gentle? Does it curve? Does it have a maximum? Does it cross a threshold? Does it approach a limit? Does it make sense only over a certain interval? These questions are much easier to answer visually.

In real life, graphs are everywhere: weather forecasts, stock charts, fitness apps, maps, business dashboards, epidemiology curves, physics simulations, hospital monitors, election data, sports analytics, engineering diagrams, and environmental reports. A student who cannot read and create meaningful graphs is at a disadvantage in a data-rich world. They may see images but not understand what the axes, scales, or shapes are actually saying.

The “why” is also about communication. Equations communicate with precision. Graphs communicate with vision. A good model uses both. Suppose a city is planning water use. The equation may estimate total demand based on population. The graph can show when demand crosses capacity. Suppose a small business is considering a price change. The equation can represent revenue. The graph can show the break-even point and possible maximum. Suppose an athlete tracks training load. The equation can model progress or fatigue. The graph can show trends and warning signs.

This objective also teaches students to be skeptical of misleading graphs. Axis scales can exaggerate or hide change. Starting a vertical axis far above zero can make small differences look dramatic. Using inconsistent intervals can distort trends. Leaving off units can make a graph impossible to interpret. In public life, graphs are used to persuade. Students need enough mathematical literacy to ask whether a graph is honest.

A deeper reason to learn this math is that equations in two variables are the gateway to functions and systems. A function is a relationship where each input has exactly one output. A system asks where two or more relationships are true at the same time. Graphs let students see intersections, zeros, maxima, minima, growth, decay, and constraints. Later mathematics—quadratics, trigonometry, logarithms, calculus, statistics, and data science—depends heavily on seeing equations as relationships with shapes.

The historical machinery behind this idea

This objective stands on one of the most important inventions in mathematics: coordinate geometry. Before coordinate systems, algebra and geometry were often treated as separate worlds. Geometry studied shapes. Algebra studied symbolic relationships and unknown quantities. In the seventeenth century, mathematicians such as René Descartes and Pierre de Fermat helped fuse these worlds. Points could be represented by ordered pairs. Curves could be represented by equations. Equations could be studied by drawing their graphs.

That fusion changed mathematics permanently. A line was no longer only a geometric object drawn with a straightedge. It could be represented by an equation like $y = mx + b$. A circle could be represented by $x^2 + y^2 = r^2$. A parabola could be represented by a quadratic equation. Algebra became visual, and geometry became symbolic.

This connection powered later developments in physics and engineering. Motion could be graphed. Forces could be modeled. Orbits could be studied with equations. Optimization problems could be visualized. Eventually, calculus grew from questions about slopes, areas, curves, and change. Modern science would be nearly impossible without the ability to represent relationships on coordinate axes.

Graphing also developed alongside measurement and data. Scientists needed ways to display observations, compare models, and see patterns. Over time, graphs became tools not just for mathematicians but for economists, doctors, engineers, governments, and businesses. Today, digital tools can graph equations instantly, but the old challenge remains: the user must know what the graph means.

The instruction to label axes and choose scales may sound small, but historically it is huge. A graph is a mathematical argument. Without clear axes, it is an ambiguous drawing. With clear axes and scales, it becomes a coordinate representation that others can interpret, test, and use. This is why the standard includes labels and scales rather than treating them as decoration.

Technical execution: how to create and graph a relationship

A strong process begins by reading the situation for quantities. A quantity is not just a number; it is a number with a meaning and usually a unit. In “a taxi charges a starting fee plus a cost per mile,” the quantities include total cost, number of miles, starting fee, and rate per mile. In “the height of a ball changes over time,” the quantities include height, time, initial height, initial velocity, and acceleration due to gravity. In “an investment grows by 6 percent per year,” the quantities include starting amount, annual growth factor, time, and account value.

After identifying quantities, choose variables. Variables should be meaningful. $t$ for time, $h$ for height, $C$ for cost, and $A$ for amount are often clearer than using $x$ and $y$ for everything. In a graph, students may still place the input on the horizontal axis and the output on the vertical axis, but the labels should preserve the context: “time (seconds)” and “height (feet),” not merely $x$ and $y$.

Next, identify the relationship type. Constant addition suggests a linear model. Repeated multiplication suggests an exponential model. Area or projectile motion often suggests a quadratic model. Inverse variation or rate formulas may suggest rational relationships. Not every context names the function type directly; the student must infer it from the structure.

Then write the equation. In a linear model, use forms such as $y = mx + b$, where $m$ is a rate and $b$ is an initial value. In a quadratic model, use forms such as $y = ax^2 + bx + c$, $y = a(x - h)^2 + k$, or factored form, depending on what information is given. In an exponential model, use $y = a(b)^x$ or $y = a(1 + r)^t$. In a formula with several variables, decide which variables are allowed to vary and which are fixed parameters.

Before graphing, determine the domain and range that make sense. If time begins at zero, the graph may start at $t = 0$. If the model stops when an object hits the ground, the graph may end when height becomes zero. If the input is number of tickets, only whole-number values make sense, even if a continuous line is used as an approximation. If the model is a physical length, negative outputs may not be meaningful.

Choose scales carefully. A graph should show the important behavior without crowding or distortion. If a projectile reaches 80 feet, a vertical scale from 0 to 100 may be useful. If a bank account grows from 500 to 800 dollars, a scale from 0 to 1,000 may be reasonable, but a scale from 500 to 800 may highlight the change. The choice depends on the purpose. Students should be able to defend scale choices.

Plot key features. For a line, two points may be enough to draw the graph, but intercepts and slope should be interpreted. For a quadratic, the vertex, intercepts, axis of symmetry, and opening direction are important. For an exponential, the initial value, growth or decay factor, intercept, and end behavior matter. For rational models, undefined values and asymptotic behavior may matter. In Math II, the focus is not on every advanced graph type, but students should begin seeing that different equations have different visual signatures.

Finally, interpret the graph. A point means a pair of related values. An intercept means one quantity is zero. A slope means rate of change for a line. A vertex means a maximum or minimum for a quadratic. A crossing of a horizontal line may mean reaching a target. A crossing of another graph may mean two models have the same output. A graph is useful only when its features are translated back into the situation.

Where this fits in the big map of mathematics

This objective is one of the main crossroads in the full math map. It connects algebraic representation, graphical representation, and contextual modeling. In algebra, students write equations. In functions, they interpret inputs and outputs. In geometry, they use the coordinate plane. In statistics, they use graphs to compare models and data. In science, they use graphs to represent laws and measurements.

It also prepares students for systems of equations. A single graph shows all solutions to one relationship. Two graphs together show where two relationships are true at the same time. That idea leads to linear-quadratic systems, optimization, constraints, and later analytic geometry.

In later courses, the same habit grows into more advanced ideas. Calculus studies slopes and areas of graphs. Statistics studies fitted models and residuals. Physics uses graphs of position, velocity, acceleration, force, and energy. Economics uses supply and demand curves. Computer science uses functions and visualizations. This objective is not a side topic. It is the visual language of quantitative reasoning.

Common student traps and how to avoid them

One common trap is using $x$ and $y$ without defining them. A variable should represent a quantity, and the graph should show the units. The fix is simple: before writing the equation, write a sentence defining each variable.

A second trap is choosing poor scales. If all the points are squeezed into a corner, the graph hides the relationship. If the scale exaggerates tiny changes, the graph may mislead. Students should choose scales that show the relevant part of the relationship clearly.

A third trap is graphing outside the meaningful domain. A quadratic projectile model may continue below the ground algebraically, but the physical situation ends when the object hits the ground. A cost model may allow fractional items algebraically, but the context may require whole numbers.

A fourth trap is treating the graph as decoration. The graph is not an afterthought. It is a representation of the solution set and a tool for interpretation. Students should be asked what points, intercepts, slopes, vertices, and curves mean.

Problem Library

Problems in the App From This Objective

189 problems across 15 archetypes in the app.

define variables, identify rate/intercept, and write equation.

12 problems Warmup Practice Mixed Review Assessment

Problem 1

Create a two-variable linear equation for context cost C is 12 dollars plus 4 dollars per item x.

Open in simulator

Problem 2

Create a two-variable linear equation for context distance d after t hours at 55 mph from starting point 20 miles away.

Problem 3

Create a two-variable linear equation for context total y from 3 dollars per notebook x plus 2 dollar fee.

Problem 4

Create a two-variable linear equation for context total cost T for a taxi ride is 2.50 dollars per mile m plus a 3 dollar base fare.

Problem 5

Create a two-variable linear equation for context number of pages P read by a student is 15 pages per hour h plus an initial 50 pages.

Problem 6

Create a two-variable linear equation for context amount of fuel F remaining in a tank starts at 100 liters and decreases by 5 liters per hour t.

Problem 7

Create a two-variable linear equation for context monthly phone bill B is 0.10 dollars per minute m used plus a 20 dollar service charge.

Problem 8

Create a two-variable linear equation for context height H of a plant in centimeters starts at 10 cm and grows 0.5 cm per day d.

Problem 9

Create a two-variable linear equation for context earnings E for selling x items at 7 dollars each plus a 50 dollar bonus.

Problem 10

Create a two-variable linear equation for context temperature T in degrees Celsius at an altitude a in kilometers starts at 20 degrees and drops 6 degrees per km.

Problem 11

Create a two-variable linear equation for context total weight W of a package is 0.2 kg per item n plus a 1.5 kg box weight.

Problem 12

Create a two-variable linear equation for context number of remaining tickets R from an initial 500 tickets after selling 25 tickets per hour h.

model area, projectile, or revenue relationship.

15 problems Warmup Practice Mixed Review Assessment

Problem 13

Create a two-variable quadratic equation for context rectangle area A with length x+3 and width x.

Problem 14

Create a two-variable quadratic equation for context projectile height h after t seconds starts at 5 with velocity 20 under -16t^2.

Problem 15

Create a two-variable quadratic equation for context revenue R from price x with demand 100-x.

Open in simulator

Problem 16

Create a two-variable quadratic equation for context rectangle area A with length 2x and width x-1.

Problem 17

Create a two-variable quadratic equation for context square area A with side length x+5.

Problem 18

Create a two-variable quadratic equation for context area A of a rectangle with perimeter 20, one side x.

Problem 19

Create a two-variable quadratic equation for context area A of a triangle with base x and height x+4.

Problem 20

Create a two-variable quadratic equation for context projectile height h after t seconds starts at 0 with velocity 50 under -16t^2.

Problem 21

Create a two-variable quadratic equation for context projectile height h after t seconds starts at 10 with velocity 32 under -16t^2.

Problem 22

Create a two-variable quadratic equation for context projectile height h after t seconds starts at 2 meters with initial velocity 15 m/s under -4.9t^2.

Problem 23

Create a two-variable quadratic equation for context revenue R from price p with demand 250-5p.

Problem 24

Create a two-variable quadratic equation for context revenue R from selling x items, where price is 50-0.5x.

Problem 25

Create a two-variable quadratic equation for context product P of two numbers where one is x and their sum is 15.

Problem 26

Create a two-variable quadratic equation for context area A of a rectangular enclosure with 100 feet of fencing, one side is x.

Problem 27

Create a two-variable quadratic equation for context height h of a ball thrown from 3 feet with initial upward velocity 40 ft/s.

identify initial value and growth/decay factor.

15 problems Warmup Practice Mixed Review Assessment

Problem 28

Create a two-variable exponential equation for context population P starts at 500 and grows by 6% each year t.

Problem 29

Create a two-variable exponential equation for context car value V starts at 20000 and decreases by 15% each year y.

Problem 30

Create a two-variable exponential equation for context temperature difference D starts at 80 and halves each hour h.

Problem 31

Create a two-variable exponential equation for context investment I starts at 1000 and grows by 8% annually a.

Problem 32

Create a two-variable exponential equation for context radioactive substance M starts at 500 grams and decays by 10% each year y.

Problem 33

Create a two-variable exponential equation for context cell population C starts at 20 and doubles every hour h.

Problem 34

Create a two-variable exponential equation for context drug concentration K starts at 100 mg and quarters each hour h.

Problem 35

Create a two-variable exponential equation for context tree height H starts at 5 feet and increases by 12% each year y.

Problem 36

Create a two-variable exponential equation for context laptop value L starts at 1500 and depreciates by 20% each year y.

Problem 37

Create a two-variable exponential equation for context algae bloom A starts at 10 square meters and triples each day d.

Problem 38

Create a two-variable exponential equation for context light intensity L starts at 1000 lux and reduces to one-third each meter m.

Open in simulator

Problem 39

Create a two-variable exponential equation for context savings S starts at 2500 and earns 4% interest monthly m.

Problem 40

Create a two-variable exponential equation for context pressure P starts at 760 mmHg and decreases by 5% each kilometer k.

Problem 41

Create a two-variable exponential equation for context virus count V starts at 50 and quadruples every hour h.

Problem 42

Create a two-variable exponential equation for context chemical concentration C starts at 200 ppm and reduces to one-fifth each step s.

define variables and preserve units.

12 problems Warmup Practice Mixed Review Assessment

Problem 43

Create a multi-variable formula for measurement context volume V of a rectangular prism with length l, width w, height h.

Problem 44

Create a multi-variable formula for measurement context density d from mass m and volume V.

Problem 45

Create a multi-variable formula for measurement context surface area S of a rectangular prism with dimensions l,w,h.

Problem 46

Create a multi-variable formula for measurement context area A of a trapezoid with bases b1,b2 and height h.

Problem 47

Create a multi-variable formula for measurement context area A of a triangle with base b and height h.

Problem 48

Create a multi-variable formula for measurement context perimeter P of a rectangle with length l and width w.

Problem 49

Create a multi-variable formula for measurement context volume V of a cylinder with radius r and height h.

Problem 50

Create a multi-variable formula for measurement context simple interest I from principal P, annual rate r, and time t in years.

Open in simulator

Problem 51

Create a multi-variable formula for measurement context force F from mass m and acceleration a.

Problem 52

Create a multi-variable formula for measurement context kinetic energy KE from mass m and velocity v.

Problem 53

Create a multi-variable formula for measurement context volume V of a cone with radius r and height h.

Problem 54

Create a multi-variable formula for measurement context surface area S of a sphere with radius r.

choose independent/dependent variables and labels.

12 problems Warmup Practice Mixed Review Assessment

Problem 55

Describe how to graph created equation C=4x+12 with context cost C for x items.

Problem 56

Describe how to graph created equation h=-16t^2+20t+5 with context height h over time t.

Problem 57

Describe how to graph created equation P=500(1.06)^t with context population after t years.

Problem 58

Describe how to graph created equation D=60t with context distance D traveled in t hours.

Problem 59

Describe how to graph created equation A=w(10-w) with context area A of a rectangle with width w.

Problem 60

Describe how to graph created equation M=100(0.5)^(t/5730) with context mass M remaining after t years.

Problem 61

Describe how to graph created equation R=500-20x with context remaining balance R after x payments.

Problem 62

Describe how to graph created equation y=(x-3)^2+1 with context a projectile's path y at horizontal distance x.

Problem 63

Describe how to graph created equation V=20000(1.03)^t with context value V of an investment after t years.

Problem 64

Describe how to graph created equation I=12/R with context current I for resistance R.

Open in simulator

Problem 65

Describe how to graph created equation y=|x-2| with context deviation y from a target value of 2 for input x.

Problem 66

Describe how to graph created equation s=sqrt(A) with context side length s of a square with area A.

use data range, intercepts, and key features.

12 problems Warmup Practice Mixed Review Assessment

Problem 67

Choose an appropriate graph scale for model context time from 0 to 10 seconds and height from 0 to 120 feet.

Problem 68

Choose an appropriate graph scale for model context population from 500 to about 2000 over 12 years.

Problem 69

Choose an appropriate graph scale for model context cost from 12 to 92 dollars for 0 to 20 items.

Problem 70

Choose an appropriate graph scale for model context temperature over a day, from -5°C to 15°C over 24 hours.

Problem 71

Choose an appropriate graph scale for model context distance traveled from 0 to 300 miles over 5 hours.

Problem 72

Choose an appropriate graph scale for model context growth of a plant from 0 to 50 cm over 10 weeks.

Problem 73

Choose an appropriate graph scale for model context number of customers in a store from 0 to 75 over 12 hours.

Problem 74

Choose an appropriate graph scale for model context profit ranging from -1000 to 5000 dollars for 0 to 100 units sold.

Problem 75

Choose an appropriate graph scale for model context speed of a car from 0 to 80 mph over 60 minutes.

Open in simulator

Problem 76

Choose an appropriate graph scale for model context amount of medicine in a bloodstream, starting at 100 mg and decreasing to 0 mg over 8 hours.

Problem 77

Choose an appropriate graph scale for model context height of a ball thrown, reaching a maximum of 64 feet and landing at 0 feet, over 4 seconds.

Problem 78

Choose an appropriate graph scale for model context monthly sales data ranging from 1500 to 3500 units over 12 months.

connect model to meaningful input/output values.

12 problems Warmup Practice Mixed Review Assessment

Problem 79

Identify domain and range restrictions for context equation C=4x+12 in x is number of purchased items.

Problem 80

Identify domain and range restrictions for context equation h=-16t^2+20t+5 in projectile height until it hits the ground.

Problem 81

Identify domain and range restrictions for context equation A=x(x+3) in rectangle dimensions x and x+3.

Problem 82

Identify domain and range restrictions for context equation C = 2.5w + 5 in w is the weight in pounds of a package being shipped, w must be at least 1 pound.

Problem 83

Identify domain and range restrictions for context equation A = pi*r^2 in r is the radius of a circular garden.

Open in simulator

Problem 84

Identify domain and range restrictions for context equation I = 500 * 0.03 * t in t is the number of years an investment of $500 earns 3% simple interest.

Problem 85

Identify domain and range restrictions for context equation V = s^3 in s is the side length of a storage cube.

Problem 86

Identify domain and range restrictions for context equation R = 25a - 500 in a is the number of attendees for a charity event, and the venue has a minimum capacity of 20 people.

Problem 87

Identify domain and range restrictions for context equation d = 60t in t is the time in hours a car travels at 60 mph, for a trip that lasts no more than 4 hours.

Problem 88

Identify domain and range restrictions for context equation H = 0.5t + 2 in t is the number of years since a tree was planted, and it was 2 feet tall when planted.

Problem 89

Identify domain and range restrictions for context equation F = 15 - 0.5h in h is the number of hours a generator has been running, starting with 15 gallons of fuel, consuming 0.5 gallons per hour, until it runs out.

Problem 90

Identify domain and range restrictions for context equation P = 3s in s is the side length of an equilateral triangle, and the side length must be at least 1 cm.

translate ordered pair to context.

12 problems Warmup Practice Mixed Review Assessment

Problem 91

Interpret point 5, 32 on graph of context equation C=4x+12.

Problem 92

Interpret point 2, 41 on graph of context equation h=-16t^2+60t+5.

Problem 93

Interpret point 3, 595.5 on graph of context equation P=500(1.06)^t.

Problem 94

Interpret point -1, 8 on graph of context equation C=4x+12.

Problem 95

Interpret point 2.5, 150 on graph of context equation d=60t.

Problem 96

Interpret point 20, 68 on graph of context equation F=1.8C+32.

Problem 97

Interpret point 5, 1276.28 on graph of context equation A=1000(1.05)^t.

Open in simulator

Problem 98

Interpret point 7, 49 on graph of context equation A=s^2.

Problem 99

Interpret point 3, 95 on graph of context equation C=15h+50.

Problem 100

Interpret point 20, 25 on graph of context equation M=100*(0.5)^(t/10).

Problem 101

Interpret point 10, 300 on graph of context equation R=500-20t.

Problem 102

Interpret point 10, 50 on graph of context equation P=-2q^2+40q-150.

recognize equivalent and non-equivalent forms.

12 problems Warmup Practice Mixed Review Assessment

Problem 103

Compare equations A=x(x+3) and A=x^2+3x for the same relationship.

Problem 104

Compare equations h=-16(t-2)^2+64 and h=-16t^2+64t for the same relationship.

Problem 105

Compare equations C=4x+12 and C=12x+4 for the same relationship.

Problem 106

Compare equations P=500(1.06)^t and P=500+0.06t for the same relationship.

Open in simulator

Problem 107

Compare equations y=(x-5)(x+5) and y=x^2-25 for the same relationship.

Problem 108

Compare equations 2x+3y=6 and y=-2/3x+2 for the same relationship.

Problem 109

Compare equations y=(x-3)^2+1 and y=(x+3)^2+1 for the same relationship.

Problem 110

Compare equations y=(x+2)^2 and y=x^2+4 for the same relationship.

Problem 111

Compare equations V=2π r(r+h) and V=2π r^2+2π rh for the same relationship.

Problem 112

Compare equations d=50t and d=50t^2 for the same relationship.

Problem 113

Compare equations y=(2x+4)/2 and y=x+2 for the same relationship.

Problem 114

Compare equations A=100(1.05)^t and A=100(1.5)^t for the same relationship.

infer model type and parameters.

12 problems Warmup Practice Mixed Review Assessment

Problem 115

Write an equation from graph description line with y-intercept 5 and slope 3.

Open in simulator

Problem 116

Write an equation from graph description parabola with vertex (2,9) opening down with a= -1.

Problem 117

Write an equation from graph description exponential graph with initial value 4 and growth factor 2.

Problem 118

Write an equation from graph description line through (0,12) and (4,20).

Problem 119

Write an equation from graph description line with slope -2 passing through (1, 7).

Problem 120

Write an equation from graph description parabola with vertex (0,0) opening up with a=1.

Problem 121

Write an equation from graph description exponential graph with initial value 100 and decay factor 0.5.

Problem 122

Write an equation from graph description line through (1, 5) and (3, 11).

Problem 123

Write an equation from graph description parabola with vertex (-1, 3) opening up with a=2.

Problem 124

Write an equation from graph description exponential graph with initial value 5 and growth factor 3.

Problem 125

Write an equation from graph description line with y-intercept 10 and slope -0.5.

Problem 126

Write an equation from graph description parabola with vertex (0, 5) opening down with a=-2.

infer pattern, write rule, and display key features.

15 problems Warmup Practice Mixed Review Assessment

Problem 127

Write an equation from table 0, 3; 1, 7; 2, 11 and describe graph features.

Problem 128

Write an equation from table 0, 2; 1, 6; 2, 18 and describe graph features.

Problem 129

Write an equation from table 0, 1; 1, 4; 2, 9; 3, 16 and describe graph features.

Problem 130

Write an equation from table 0, 5; 1, 3; 2, 1 and describe graph features.

Problem 131

Write an equation from table 0, -1; 1, 2; 2, 5 and describe graph features.

Problem 132

Write an equation from table 0, 1; 2, 2; 4, 3 and describe graph features.

Problem 133

Write an equation from table 1, 7; 2, 10; 3, 13 and describe graph features.

Problem 134

Write an equation from table 0, 16; 1, 8; 2, 4 and describe graph features.

Problem 135

Write an equation from table 0, 3; 1, 12; 2, 48 and describe graph features.

Problem 136

Write an equation from table 0, 4; 1, 6; 2, 9 and describe graph features.

Open in simulator

Problem 137

Write an equation from table 0, 100; 1, 10; 2, 1 and describe graph features.

Problem 138

Write an equation from table 0, 0; 1, 1; 2, 4; 3, 9 and describe graph features.

Problem 139

Write an equation from table 0, 0; 1, -1; 2, -4; 3, -9 and describe graph features.

Problem 140

Write an equation from table 0, 2; 1, 1; 2, 2; 3, 5 and describe graph features.

Problem 141

Write an equation from table 0, 0; 1, 2; 2, 8; 3, 18 and describe graph features.

critique axis choices and omitted features.

12 problems Warmup Practice Mixed Review Assessment

Problem 142

Explain why graph window or scale y-axis starts at 90 for data from 90 to 100 is misleading for model cost comparison.

Problem 143

Explain why graph window or scale projectile graph window ends before the x-intercepts is misleading for model height over time.

Problem 144

Explain why graph window or scale exponential graph shown only from x=0 to 2 is misleading for model long-term growth.

Problem 145

Explain why graph window or scale quadratic graph window excludes the vertex is misleading for model profit model.

Problem 146

Explain why graph window or scale rational function graph window from x=-1 to x=1 is misleading for model population growth approaching a carrying capacity.

Problem 147

Explain why graph window or scale quadratic function graphed over x values from -100 to 100 is misleading for model parabolic trajectory.

Open in simulator

Problem 148

Explain why graph window or scale y-axis for temperature data from 20 to 25 degrees Celsius is misleading for model temperature change.

Problem 149

Explain why graph window or scale sine wave graphed only from x=0 to x=0.5π is misleading for model seasonal sales cycle.

Problem 150

Explain why graph window or scale y-axis for stock price from $100.00 to $100.50 is misleading for model daily stock price fluctuation.

Problem 151

Explain why graph window or scale step function graphed with a window that only shows a single step interval is misleading for model tax bracket calculation.

Problem 152

Explain why graph window or scale two linear functions graphed with a window that doesn't include their intersection point is misleading for model break-even analysis.

Problem 153

Explain why graph window or scale exponential growth data plotted on a linear y-axis with a very large range is misleading for model spread of a virus.

represent possible combinations and boundary line/curve.

12 problems Warmup Practice Mixed Review Assessment

Problem 154

Build and graph a constraint equation or inequality for context budget of 100 dollars with x items at 4 dollars and y items at 5 dollars.

Problem 155

Build and graph a constraint equation or inequality for context rectangle area at most 60 with sides x and y.

Open in simulator

Problem 156

Build and graph a constraint equation or inequality for context capacity of 30 people split between adults x and children y.

Problem 157

Build and graph a constraint equation or inequality for context total of 20 hours to spend on studying math x and science y.

Problem 158

Build and graph a constraint equation or inequality for context recipe requires at most 12 cups of flour for x batches of cookies (1 cup/batch) and y batches of bread (2 cups/batch).

Problem 159

Build and graph a constraint equation or inequality for context truck has a maximum load capacity of 5000 pounds for x boxes weighing 50 pounds each and y crates weighing 100 pounds each.

Problem 160

Build and graph a constraint equation or inequality for context student needs to score at least 70 points on a test with x multiple-choice questions (2 points each) and y essay questions (10 points each).

Problem 161

Build and graph a constraint equation or inequality for context total distance covered by running x miles and cycling y miles is at least 15 miles.

Problem 162

Build and graph a constraint equation or inequality for context a farmer wants to spend no more than 300 dollars on x pounds of fertilizer A at 2 dollars per pound and y pounds of fertilizer B at 3 dollars per pound.

Problem 163

Build and graph a constraint equation or inequality for context a factory has 400 labor hours available to produce x units of product A (2 hours/unit) and y units of product B (4 hours/unit).

Problem 164

Build and graph a constraint equation or inequality for context a person wants to consume at most 2000 calories from x servings of food A (150 calories/serving) and y servings of food B (200 calories/serving).

Problem 165

Build and graph a constraint equation or inequality for context a theater has a maximum capacity of 250 seats for adults x and children y.

model two functions and interpret intersection/relative behavior.

12 problems Warmup Practice Mixed Review Assessment

Problem 166

Create equations for related quantities in context plan A costs 10+3x and plan B costs 25+1.5x and compare graphs.

Problem 167

Create equations for related quantities in context linear savings L=20+5t and exponential savings E=10(1.2)^t and compare graphs.

Problem 168

Create equations for related quantities in context revenue R=x(50-x) and cost C=10x+100 and compare graphs.

Problem 169

Create equations for related quantities in context Company A charges a flat fee of $50 plus $5 per hour. Company B charges a flat fee of $20 plus $10 per hour. and compare graphs.

Open in simulator

Problem 170

Create equations for related quantities in context A product's cost is C=200+5x, and its revenue is R=20x-0.5x^2. and compare graphs.

Problem 171

Create equations for related quantities in context Population A starts at 100 and doubles every 5 years, while Population B starts at 500 and increases by 50 people per year. and compare graphs.

Problem 172

Create equations for related quantities in context Ball 1's height is h1 = -16t^2 + 64t, and Ball 2's height is h2 = -16t^2 + 48t + 10. and compare graphs.

Problem 173

Create equations for related quantities in context The distance from a central point is d1 = |x-3|. A linear path is given by d2 = 0.5x + 1. and compare graphs.

Problem 174

Create equations for related quantities in context Investment A starts with $1000 and grows by 7% annually. Investment B starts with $500 and grows by 10% annually. and compare graphs.

Problem 175

Create equations for related quantities in context The cost of manufacturing x units with process 1 is C1 = 10x + 50. The cost with process 2 is C2 = 100*sqrt(x). and compare graphs.

Problem 176

Create equations for related quantities in context The daily profit P from selling x items is P = -2x^2 + 80x - 300. The daily overhead cost is a constant C = 200. and compare graphs.

Problem 177

Create equations for related quantities in context Cell phone plan X costs $30 base fee plus $0.10 per minute. Plan Y costs $15 base fee plus $0.20 per minute. and compare graphs.

identify variable, unit, scale, or parameter mismatch.

12 problems Warmup Practice Mixed Review Assessment

Problem 178

Correct the equation or graph-labeling error in model graph of cost C=4x+12 labels y-axis as number of items.

Problem 179

Correct the equation or graph-labeling error in model population growth by 6% is graphed as a line P=500+6t.

Problem 180

Correct the equation or graph-labeling error in model height model uses negative time values as viable context.

Problem 181

Correct the equation or graph-labeling error in model rectangle area A=x(x+3) is labeled as perimeter.

Open in simulator

Problem 182

Correct the equation or graph-labeling error in model volume of a cylinder V = pi*r^2*h is labeled in cm^2.

Problem 183

Correct the equation or graph-labeling error in model account balance with 3% annual compound interest is modeled by B = P + 0.03*P*t.

Problem 184

Correct the equation or graph-labeling error in model graph showing speed v=10-2t but y-axis is labeled as distance (m).

Problem 185

Correct the equation or graph-labeling error in model side length x for a square area A=x^2 allows x=-5.

Problem 186

Correct the equation or graph-labeling error in model linear function y = 3x + 5 is described as starting at 3.

Problem 187

Correct the equation or graph-labeling error in model a car's speed is given as 60 meters.

Problem 188

Correct the equation or graph-labeling error in model half-life decay of a substance is modeled as M = M0 - k*t.

Problem 189

Correct the equation or graph-labeling error in model graph of temperature T=20+5t labels x-axis as temperature (Celsius).