Chapter 1 A Physics Toolkit

A Physics Toolkit 1 In this chapter you will: Use mathematical tools to measure and predict. Apply accuracy and precision when measuring. Display and evaluate data graphically.

Mathematics and Physics Section Mathematics and Physics 1.1 In this section you will: Be able to answer the question, what is physics? Review the algebra required for this class. Learn the GUESS problem solving strategy. Use and do conversions with SI units. Evaluate answers using dimensional analysis. Perform arithmetic operations using scientific notation. Learn about and use significant figures.

Mathematics and Physics Section Mathematics and Physics 1.1a What is Physics? Physics is a branch of science that involves the study of the physical world: energy, matter, and how they are related. Learning physics will help you to understand the physical world. Physics is considered the basis for all other sciences: - Biology, Chemistry, Astronomy, Geology, etc. Physics is the fundamental science.

Mathematics and Physics Section Mathematics and Physics 1.1 What does one do as a physicist? Many research physicists work in environments where they perform basic research in industry, research universities, and astronomical observations. Physicists who find new ways to use physics are often employed by engineering, business, law, and consulting firms. Physicists are also extremely valuable in areas such as computer science, medicine, communications, and publishing. Finally, many physicists who love to see young people get excited about physics become teachers.

Mathematics and Physics Section Mathematics and Physics 1.1 What jobs do non-physicists hold that use physics every day? Every job has some relation to physics! Athletes - The laws of motion to lift, throw, push, hit, tackle, run, drag, jump, and crawl. - The more an athlete and coach understand and use their knowledge of physics in their sport, the better the athlete will become.

Mathematics and Physics Section Mathematics and Physics 1.1 Automotive mechanics ◦ areas such as optics, electricity and magnetism, thermodynamics, and mechanics play greater roles as vehicles become more and more complex. CAT scan, computed tomography, and magnetic resonance imaging (MRI) ◦ technicians in hospitals and medical clinics who use such technology must have an understanding of what X- rays and magnetic resonance imaging are, how they behave, and how such high-tech instruments are to be used.

Mathematics and Physics Section Mathematics and Physics 1.1 Mathematics in Physics Mathematics is the language of science. In physics, equations are important tools for modeling observations and for making predictions.

Mathematics and Physics Section Mathematics and Physics 1.1 Order Of Operations Simply: 2 + 5 x √ Correct: 2 + 5 x X Incorrect: ( 2 + 5 ) x = 7 x

Mathematics and Physics Section Mathematics and Physics 1.1 Order Of Operations : PEMA P arentheses and Brackets E xponents M ultiplication and Division (from left to right) A ddition and Subtraction (from left to right)

Mathematics and Physics Section Mathematics and Physics 1.1 Order Of Operations : Example 1 Step 1: parentheses Step 2: exponent Step 3: multiplication Step 4: addition Step 5: solution

Order Of Operations : Example 2 Section 1.1 Order Of Operations : Example 2 Step 1: Step 2: distribute 8 into (x + 1) Step 3: remove 1st parenthesis Step 4: combine like terms Step 5: within parenthesis, left to right, first comes division Step 6: then multiplication Step 7: simplify exponent Step 8: solution

Mathematics and Physics Section Mathematics and Physics 1.1 Algebra Review with Physics Variables are used to represent concepts. ex: d for displacement, t for time, p for momentum Units also are abbreviated. ex: m for meters, s2 for seconds squared Do not confuse variables with units. Subscripts are used to give more information about a variable. ex: vave : average velocity vi : initial velocity

Mathematics and Physics Section Mathematics and Physics 1.1 Problem Solving Strategy Given: Write down given or known quantities. Draw a picture. Unknown: Write down the unknown variable. Equation: Find an applicable equation. Isolate the unknown variable. Substitute numbers with units into the equation. Solve. Sense: Are the units correct? Does the answer make sense?

Mathematics and Physics Section Mathematics and Physics 1.1 Example: Electric Current The voltage across a circuit equals the current multiplied by the resistance in the circuit. That is, V (volts) = I (amperes) × R (ohms). What is the resistance of a light bulb that has a 0.75 amperes current when plugged into a 120-volt outlet? Step 1: Given - Write the given quantities with units. Step 2: Unknown - Identify unknown variable. Given: I = 0.75 amperes V = 120 volts Unknown: R = ?

Mathematics and Physics Section Mathematics and Physics 1.1 Electric Current Step 3: Equation. Rewrite the equation so that the unknown value is alone on the left. V = IR IR = V Reflexive property of equality. Divide both sides by I. Step 4: Substitute 120 volts for V, 0.75 amperes for I. Solve. 120 volts R = 0.75 amperes R = 160  Resistance will be measured in ohms.

Mathematics and Physics Section Mathematics and Physics 1.1 Electric Current Step 5: Sense Are the units correct? 1 volt = 1 ampere-ohm, so the answer in volts/ampere is in ohms, as expected. Does the answer make sense? 120 is divided by a number a little less than 1, so the answer should be a little more than 120.

Mathematics and Physics Section Mathematics and Physics 1.1 SI Units Units are CRITICAL in physics. It is the units that give meaning to the numbers. It is helpful to use units that everyone understands. Scientific institutions have been created to define and regulate measures. The worldwide scientific community and most countries currently use an adaptation of the metric system to state measurements.

Mathematics and Physics Section Mathematics and Physics 1.1 SI Units The Système International d’Unités, or SI, uses seven base quantities, also called the fundamental units.

Mathematics and Physics Section Mathematics and Physics 1.1 SI Units The base quantities were originally defined in terms of direct measurements. Other units, called derived units, are created by combining the base units in various ways. ex: speed is measured in meters per second (m/s) The SI system is regulated by the International Bureau of Weights and Measures in Sèvres, France. This bureau and the National Institute of Science and Technology (NIST) in Gaithersburg, Maryland, keep the standards of length, time, and mass against which our metersticks, clocks, and balances are calibrated.

Mathematics and Physics Section Mathematics and Physics 1.1 SI Units Measuring standards for kilogram and meter are shown below.

Mathematics and Physics Section Mathematics and Physics 1.1 SI Units You probably learned in math class that it is much easier to convert meters to kilometers than feet to miles. The ease of switching between units is another feature of the metric system. To convert between SI units, multiply or divide by the appropriate power of 10.

Mathematics and Physics Section Mathematics and Physics 1.1 Prefixes are used to change SI units by powers of 10, as shown in the table below.

Mathematics and Physics Section Mathematics and Physics 1.1 Scientific Notation A number in the form a x 10n is written in scientific notation where 1 ≤ a < 10, and n is an integer. (An integer is a whole number, not a fraction, that can be positive, negative, or zero.) When moving the decimal point to the right, you reduce the exponent when using scientific notation. Right – REDUCE When moving the decimal point to the left, you make the exponent larger when using scientific notation. LEFT – LARGER Common powers of ten include 100 = 1, 101 = 10, 102 = 100, etc.

Mathematics and Physics Section Mathematics and Physics 1.1 Scientific Notation: Example 1 Write 7,530,000 in scientific notation. The value for a is 7.53 (The decimal point is to the right of the first non-zero digit.) So the form will be 7.53 x 10n. 7,530,000. = 7.53 x 106 (Move the decimal point 6 places to the left; exponent gets larger.)

Mathematics and Physics Section Mathematics and Physics 1.1 Scientific Notation: Example 2 Write 0.000000285 in scientific notation. The value for a is 2.85 (The decimal point is to the right of the first non-zero digit.) So the form will be 2.85 x 10n. 0.000000285 = 2.85 x 10-7 (Move the decimal point to the right 7 places; exponent gets smaller.)

Mathematics and Physics Section Mathematics and Physics 1.1 Dimensional Analysis You often will need to use different versions of a formula, or use a string of formulas, to solve a physics problem. To check that you have set up a problem correctly, write the equation or set of equations you plan to use with the appropriate units. Before performing calculations, check that the answer will be in the expected units. For example, if you are finding a speed and you see that your answer will be measured in s/m or m/s2, you know you have made an error in setting up the problem. This method of treating the units as algebraic quantities, which can be cancelled, is called dimensional analysis.

Mathematics and Physics Section Mathematics and Physics 1.1 Dimensional Analysis example: Calculate the distance a car travels when it is moving at a velocity of 20 meters per second for 10 seconds. Use the formula: distance = velocity x time ? Use dimensional analysis: meters = meters x seconds second Treat the units as if they were algebraic quantities. meters = meters x seconds Seconds in the numerator cancel seconds in the denominator. The formula is therefore dimensionally correct. meters = meters 

Mathematics and Physics Section Mathematics and Physics 1.1 Dimensional Analysis Dimensional analysis also is used in choosing conversion factors. This is also known as the factor-label method. A conversion factor is a multiplier equal to 1. For example, because 1 kg = 1000 g, you can construct the following conversion factors:

Mathematics and Physics Section Mathematics and Physics 1.1 Factor-Label Method Choose a conversion factor that will make the units cancel, leaving the answer in the correct units. For example, to convert 1.34 kg of iron ore to grams, do as shown below: 1.34 kg 1000 g = 1,340 g 1 kg

Mathematics and Physics Section Mathematics and Physics 1.1 8760 hours/year Factor-Label Method: Example 2 Convert 1 year to hours. 1 year 365 days 24 hours = 8,750 hours 1 year 1 day

Mathematics and Physics Section Mathematics and Physics 1.1 8760 hours/year Warmup Problem Pressure is defined as force divided by area of contact ( P = F / A). What pressure must you apply to your pen in order to create a force of 0.25 N on a piece of paper, if the tip of the pen has a surface area of 3 mm2 touching the paper?

Mathematics and Physics Section Mathematics and Physics 1.1 Rules for Rounding Off 1. In a series of calculations, carry the extra digits through to the final answer, then round off. Rounding only occurs ONCE in a calculation! 2. If the digit to be removed is:  <5, the preceding stays the same. example: 1.33 rounds to 1.3  5 or greater, the preceding digit increases by 1. example: 1.36 rounds to 1.4. Example: Round 62.5347 to four significant figures. Look at the fifth figure. It is a 4, a number less than 5. Therefore, simply drop every figure after the fourth, and the number becomes 62.53

Mathematics and Physics Section 1.1 Mathematics and Physics Significant Digits (Significant Figures or Sig Figs) Definition: All the valid digits in a measurement, the number of which indicates the measurement’s precision (degree of exactness). Do not count sig figs for non-measurement quantities such as counting (4 washers) or exact conversion factors (24 hrs. in 1 day). Use the Atlantic & Pacific Rule to determine the sig figs. PACIFIC OCEAN ATLANTIC OCEAN

The Atlantic /Pacific Rule for Sig Figs If the… Section 1.1 Mathematics and Physics The Atlantic /Pacific Rule for Sig Figs If the… Decimal is Present Count all digits from the Pacific side from the first non-zero digit. Decimal is Absent Count from the Atlantic side from the first non-zero digit. Trailing zeros are indeterminate; they may or may not be significant. Use scientific notation to remove the ambiguity.

Mathematics and Physics Section 1.1 Mathematics and Physics How many sig figs are there? Count all the digits from the first non-zero digit. PACIFIC OCEAN Decimal point present ATLANTIC OCEAN Decimal point absent 3 sig figs (and 1 indeterminate) 5 sig figs 705.00 g 2130 m 523.0 g 706, 000 g 0.0098070 mm 9,010, 000 km 4 sig figs 3 sig figs (and 3 indeterminate) 5 sig figs 3 sig figs (and 4 indeterminate)

Mathematics and Physics Section Mathematics and Physics 1.1 Significant Digits Using the Atlantic rule, we can’t be sure if trailing zeros are significant or not. To specify the exact number of sig figs, use scientific notation. Exponents do not count towards significant digits. Example: 9,010, 000 km (3 sig figs and 4 indeterminate). To write this number indicating: 3 significant digits: 4 significant digits: 5 significant digits: 9.01 x 106 km 9.010 x 106 km 9.0100 x 106 km

Mathematics and Physics Section Mathematics and Physics 1.1 Operations with Significant Digits When you perform any arithmetic operation, it is important to remember that the result never can be more precise than the least- precise measurement. Add / Subtract: Round to the least number of DECIMAL places as determined by the original calculation. (All numbers must be the same power of 10). Example: 23.1 4.77 125.39 + 3.581 156.841 Round to 156.8 (one decimal place)

Mathematics and Physics Section Mathematics and Physics 1.1 Significant Digits – Addition / Subtraction Example: Add 5.861 dL + 2.614 L + 3.5 mL Convert to the same units and power of 10. Add in column form. 5.861 dL = 5.861 x 10-1 L = 0.5861 L 2.614 L 3.5 mL = 3.5 x 10-3 L = 0.0035 L 3.2036 L Round to the least amount of decimal places 3.204 L  fewest decimal places

Mathematics and Physics Section Mathematics and Physics 1.1 Significant Digits – Multiplication / Division Multiplication / Division: Round to the fewest number of SIGNIFICANT FIGURES. Example: (3.64928 x 105) (7.65314 x 107) (5.2 x 10-3) (5.7254 x 105) least precise measurement = (3.64928 x 105) x (7.65314 x 107) ÷ (5.2 x 10-3) ÷ (5.7254 x 105) = 9.3808 x 109 = 9.4 x 109 because the least precise measurement has 2 sig figs.

Mathematics and Physics Section Mathematics and Physics 1.1 Significant Digits – Combination Operations When doing a calculation that requires a combination of addition/subtraction and multiplication/division, use the multiplication rule. Example: slope = 70.0 m – 10.0 m = 3.3 m/s 29 sec – 11 sec 29 sec and 11 sec only have two significant digits each, so the answer should only have two significant digits

Mathematics and Physics Section Mathematics and Physics 1.1 Multistep Calculations   Do not round to 580 N2 and 1300 N2 Do not round to 1800 N2 Final answer, so it should be rounded to two sig figs.

Section Check 1.1 Question 1 The potential energy, PE, of a body of mass, m, raised to a height, h, is expressed mathematically as PE = mgh, where g is the gravitational constant. If m is measured in kg, g in m/s2, h in m, and PE in joules, then what is 1 joule described in base unit? 1 kg·m/s 1 kg·m/s2 1 kg·m2/s 1 kg·m2/s2

Section Section Check 1.1 Answer 1 Answer: D Reason:

Section Check 1.1 Question 2 A car is moving at a speed of 90 km/h. What is the speed of the car in m/s? (Hint: Use Dimensional Analysis) 2.5×101 m/s 1.5×103 m/s 2.5 m/s 1.5×102 m/s

Section Section Check 1.1 Answer 2 Answer: A Reason:

Section Check 1.1 Question 3 Which of the following representations is correct when you solve 0.03 kg + 3.333 kg? 3 kg 3.4 kg 3.36 kg 3.363 kg

Section Check 1.1 Answer 3 Answer: C Reason: When you add or subtract, round to the least number of decimal places.

Section 1.2 Measurement

Measurement 1.2 In this section you will: Distinguish between accuracy and precision. Determine the precision of measured quantities. Understand the use of error bars in analyzing data. Learn how to calculate percent error and percent deviation. Understand that “sources of error” is different from making errors.

Measurement 1.2 What is a Measurement? Section Measurement 1.2 What is a Measurement? A measurement is a comparison between an unknown quantity and a standard. Measurements quantify observations. Careful measurements enable you to derive the relation between any two quantities.

Measurement 1.2 Comparing Results Section Measurement 1.2 Comparing Results When a measurement is made, the results often are reported with an uncertainty. Therefore, before fully accepting a new data, other scientists examine the experiment, looking for possible sources of errors, and try to reproduce the results. A new measurement that is within the margin of uncertainty confirms the old measurement.

Measurements by Three Students Section Measurement 1.2 Measurements by Three Students

Measurement 1.2 Precision vs. Accuracy Section Measurement 1.2 Precision vs. Accuracy precision – a characteristic of a measured value describing the degree of exactness of a measurement. - The number of decimal places indicates the precision of a measurement. - The grouping of data also indicates precision. A tight grouping has high precision. accuracy – a characteristic of a measured value that describes how well the results of a measurement agree with the “real” value, which is the accepted value.

Click image to view the movie. Section Measurement 1.2 Precision Versus Accuracy Click image to view the movie.

Section Measurement 1.2

Measurement 1.2 Calculating Accuracy and Precision Section Measurement 1.2 Calculating Accuracy and Precision Percent error is used to determine accuracy, or the variation of a measurement compared to a known value. Percent error = |measured value – accepted value | × 100% accepted value Example: The accepted value for the acceleration due to gravity is 9.80 m/s2. The experimental results on the first trial was 8.50 m/s2. What was the percent error? 8.50 m/s2 – 9.80 m/s2 x 100% = 13.3% 9.80 m/s2

Measurement 1.2 Calculating Accuracy and Precision Section Measurement 1.2 Calculating Accuracy and Precision Percent deviation is used to determine precision, or a measure of the difference between a single measurement and the average of all measurements. ◦ Percent deviation can be an indicator of how carefully you have taken and recorded your measurements. Percent deviation = | measured value – average value | × 100% average value

Measurement 1.2 Techniques of Good Measurement Section Measurement 1.2 Techniques of Good Measurement To assure precision and accuracy, instruments used to make measurements need to be used correctly. This is important because one common source of error comes from the angle at which an instrument is read. The precision of a measurement is one-half of the smallest division of the instrument. Read to the line and guess the last digit.

Measurement 1.2 Sources of Error Section Measurement 1.2 Sources of Error A source of error is any factor that may affect the outcome of an experiment. An “error” to a scientist does not mean “mistake”; it more closely means “uncertainty”. When explaining sources of error in your experiment, do not list “human error”. This term is vague and lazy. Instead, think about specific things that happened during the experiment where the results may have been affected. Example: Suppose you where measuring the acceleration due to gravity of a falling object. Air resistance introduces error in your calculations.

Measurement 1.2 Sources of Error – More Examples Section Measurement 1.2 Sources of Error – More Examples When measuring the magnetic field of a small magnet, you may need to account for the Earth’s magnetic field. A draft may affect the fall of your paper helicopter. The temperature of the room may rise during an experiment due to hot plates, climate, body heat. Sources of Error – Non-Examples I misread the ruler (or scale, or graduated cylinder, etc.) I miscalculated. (Check your work; then have a lab partner check it.) Human error. (Never list this in your lab report !!)

Section Check 1.2 Question 1 Ronald, Kevin, and Paul perform an experiment to determine the value of acceleration due to gravity on the Earth (980 cm/s2). The following results were obtained: Ronald - 961 ± 12 cm/s2, Kevin - 953 ± 8 cm/s2, and Paul - 942 ± 4 cm/s2. Justify who gets the most accurate and precise value. Kevin got the most precise and accurate value. Ronald’s value is the most accurate, while Kevin’s value is the most precise. Ronald’s value is the most accurate, while Paul’s value is the most precise. Paul’s value is the most accurate, while Ronald’s value is the most precise.

Section Check 1.2 Answer 1 Answer: C Reason: Ronald’s answer is closest to 980 cm/s2 and hence his result is the most accurate. Paul’s measurement is the most precise within 4 cm/s2.

Section Check 1.2 Question 2 What is the precision of an instrument? The smallest division of an instrument. The least count of an instrument. One-half of the least count of an instrument. One-half of the smallest division of an instrument.

Section Check 1.2 Answer 2 Answer: D Reason: Precision depends on the instrument and the technique used to make the measurement. Generally, the device with the finest division on its scale produces the most precise measurement. The precision of a measurement is one-half of the smallest division of the instrument.

Section Check 1.2 Question 3 A 100-cm long rope was measured with three different scales. The answer obtained with the three scales were: 1st scale - 99 ± 0.5 cm, 2nd scale - 98 ± 0.25 cm, and 3rd scale - 99 ± 1 cm. Which scale has the best precision? 1st scale 2nd scale 3rd scale Both scale 1 and 3

Section Check 1.2 Answer 3 Answer: B Reason: Precision depends on the instrument. The measurement of the 2nd scale is the most precise within 0.25 cm.

Section Check 1.2 Question 4 Which of the following could be considered a legitimate source of error in your lab? The wheels on the cart wobbled a bit, which slowed it down. It was hard to get an exact reading from the spring scale because my hand shook. The thermometer may have not yet reached thermal equilibrium. All of them

Section Check 1.2 Answer 4 Answer: D Reason: Remember, “error” is not errors that you made in lab. It is the unwanted factors make your results different from the expected results. It is why your experiment will not exactly match the formula.

Bulls-eye Lab Activity Section 1.2 Bulls-eye Lab Activity Name:_______________________________________ Date:____________ Period:_______ c = 4.31cm d = 3.05 cm c d

Chapter 1 A Physics Toolkit 1.3 Graphing Data

1.3 Section Original Speed (m/s) Braking Distance (m) 11 18 16 32 20 49 25 68 29 92

Domestic Car Sales 1.3 Section 1940's 1950's 1960's 1970's 1980's   1940's 1950's 1960's 1970's 1980's 1990's 2000's Car Sales in Thousands (Domestic) 4,800 7,200 11,120 8,987 11,653 11,985 12,087

Section 1.3

Section 1.3

Section 1.3

Graphing Data 1.3 In this section you will: Graph the relationship between independent and dependent variables. Interpret graphs. Recognize common relationships in graphs.

Graphing Data 1.3 Identifying Variables Section Graphing Data 1.3 Identifying Variables A variable is any factor that might affect the behavior of an experimental setup. It is the key ingredient when it comes to plotting data on a graph. The independent variable is the factor that is changed or manipulated during the experiment. The dependent variable is the factor that depends on the independent variable. Question: What is the independent variable, x or y? Answer: x

Section Graphing Data 1.3 This is a scatter plot graph. What is the independent variable? Dependent variable?

Graphing Data 1.3 Linear Relationships Section Graphing Data 1.3 Interpolation is a method of constructing new data points within the range of a set of known data points. “reading between the points” Extrapolation is the process of constructing new data points outside a set of known data points. “reading beyond the points” Linear Relationships Scatter plots of data may take many different shapes, suggesting different relationships.

Graphing Data 1.3 Linear Relationships Section Graphing Data 1.3 Linear Relationships When the line of best fit is a straight line, as in the figure, the dependent variable varies linearly with the independent variable. This relationship between the two variables is called a linear relationship. The relationship can be written as an equation:

Graphing Data 1.3 Linear Relationships Section Graphing Data 1.3 Linear Relationships The slope is the ratio of the vertical change to the horizontal change. To find the slope, select two points, A and B, far apart on the line. The vertical change, or rise, Δy, is the difference between the vertical values of A and B. The horizontal change, or run, Δx, is the difference between the horizontal values of A and B.

Graphing Data 1.3 Linear Relationships Section Graphing Data 1.3 Linear Relationships As presented in the previous slide, the slope of a line is equal to the rise divided by the run, which also can be expressed as the change in y divided by the change in x. If y gets smaller as x gets larger, then Δy/Δx is negative, and the line slopes downward. The y-intercept, b, is the point at which the line crosses the y-axis, and it is the y-value when the value of x is zero.

Graphing Data 1.3 Nonlinear Relationships Section Graphing Data 1.3 Nonlinear Relationships When the graph is not a straight line, it means that the relationship between the dependent variable and the independent variable is not linear. There are many types of nonlinear relationships in science. Two of the most common are the quadratic and inverse relationships. Check this vocabulary term.

Graphing Data 1.3 Nonlinear Relationships Section Graphing Data 1.3 Nonlinear Relationships The graph shown in the figure is a quadratic relationship. A quadratic relationship exists when one variable depends on the square of another. A quadratic relationship can be represented by the following equation:

Graphing Data 1.3 Nonlinear Relationships Section Graphing Data 1.3 Nonlinear Relationships The graph in the figure shows how the current in an electric circuit varies as the resistance is increased. This is an example of an inverse relationship. In an inverse relationship, a hyperbola results when one variable depends on the inverse of the other. An inverse relationship can be represented by the following equation:

Graphing Data 1.3 Nonlinear Relationships Predicting Values Section Graphing Data 1.3 Nonlinear Relationships There are various mathematical models available apart from the three relationships you have learned. Examples include: sinusoids—used to model cyclical phenomena; exponential growth and decay—used to study radioactivity Combinations of different mathematical models represent even more complex phenomena. Predicting Values Relations, either learned as formulas or developed from graphs, can be used to predict values you have not measured directly. Physicists use models to accurately predict how systems will behave: what circumstances might lead to a solar flare, how changes to a circuit will change the performance of a device, or how electromagnetic fields will affect a medical instrument.

Section Check 1.3 Question 1 Which type of relationship is shown following graph? Linear Inverse Parabolic Quadratic Answer: B Reason: In an inverse relationship a hyperbola results when one variable depends on the inverse of the other.

Section Check 1.3 Question 2 What is line of best fit? The line joining the first and last data points in a graph. The line joining the two center-most data points in a graph. The line drawn close to all data points as possible. The line joining the maximum data points in a graph. Answer: C Reason: The line drawn closer to all data points as possible, is called a line of best fit. The line of best fit is a better model for predictions than any one or two points that help to determine the line.

Section Check 1.3 Question 3 Which relationship can be written as y = mx? Linear relationship Quadratic relationship Parabolic relationship Inverse relationship Answer: A Reason: Linear relationship is written as y = mx + b, where b is the y intercept. If y-intercept is zero, the above equation can be rewritten as y = mx.

Section Check 1.3 Question 4 What is this relationship? Linear relationship Quadratic relationship Parabolic relationship Inverse relationship Answer: B

Section Check 1.3 Question 5 What relationship has the equation y = a/x? Linear relationship Quadratic relationship Parabolic relationship Inverse relationship Answer: D

Section Check 1.3 Question 6 What is the difference between interpolation and extrapolation? Answer: Interpolation is reading between the data points. Extrapolation is reading beyond the data points.

Chapter Summary 1.1 Chapter 1 Test The test is worth 50 points. Section Chapter Summary 1.1 Chapter 1 Test The test is worth 50 points. - Multiple Choice (13 worth 1 pt. each) - Matching (25 worth 1 pt. each) - Problems ( 2 worth 6 pts. Each)