OA study guide
Given the graph, which variable would you expect is the independent variable in this situation?
"Number of Employees" appears on the horizontal axis, which is typically where the independent variable goes.
The slope formula
"Rise over run," or said another way, the difference between two points' y-values, divided by the difference of those same two points' x-values.
What is the coordinate of point A in the following graph?
(2, 2)
In this module you learned how to convert Celsius into Fahrenheit. Now suppose that you graph the function converting the other way, from Fahrenheit to Celsius. You know that 20⁰C is the same as 68⁰F. What would the coordinates of this point be on the graph going from Fahrenheit to Celsius?
(68, 20)
Inverse functions (written as f⁻¹(x))
- Write out the equation as x = f(y) - Rearrange the equation to make y the subject - Replace y with f⁻¹(x) is written f−1(x) and read "f inverse of x." In terms of the graphs of a function and its inverse, a "flipping" occurs which is noticeable in both the coordinates and the graphs of the functions. For example, notice below how the associated coordinates "flip" for the two functions S and R.
Follow these steps to solve a polynomial function using a graph:
1. Determine the output value you are looking for; usually, it will be specifically stated in the problem. 2. Starting with the specific output value you identified, trace that value on the dependent variable axis to any associated coordinates on the graph. 3. Trace from these associated coordinates to their corresponding values on the independent variable axis. 4. Estimate these values on the independent variable axis. 5. Check your solutions by plugging them back into the polynomial function and verify that you get the output value you identified in step 1. Polynomial equations can have multiple solutions at times. 2. Multiple inputs can give the same output, but each input can only give one output. 3. Interpreting the solution to a polynomial equation means remembering which function you found a solution to and then interpreting the corresponding values (independent and dependent variables) in a real-world context.
possible outlier
A data point that "lies" significantly "outside" the general trend in a set of data; it could prove to be a true outlier that should be removed or it could be legitimate data that should remain in the data set
A function of best fit
A function that best represents given data points
linear functions
A function that produces a straight line (linear means "straight"), indicating a constant increase or decrease.
polynomial
A function with real non-negative numbers - constants, variables, and exponents - that can be combined using addition, subtraction, multiplication and division. The word comes from poly- (meaning "many") and -nomial (meaning "terms").
All of these everyday examples describe intervals of numbers.
A plumber says he will visit your apartment between 10 a.m. and noon. A meteorologist calls for between one and two inches of snow. A pollster predicts that a candidate has between 35% and 42% support from the electorate.
regression function
A proposed function to model a data set, based on a regression analysis.
Interval
A range of numbers that include all the real numbers between its endpoints.
Trends in Data
A set of data can trend up (increase), down (decrease), vary up and down, or just remain constant. Note that trends generally address patterns in the dependent variable, not the independent variable.
Tostle Auto's Series A model can hold 10 gallons of gas, and it consumes 0.04 gallons of gas per mile; Series B model can hold 12 gallons of gas, and it consumes 0.03 gallons of gas per mile. Let A(m) model the amount of gas, in gallons, in a Series A model's tank, and B(m) in a Series B model's tank, where m is the number of miles to drive. Assuming both functions start with a full tank. Which statement is correct?
A(m)'s line decreases faster than B(m) . It implies the gas in a Series A auto decreases faster.
Which of the following functions is increasing with the steepest slope?
A(x)=30x+6 This line has the steepest incline since it has the largest slope.
Which of the following functions is increasing with the steepest slope?
A(x)=30x+6 This line has the steepest incline since it has the largest slope.
Original Graph:
Algebraically, what you are saying is "swap the x- and y-values," which is why the line y = x becomes the "flipping line" for the graph of inverse functions.
starting values
Also "initial value" or "starting point." The output value of a function when the input value is 0.
y-intercept
Also "starting value," "starting point," or "initial value." The output value of a function when the input value is 0. You now know that b is the starting value of a function and m is the slope. In math terms, the value b is called the y-intercept
y-intercept
Also "starting value," "starting point," or "initial value." The output value of a function when the input value is 0. You now know that b is the starting value of a function and m is the slope. In math terms, the value b is called the y-intercept of a line; b tells you the value of y when x = 0.
Inequality
An expression that compares two numerical values when they are not equal.
An epidemic of influenza spread through students at Perry High School as modeled by the function in the following graph.
An instantaneous rate of change of zero says that the number of cases was neither increasing nor decreasing. It tells when the number of influenza cases per day is not changing.
Given the function C(p)=20-p, which of these would be a coordinate on the graph?
Answer: (5, 15) If you plug 5 into the function, you do indeed get 15. Said another way, C(5)=20-5=15.
Consider this next example: Sunrise Sky Spa and Retreat Spa are both located in the same small city and they compete for clients, or members. Their memberships since 2000 can be modeled by the following two functions, respectively: Sunrise Sky Spa: s(t)=7t³−174t²+1200t+1300, Retreat Spa: r(t)=−50.6t²+910.8t+1101.4, where t is the number of years since 2000. The following graph represents data for two competing spas, Retreat Spas (r) and Sunrise Sky Spas (s). The data shows their number of memberships over time (t), measured in years since 2000. If an investor were trying to decide which spa to invest in, which would seem like the better choice at the start of 2002?
At the start of 2002 (t = 2), it would appear that both spas are growing, but with Sunrise Sky Spa having more members, since $r(2)\approx2650$and $s(2)\approx3050$. Based on data up to that point in time, Sunrise Sky Spa would seem the better choice for the investment.
Your daughter babysits and charges $8 per hour. Select the function that models the amount of money she earns when babysitting for h hours.
B(h)=8h Since she earns $8 for each hour she babysits, multiply 8 times the number of hours.
A chorus rents tuxedos for its singers at $85.74 per tuxedo with a $25 processing fee, so the cost function is C(x)=85.74x+25. How much does it cost the chorus to rent tuxedos for 14 singers?
C(85.74(14)+25=1225.36 .
Which graph includes the coordinate (5, 10)?
Correct graph is attached
Which of the following functions has the smallest decline?
D(x)=-6x-40 This line is declining and is the smallest decline of all the lines here.
function, R(a)=0.30a
Did you know that most people download an app, use it once or twice, and then forget about it? If only 30% of people continue to use an app after downloading it, the function can be modeled, or represented, by: R(a)=0.30a where R represents the number of returning users and a is the number of people who download an app.
If your equation is not in the form of f(x)=mx+b, you can rearrange it using the commutative property.
For example, if P(t) was given as P(t)=−60+3t, it is a good habit to change it to P(t)=3t−60 so it is easy to identify the slope and y-intercept.
Inequality example
For example, you can write the range of acceptable bike tire pressures as 60 ≤ x ≤ 70, where x represents the bike tire pressure. The inequality is saying that if x is any value between 60 and 70, including the values 60 and 70, then you are good. The less-than-or-equal-to sign (≤) means that values are okay as long as they are just that: less than or equal to the stated value.
The amount of gasoline remaining in your gas tank, G(h), is a linear function of driving time, h, in hours. At time h = 0 (before any driving), you have 10 gallons in your tank. After driving 3 hours, you have 5.5 gallons in your tank. Find the equation of G(h).
G(h) = -1.5x + 10
A function doubles each input. What does the inverse function do?
Halves each input
Outliers also change the equation and the graph of the regression function. 2 things to note
If an outlier is visually spotted in the data or is known ahead of time, it is generally removed from the data set to improve the fit of the regression function. If you are given the choice among a scatterplot of real-world data, one with outliers, and one without outliers, you should generally favor the one without outliers.
Recall that the y-intercept is b in f(x)=mx+b.
In P(h)=−20h+100, the y-intercept is 100, which means the battery power is 100% when Sam first turns on the laptop.
what if your company needs to generate an additional $1,000 of revenue next month to help stay on target for annual goals? The question then becomes, "How many items do you need to sell for a revenue of $1,000?" Units Sold (S) Revenue (R) 50 250 100 500 120 600 150 750 200. 1000
In this case, you are looking for S(1000)since S returns the number of items sold for an associated revenue. This turns out to be S(1000)=200, which is similar to what you saw before. Next is the graph of the function S from this perspective—that is, input a revenue, and output the associated number of items to sell.
Which line in the following graph has the largest slope?
Line A has the largest slope. (X axis)
Of the three lines in the following graph, which line decreases the fastest?
Line C
To convert from Celsius to Fahrenheit,
Multiply by 9/5 then add 32
Interpreting the value of the model
Note that the value is where the point meets at the opposite axis (the y axis) For example, when t=8, the value of r(t) is approximately 5,150, and s(t)'s value is approximately 3,350. In function notations, you can write r(8)≈5,150, s(8)≈3,350. In coordinate notation, you can write (8, 5150) and (8, 3350) for r and s, respectively.
solve quadratic equations EXAMPLE u(t)=−0.0125t²+t , you would not do multiplication first, because the order of operations says you have to do the exponent operation before multiplication:
Notice that in the expression, −0.0125(10)² the exponent operation, (10)²=100, must be done before the multiplication operation. This result implies that the application would be using 8.75% of CPU resource 10 seconds after it started running. In the graph, it does look like the function crosses the point (10, 8.75), which confirms the calculation.
Examine the following graph. Which game attracted more gamers on June 1?
On June 1st, 2010, Instinct Fighters attracted more gamers. This is because i(5)≈3300 is greater than z(5)≈3050. Keep in mind that since the time is measured since January 2010, June 1st corresponds to t = 5.
Red Hot PC Fix fixes personal computers (PCs). It can fix 25 PCs per day with the current staff. The following table shows the number of PCs the company can fix with different numbers of extra employees. Let the function P(e) model the number of PCs the company can fix per day, where e is the number of extra employees. Write the equation of this function.
P(e)=6e+25 To calculate the actual employee count, minus the difference between the number of PCs fixed per day (25-31 = 6 & 31 - 37 = 6)
The slope is m in f(x)=mx+b.
P(h)=−20h+100, the slope is −20, which means the battery power decreases by 20% every hour.
Two laptop companies are racing to decrease the weight of their laptops. The weight, in pounds, of the lightest laptop released by Proxatech Company and Alta-Comp Inc. can be modeled by P(t)=4−0.2t and A(t)=4.8−0.4t, respectively, where t is the number of years since 2000. Since the equations are not written in the format of f(x)=mx+b, it is a good habit to rewrite them into:
P(t)=−0.2t+4 and A(t)=−0.4t+4.8. Examine their graphs:
Example if inverse function
Remember the example above about the conversion of temperatures from Celsius to Fahrenheit? You can undo that and go from Fahrenheit to Celsius, too.
One Common Function y=f(x).
Remember: this does not mean multiplying f times x. The notation is read as "the value of f at x" or just "f of x." That is, for a given input of x, you want to know the function's output, f(x).
The distance from Boston, MA, to Albany, NY, along Interstate 90 is 169.5 miles, and the maximum speed in both Massachusetts and New York is 65 miles per hour. You left Boston 90 minutes ago, driving at the speed limit. If the function that gives the remaining distance is d(x)=169.5-65x, where x is the number of hours you have already traveled, how much farther is it to Albany?
Since 90 minutes is 1.5 hours, substitute x = 1.5 into C(x)=169.5−65x to get C(1.5)=169.5−65(1.5)=72
Notice that you may not have an algebraic formula for the function f; that is defined by the graph. There will be many times in real life when this happens. In these cases, you still use the same function notation. Also notice that the years are on the horizontal axis while the Zillow Home Value Index is on the vertical axis. This implies that the year t is probably the independent variable and the home value index z is probably the dependent variable.
Sometimes you can define the relationship between two variables by the graph or by a function. The following applet is such an example that converts temperature in degrees Celsius (C) to degrees Fahrenheit (F). The function is F(C)=1.8C+32. Suppose you wanted to figure out 25⁰C in degrees Fahrenheit. You could move the slider above to C = 25 and you would see the coordinate (25, 77). This tells you that 25⁰C is equivalent to 77⁰F. You can also see this by using the formula above, F(C)=1.8C+32. For example, F(25)=1.8(25)+32=45+32=77. Either way, 25⁰C is equivalent to 77⁰F.
Sales tax in your state is 7%. Which function models the amount of sales tax, T, when you spend x dollars?
T(x)=0.07(x) 7% of total spent is written as 0.07x.
What is a dependent variable?
The dependent variable is the variable that is measured and a dependent variable on the vertical axis
The function in the following graph models the amount of coffee sold at Mugs & Cups on a certain day. Compare the average rate of change from 8:00 a.m. to 10:00 a.m., and from 10:00 a.m. to 12:00 p.m. Which statement is correct?
The function increased more from x = 0 to x = 2 than from x = 2 to x = 4. From 8:00 a.m. to 10:00 a.m., more coffee was sold per hour than from 10:00 a.m. to 12:00 p.m.
The number of financial applications at Macintosh Store grows by a certain fixed amount every year according to the following function: M(t)=2,200t+5,000, where t is the number of years since 2000. Which statement is correct?
The function's slope is 2,200. It implies that the number of financial applications increases by 2,200 per year. The slope is m in f(x)=mx+b.
The number of financial applications at Macintosh Store grows by a certain fixed amount every year according to the following function: M(t)=2,200t+5,000, where t is the number of years since 2000. Which statement is correct?
The function's y-intercept is 5,000. It implies that there were 5,000 financial applications at Macintosh Store in 2000.
What is the independent variable?
The independent variable is the variable that is changed independent variable on the horizontal axis
Risa is a member of a nature conservatory society. In recent years, the rabbit population in Greystone Park has been studied to learn if it may be nearing overpopulation. Risa's group was asked to determine how the instantaneous rate of change varied from month to month.
The instantaneous rate of change for the rabbit population changed between positive values, 0, and negative values.
The Least Squares Regression (LSR)
The least squares regression method works by minimizing the sum of the square of the errors as small as possible, hence the name least squares. Basically the distance between the line of best fit and the error must be minimized as much as possible. This is the basic idea behind the least squares regression method.
Output Variables
The output, represented by one or more letters, such as m or d(t), produced by a function, based on the input variable.
the Information Age
The present time, when information is widely available to many people through computer technology.
data regression
The process of examining data points to determine a valid equation. "Regression" refers to bringing different data points down toward a normal average - "regressing" them.
An action camera company has fixed costs of $9,000 per month and material cost of $500 to produce each camera. The function modeling cost per month is: C(x)=9,000+500x What is the function's slope? What does it mean?
The slope is 500. It implies the cost of producing each camera is $500. In f(x)=mx+b, m is the slope.
With the Scarlet Dragon scenario in mind, solve for t in c(t)=10 and interpret its meaning in this context.
The solutions are t ≈ 0.18 and t ≈ 10.3. These solutions imply that the restaurant has 10 customers at about 10:11 a.m. and at about 8:18 p.m. on a typical day.
Deal D, which is D(t)=300t, has the only line crossing the origin because its y-intercept is 0. This is because the starting value for this deal has Seth paying 0 dollars.
The steeper the line is, the greater the magnitude of its slope. In the graph, Deal D, or D(t), is the most steep, with a slope of $300 per month, and Deal A, or A(t), is the least steep with a slope of $150 per month.
polynomial's degree
The value of a polynomial's largest exponent; that is, if the largest exponent in a polynomial function is 3, the polynomial is "of the third degree." (Largest exponent)
An action camera company has fixed costs of $9,000 per month and material cost of $500 to produce each camera. The function modeling cost per month is: C(x)=9,000+500x What is the function's y-intercept? What does it mean?
The y-intercept is 9,000. It implies the company has fixed cost of $9,000 per month, before producing any cameras. The function can be written as C(x)=500x+9,000, which matches the format of f(x)=mx+b, where b is the y-intercept.
An extreme extrapolation value has a lot of uncertainty.
True
An extreme extrapolation value should not be interpreted except by a trained statistician.
True
Cartesian Coordinates come in pairs in the form (x, y).
True
Creating graphs using the Cartesian Coordinate System is fairly straightforward, with the first value representing the distance traveled on the x-axis and the second value representing distance traveled on the y-axis.
True
Extrapolation occurs where you do not have data. Before your first point and after the last point
True
Extreme extrapolation values are those beyond 50% of the range from the minimum or maximum data value.
True
For any line, any two slope triangles will give you the same slope.
True
In a table of data for a function, if the y-value increases or decreases at the same rate, the function is linear.
True
In the last examples, you have seen functions of two forms: f(x)=mx and f(x)=mx+b. You can consider these to be the same type of function for both of these examples; the first version is just the second when b = 0.
True
Interpolation values are generally more certain than extrapolation values.
True
Intervals of numbers are represented by shaded segments on the number line with special notation for endpoints. An empty, or open, endpoint means the endpoint is excluded while a filled in, or closed, endpoint means the endpoint is included.
True
Intervals of numbers are represented in writing by listing the pair of endpoints enclosed in either parenthesis (if the endpoints are not included) or in brackets (if the endpoints are included).
True
The curve of best fit is just a "best guess." It is normal to see a difference between the estimated value and the real value.
True
The farther the extrapolation value is from the known data values, the less trust should be placed in the predicted value.
True
Trends in data can be apparent in tables in similar ways that trends are apparent in graphs.
True
Using any two coordinates on the graph of a line, you can use the slope formula to calculate the line's slope.
True
Solving Polynomial Equations Using the Graph
Use the input (x) inside the parenthesis, to solve for the output, on the opposite axis where the line meets the point. You will look for the independent values that meet at identical points. For example: c(t)=60 From this graph, it looks like the corresponding independent variable values would be about t ≈ 7.1 and t ≈ 9.25. This means that the graph passes through the points (7.1, 60) and (9.25, 60). In function notation, this would mean that c(7.1)≈60 and c(9.25)≈60.
Example of function
What comes out of a vending machine based on your selection
r(m)=15m²−120m+330, where m is the number of months since September. The following graph represents this function.
What was the shop's revenue on October 1, and on January 1 of the following year? October 1 implies that m = 1 in r(m)=15m2−120m+330, because m stands for the number of months since September 1. Substitute m =1 into r(m), and you have: r(1)=15(1)²−120(1)+330=15(1)−120+330=15−120+330=225. Note that the exponent operation must be done before the multiplication operation The result implies that the ice cream shop's revenue on October 1 was $225.00. According to the graph, the function does indeed cross the point (1, 225).
Suppose a short-distance towing company uses the function P(m)=1.8m+32 to determine the price, P, of towing a car m miles. What would the price be for the coordinate (10, P)?
With m = 10, you calculate P(10)=1.8(10)+32=50.
Suppose someone wants to look at how student economic indicators (average student debt, median earnings, and debt to earnings ratio) vary over time. In this context, what is the independent variable for the data presented above?
Year Time (in years) is the independent variable in this context since you want to know how student economic indicators vary over time.
Seth has started driving for a trip from home. His speedometer tells him he is traveling at 65 miles per hour. If he travels at this speed for three hours, how far will he travel?
You can calculate this as 3×65=195 miles . This also means Seth's trip odometer will show he has traveled 195 miles at three hours into the trip. Here is another way of looking at this: In the following graph, the distance that Seth has traveled is given by the function d(t)=65t, where t, the input variable, is the time he has been traveling. Then d(t), the output variable, is the distance the car has traveled in t hours.
Inverse Functions example Examine the following more practical example. This a model of Revenue, R, and Units Sold, S, for a company and the corresponding graph of the data. Units Sold (S) Revenue (R) 50 250 100 500 120 600 150 750 200. 1000
You could ask, "How much revenue do you get from the sale of 200 items?" This would be helpful for predicting revenue from sales. From a mathematical perspective, you are asking what the value of R(200) would be. Said in the context of the problem, R(200) is the revenue generated from 200 units sold. For this data, R(200)=1000.
Your company has given your team a budget of no more than $1,000 for a project. It is totally acceptable for you to spend no money on this project if you can creatively find a way to do that. Which interval represents the amount of money you could spend on the project?
[0, 1,000] You could spend no money on the project (thus the bracket on the "0") or you could spend up to $1,000, meaning you could spend exactly $1,000 (thus the bracket on the "1,000").
Inverse function
are basically functions that "reverse" each other. Whatever the function does, its inverse undoes. Therefore, whenever you undo something, you are using an inverse function. To find the input and output of the inverse function, you simply swap the input and output of the original function.
Hank works for Riveting Marketing, a company that uses a 500 gigabyte (GB) hard disk as its server. The following graph depicts the remaining space at the server each month over a 60-month period. Hank's boss at Riveting Marketing wants to know if there were any similarities in remaining storage space between what happened during the first two years and what happened over the next two years.
been m=332−17624−48=−6.5 .
Concave down can happen in two ways
either when a function's values decrease faster and faster or when they increase slower and slower.
quadratic polynomial
equation that includes one variable raised to the second power (like x^2) and no greater exponent (3 or more); its name comes from Latin (quadratus) because the variable is squared (exponent of 2)
Which lines have the fastest rate of increase?
f(x)=3x−100 The slopes for the functions f, g, and h are 3, 1, and 0.5, respectively. This means f(x) has the fastest rate of increase. (With the 'X')
function
is a mathematical relationship between two variables where every input value is matched with exactly one output value. Think of a function as a "machine" that takes an input and produces an output based on the input. For instance, think of an automated teller machine (ATM).
a diminishing return
is often seen in computer projects in real life; adding staff increases productivity to a point, but there is an upper limit to that increase as each staff member also requires training and supervision. This is just one way that interpreting data can be more complicated than it appears at first glance.
To do more complex input/output problems, it is vital to use the order of operations:
parentheses, exponents, multiplication and division (from left to right), and addition and subtraction (from left to right).
Estimate s(t)'s value at t = 12.
s(12)≈2,750
There are two important aspects of linear functions
starting values and slopes.
This second common function can be modeled by: f(x) = mx +b
where m and b are constants (that is, numbers that do not change) and x is the input variable. For example, a software sales representative makes a base salary of $40,000 a year, plus 10% commission on her sales; her annual earnings are modeled by this function: E(x)=40,000+0.10x, where E represents her earnings and x is the amount of sales.
The most basic type of function is f(x) = mx
where m is a number multiplied by the input, x. You actually use this function every day, though you might not think of it in these mathematical terms just yet Consider these examples: A real estate agent generally gets paid 5% commission based upon the sale price of a property. The agent's pay can be modeled as: C(p)=0.05p, where C represents the commission and p is the price of the property.
Solving linear equation f(t)=−0.0125t+1
you multiply first and then add.
