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Using tabular form to write an equation for a linear function

[link] relates the number of rats in a population to time, in weeks. Use the table to write a linear equation.

w , number of weeks 0 2 4 6
P(w) , number of rats 1000 1080 1160 1240

We can see from the table that the initial value for the number of rats is 1000, so b = 1000.

Rather than solving for m , we can tell from looking at the table that the population increases by 80 for every 2 weeks that pass. This means that the rate of change is 80 rats per 2 weeks, which can be simplified to 40 rats per week.

P ( w ) = 40 w + 1000

If we did not notice the rate of change from the table we could still solve for the slope using any two points from the table. For example, using ( 2 , 1080 ) and ( 6 , 1240 )

m = 1240 1080 6 2    = 160 4    = 40
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Is the initial value always provided in a table of values like [link] ?

No. Sometimes the initial value is provided in a table of values, but sometimes it is not. If you see an input of 0, then the initial value would be the corresponding output. If the initial value is not provided because there is no value of input on the table equal to 0, find the slope, substitute one coordinate pair and the slope into f ( x ) = m x + b , and solve for b .

A new plant food was introduced to a young tree to test its effect on the height of the tree. [link] shows the height of the tree, in feet, x months since the measurements began. Write a linear function, H ( x ) , where x is the number of months since the start of the experiment.

x 0 2 4 8 12
H ( x ) 12.5 13.5 14.5 16.5 18.5

H ( x ) = 0.5 x + 12.5

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Access this online resource for additional instruction and practice with linear functions.

Key equations

slope-intercept form of a line f ( x ) = m x + b
slope m = change in output (rise) change in input (run) = Δ y Δ x = y 2 y 1 x 2 x 1
point-slope form of a line y y 1 = m ( x x 1 )

Key concepts

  • The ordered pairs given by a linear function represent points on a line.
  • Linear functions can be represented in words, function notation, tabular form, and graphical form. See [link] .
  • The rate of change of a linear function is also known as the slope.
  • An equation in the slope-intercept form of a line includes the slope and the initial value of the function.
  • The initial value, or y -intercept, is the output value when the input of a linear function is zero. It is the y -value of the point at which the line crosses the y -axis.
  • An increasing linear function results in a graph that slants upward from left to right and has a positive slope.
  • A decreasing linear function results in a graph that slants downward from left to right and has a negative slope.
  • A constant linear function results in a graph that is a horizontal line.
  • Analyzing the slope within the context of a problem indicates whether a linear function is increasing, decreasing, or constant. See [link] .
  • The slope of a linear function can be calculated by dividing the difference between y -values by the difference in corresponding x -values of any two points on the line. See [link] and [link] .
  • The slope and initial value can be determined given a graph or any two points on the line.
  • One type of function notation is the slope-intercept form of an equation.
  • The point-slope form is useful for finding a linear equation when given the slope of a line and one point. See [link] .
  • The point-slope form is also convenient for finding a linear equation when given two points through which a line passes. See [link] .
  • The equation for a linear function can be written if the slope m and initial value b are known. See [link] , [link] , and [link] .
  • A linear function can be used to solve real-world problems. See [link] and [link] .
  • A linear function can be written from tabular form. See [link] .

Questions & Answers

find the equation of the line if m=3, and b=-2
Ashley Reply
graph the following linear equation using intercepts method. 2x+y=4
ok, one moment
how do I post your graph for you?
it won't let me send an image?
also for the first one... y=mx+b so.... y=3x-2
y=mx+b you were already given the 'm' and 'b'. so.. y=3x-2
Please were did you get y=mx+b from
y=mx+b is the formula of a straight line. where m = the slope & b = where the line crosses the y-axis. In this case, being that the "m" and "b", are given, all you have to do is plug them into the formula to complete the equation.
"7"has an open circle and "10"has a filled in circle who can I have a set builder notation
Fiston Reply
x=-b+_Гb2-(4ac) ______________ 2a
Ahlicia Reply
I've run into this: x = r*cos(angle1 + angle2) Which expands to: x = r(cos(angle1)*cos(angle2) - sin(angle1)*sin(angle2)) The r value confuses me here, because distributing it makes: (r*cos(angle2))(cos(angle1) - (r*sin(angle2))(sin(angle1)) How does this make sense? Why does the r distribute once
Carlos Reply
so good
this is an identity when 2 adding two angles within a cosine. it's called the cosine sum formula. there is also a different formula when cosine has an angle minus another angle it's called the sum and difference formulas and they are under any list of trig identities
strategies to form the general term
consider r(a+b) = ra + rb. The a and b are the trig identity.
How can you tell what type of parent function a graph is ?
Mary Reply
generally by how the graph looks and understanding what the base parent functions look like and perform on a graph
if you have a graphed line, you can have an idea by how the directions of the line turns, i.e. negative, positive, zero
y=x will obviously be a straight line with a zero slope
y=x^2 will have a parabolic line opening to positive infinity on both sides of the y axis vice versa with y=-x^2 you'll have both ends of the parabolic line pointing downward heading to negative infinity on both sides of the y axis
y=x will be a straight line, but it will have a slope of one. Remember, if y=1 then x=1, so for every unit you rise you move over positively one unit. To get a straight line with a slope of 0, set y=1 or any integer.
yes, correction on my end, I meant slope of 1 instead of slope of 0
what is f(x)=
Karim Reply
I don't understand
Typically a function 'f' will take 'x' as input, and produce 'y' as output. As 'f(x)=y'. According to Google, "The range of a function is the complete set of all possible resulting values of the dependent variable (y, usually), after we have substituted the domain."
Sorry, I don't know where the "Â"s came from. They shouldn't be there. Just ignore them. :-)
It is the  that should not be there. It doesn't seem to show if encloses in quotation marks. "Â" or 'Â' ... Â
Now it shows, go figure?
what is this?
unknown Reply
i do not understand anything
lol...it gets better
I've been struggling so much through all of this. my final is in four weeks 😭
this book is an excellent resource! have you guys ever looked at the online tutoring? there's one that is called "That Tutor Guy" and he goes over a lot of the concepts
thank you I have heard of him. I should check him out.
is there any question in particular?
I have always struggled with math. I get lost really easy, if you have any advice for that, it would help tremendously.
Sure, are you in high school or college?
Hi, apologies for the delayed response. I'm in college.
how to solve polynomial using a calculator
Ef Reply
So a horizontal compression by factor of 1/2 is the same as a horizontal stretch by a factor of 2, right?
The center is at (3,4) a focus is at (3,-1), and the lenght of the major axis is 26
Rima Reply
The center is at (3,4) a focus is at (3,-1) and the lenght of the major axis is 26 what will be the answer?
I done know
What kind of answer is that😑?
I had just woken up when i got this message
Can you please help me. Tomorrow is the deadline of my assignment then I don't know how to solve that
i have a question.
how do you find the real and complex roots of a polynomial?
@abdul with delta maybe which is b(square)-4ac=result then the 1st root -b-radical delta over 2a and the 2nd root -b+radical delta over 2a. I am not sure if this was your question but check it up
This is the actual question: Find all roots(real and complex) of the polynomial f(x)=6x^3 + x^2 - 4x + 1
@Nare please let me know if you can solve it.
I have a question
hello guys I'm new here? will you happy with me
The average annual population increase of a pack of wolves is 25.
Brittany Reply
how do you find the period of a sine graph
Imani Reply
Period =2π if there is a coefficient (b), just divide the coefficient by 2π to get the new period
if not then how would I find it from a graph
by looking at the graph, find the distance between two consecutive maximum points (the highest points of the wave). so if the top of one wave is at point A (1,2) and the next top of the wave is at point B (6,2), then the period is 5, the difference of the x-coordinates.
you could also do it with two consecutive minimum points or x-intercepts
I will try that thank u
Case of Equilateral Hyperbola
Jhon Reply
f(x)=4x+2, find f(3)
f(3)=4(3)+2 f(3)=14
pre calc teacher: "Plug in Plug in...smell's good" f(x)=14
Explain why log a x is not defined for a < 0
Baptiste Reply
the sum of any two linear polynomial is what
Esther Reply
Practice Key Terms 7

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