Graph Transformations
There are many times when you’ll know very well what the graph of a
particular function looks like, and you’ll want to know what the graph of a
very similar function looks like. In this chapter, we’ll discuss some ways to
draw graphs in these circumstances.
Transformations “after” the original function
Suppose you know what the graph of a function f(x) looks like. Suppose
d 2 R is some number that is greater than 0, and you are asked to graph the
function f(x) + d. The graph of the new function is easy to describe: just
take every point in the graph of f(x), and move it up a distance of d. That
is, if (a, b) is a point in the graph of f(x), then (a, b + d) is a point in the
graph of f(x) + d.
As an explanation for what’s written above: If (a, b) is a point in the graph
of f(x), then that means f(a) = b. Hence, f(a) + d = b + d, which is to say
that (a, b + d) is a point in the graph of f(x) + d.
The chart on the next page describes how to use the graph of f(x) to create
the graph of some similar functions. Throughout the chart, d > 0, c > 1, and
(a, b) is a point in the graph of f(x).
Notice that all of the “new functions” in the chart di↵er from f(x) by some
algebraic manipulation that happens after f plays its part as a function. For
example, first you put x into the function, then f(x) is what comes out. The
function has done its job. Only after f has done its job do you add d to get
the new function f(x) + d. 67Because all of the algebraic transformations occur after the function does
its job, all of the changes to points in the second column of the chart occur
in the second coordinate. Thus, all the changes in the graphs occur in the
vertical measurements of the graph.
New How points in graph of f(x) visual e↵ect
function become points of new graph
f(x) + d (a, b) 7! (a, b + d) shift up by d
f(x) Transformations before and after the original function
As long as there is only one type of operation involved “inside the function”
– either multiplication or addition – and only one type of operation involved
“outside of the function” – either multiplication or addition – you can apply
the rules from the two charts on page 68 and 70 to transform the graph of a
function.
Examples.
• Let’s look at the function • The graph of 2g(3x) is obtained from the graph of g(x) by shrinking
the horizontal coordinate by 1
3, and stretching the vertical coordinate by 2.
(You’d get the same answer here if you reversed the order of the transfor-
mations and stretched vertically by 2 before shrinking horizontally by 1
3. The
order isn’t important.)
74
7:—
(x) 4,
7c’
‘I
II
‘I’
-I
Which equation best represents the weight, w, of this alligator, n, weeks after hatching?
Answer:pretty sure its 0.25n+6
Step-by-step explanation: that seems the most reasonable
Someone please help me with this question
Answer:
Jinsoul from Loona
Step-by-step explanation:
In order to solve this problem, you must do the famous math operation: Jinsoul from Loona
Have a nice day!
Please help me.
What is the equation of the line?
Answer:
[tex]y = \frac{-7}{3}x -12[/tex]
Step-by-step explanation:
Point-slope form is [tex]y-y_1= m (x-x_1)[/tex]. The [tex]m[/tex], [tex]x_1[/tex], and [tex]y_1[/tex] have to be substituted for with real values in order to make an equation of a line. From there, you can isolate [tex]y[/tex] to turn it into slope-intercept form.
1) First, let's find the slope of the line, which is represented by the variable [tex]m[/tex]. Use the slope formula, [tex]\frac{y_2-y_1}{x_2-x_1}[/tex], to find the slope. [tex]x_1[/tex] and [tex]y_1[/tex] are the x and y values of one point, [tex]x_2[/tex] and [tex]y_2[/tex] are the x and y values of another point. For this problem, let's use the given points (-3, -5) and (-6, 2):
[tex]\frac{(2)-(-5)}{(-6)-(-3)} \\= \frac{2+5}{-6+3} \\= \frac{7}{-3}[/tex]
Thus, the slope is [tex]\frac{7}{-3}[/tex].
2) Now that we have the slope, let's substitute it into the point-slope formula. The [tex]x_1[/tex] and [tex]y_1[/tex] represent the x and y values of a point that the line intersects so you could either use (-3, -5) or (-6,2), but for this answer, I used (-6,2). (The answer will always be the same no matter which point you choose.) Substitute these values then isolate y to put it in slope-intercept form:
[tex]y-(2) = \frac{-7}{3} (x-(-6))\\y -2 = \frac{-7}{3} (x+6)\\y-2 = \frac{-7}{3}x-\frac{42}{3} \\y -2 = \frac{-7}{3}x-14\\y = \frac{-7}{3}x-12[/tex]
So, [tex]y = \frac{-7}{3}x -12[/tex] is the answer.
Which of the following proportions is true?
12 pizzas
18 people
9 pizzas
24 people
24 pizzas
48 people,
8. pizzas
20 people
20 pizzas
60 people
36 pizzas
96 people
48 pizzas
24 people
48 pizzas
60 people
Consider the complementary angles shown below. Find the value of x. X + 4 3x + 2
Answer:
x = 21
Step-by-step explanation:
For the complementary angles, the sum of two angles is equal to 90 degrees.
ATQ,
3x+2+x+4 = 90
Taking like terms together,
3x+x+6=90
4x+6=90
4x = 84
x = 21
So, the value of x is equal to 21.
A fish swims 30 miles every 5 hours. How many miles does the fish swim each hour?
A. 15 miles
B. 5 miles
C. 150 miles
D. 6 miles
Answer:
6 miles
Step-by-step explanation:
30 divided by 5 equals six, therefore six miles.
Answer:
i think it is 6
Step-by-step explanation:
you divided 30 by 5
Jorge can travel 50 miles on his
motorized scooter using 1 gallon of gas.
How many miles can he travel using
8 gallons of gas?
Answer:
its c
Step-by-step explanation:
Answer:
400 miles
Step-by-step explanation:
50 miles.....1 gallon
x miles......8 gallons
50*8=400
what is 25% of 12???
Answer:
3
Step-by-step explanation:
%25 is 1/4
1/4 of 12 is 3
Answer:
3
Step-by-step explanation:
25% is equal to 1/4. Of means multiply. Therefore, you would just multiply 1/4 and 12. Your answer should be 3.
What's 27.63 subtracted by 60
Answer:
-32.37
Step-by-step explanation:
Answer:
-32.37
Step-by-step explanation:
HELPPPPPPP PLEASSEEE
Waiting overnight
You should never wait overnight before making a big purchase if there is only one item left. Teens have cited ʺfriendsʺ as the strongest influence over their purchase decisions.
Answer:
the first one
Step-by-step explanation:
because my mom shop and she do not NEVER go to sleep it will mess up your purachase
Leon's older brother is 4 3/4 feet tall. Leon is 1/3 foot shorter than his brother . How tall is leon?
Let
x------> height of Leon's older brother
y------> height of Leon
Convert mixed number to an improper fraction
4\frac{3}{4}\ ft=\frac{4*4+3}{4}= \frac{19}{4}\ ft443 ft=44∗4+3=419 ft
we know that
x=\frac{19}{4}\ ftx=419 ft -----> equation A
y=x-\frac{1}{3}y=x−31 -----> equation B
substitute equation A in equation B
y=\frac{19}{4}-\frac{1}{3}=\frac{53}{12}\ fty=419−31=1253 ft
Convert to mixed number
\frac{53}{12}=\frac{48}{12}+\frac{5}{12}=4\frac{5}{12}\ ft1253=1248+125=4125 ft
therefore
the answer is
4\frac{5}{12}\ ft4125 ft
Answer:
4 3/4 - 1/3 = 4 5/12
Step-by-step explanation:
4 9/12 - 4/12 = 4 5/12
Andrea paid $60.75 for 9 sandwiches. Each sandwich costs the same amount.
Use the drop down menus to write an equation for the price, p, of one sandwich.
P x _______=____??
Answer:
Px9=60.75
Step-by-step explanation:
Y=2x^2 - 8x+3 graph the equation
Answer:
Graph is below.
Step-by-step explanation:
The solutions of the equations are : x = [8 ± √ 40] / 4
Given,
Y=2x² - 8x+3
Now,
To get the solution of equation equate it to 0 .
2x² - 8x+3 = 0
By quadratic formula,
x = -b ± √ b² - 4ac/2a
Substitute the values in the formula,
x = [8 ± √ 64 - 4(2)(3)] / 2* 2
x = [8 ± √ 40] / 4
Now plot the points in the graph. The graph is attached below.
Know more about equations,
https://brainly.com/question/17181370
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An angle measures 80° less than the measure of its complementary angle. What is the measure of each angle
Answer:
let the angle be x and the complementary angle be y
so, x+y=90 and given,
x=y-80
solving both the equations,
x = 5
y = 85
so the angles are 5 and 85
Find the perimeter of the figure to the right
6 yards
8 4/5 yard
9 2/3 yards
8 yards
2.7.–2.5 = 6.75
A . agree
b. disagree; a negative times a negative is positive
C. disagree; a positive times a negative is negative.
PLEASE answer quick
HELP PLZ !!!! The graph and the table below show the relationship between the number of servings and the amount of protein in two different snacks.
How many servings of wheat crackers have the same number of grams of protein as one serving of mixed nuts?
Answer:
exactly 1 (I Think)
Step-by-step explanation:
3
Answer:
exactly 1
Step-by-step explanation:
it is
Can someone help me I will give the brainliest!!
Answer & Step-by-step explanation:
2. 4 times (*) f equals (=) 1/2
Equation: 4f=1/2
4 times (*) 1/8 equals (=) 1/2
f = 1/8
3. 832 divided by (/) n equals (=) 16
Equation: 832/n=16
832 divided by (/) 52 equals (=) 16
n = 52
4. 10 subtracted from (-) x equals (=) 6
Equation: 10-x=6
10 - 4 = 6
x = 4
Hope this helps ;)
Consider the monthly payment formula M=Pr(1+r)^n/(1+r)^n-1. what is the name of the formula you get if you solve for P? what about if you solve for n? what special function do you need to use when solving for n?
WRITTEN RESPONSE QUESTION PLEASE ANSWER ASAP WILL AWARD BRAINLIEST.
Answer:
0.0061
Step-by-step drip check:
The name of the formula you get if you solve for P could be "Principal (loan amount)" "Term of the loan (in months)" or "Number of monthly payments"
Logarithms;
[tex]n = \dfrac{ ln[1 - \dfrac{Pr}{M} ]}{ln(1+r)}[/tex]
What is Monthly Payment Formula?Consider the monthly payment formula
[tex]M= \dfrac{Pr(1+r)^n}{(1+r)^n-1}[/tex]
where: M = monthly payment.
P = principal loan amount.
r = interest rate per month (in decimal form).
n = term of the loan (in months).
If we will solve for P, the name of the formula is "Principal (loan amount)".
If we solve for n, the formula is "Term of the loan (in months)" or "Number of monthly payments".
When solving for n, we have to use logarithms:
[tex]M (1+r)^n-1=Pr(1+r)^n\\\\\dfrac{1}{ (1+r)^n} = 1 - \dfrac{Pr}{M} \\\\ln\dfrac{1}{ (1+r)^n} = ln[1 - \dfrac{Pr}{M} ]\\\\\\n = \dfrac{ ln[1 - \dfrac{Pr}{M} ]}{ln(1+r)}[/tex]
Learn more about monthly payments questions;
https://brainly.com/question/29766847
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make b the subject of the formula a=squareroot of b+16
Step-by-step explanation:
a=
[tex] a = \sqrt{b + 16} \\ a { }^{2} = b + 16 \\ b = a {}^{2} - 16[/tex]
Each letter in SYNONYM is written on a piece of paper and randomly chosen from a bag.What are the odds for choosing an Y?
2/7 or 2:7 or about 28.5%
Answer: 2/7 or 2 out of 7
Step-by-step explanation: because there are 2 y's in the word synonym
Look at Jerrys monthly budget below. What percentage of Jerrys net income is spent on transportation?
Net Monthly Income: 2,970.00
Housing: $750.00
Utilities: $120.00
Food: $450.00
Transportation: $650.00
Savings: $300.00
Other Expenses: $500.00
Total Monthly Expenses: 2,970.00
A. 28%
B. 25.3%
C. 86.7%
D. 21.9%
Answer:
D. 21.9%
Step-by-step explanation:
(650÷2970)x100= 21.88% rounded off 21.9%
Help pls! 37 points.
Answer:
7 squares
Step-by-step explanation:
Key skills: Counting, Averages
1) Based on what I counted, there are 11 columns, and 53 squares. If you want an average of 5 squares per column and you have 12 columns, the total number of squares you want is 60.
2) Since there are already 53 squares present, you need 60-53 = 7 squares more.
Hope you understood and have a nice day!!
answer:
should be 7 squares.
i need help on question 4 A. Can someone please show working out too so I can understand it, thank you!
Answer:
[tex]Q=(9,11)[/tex]
Step-by-step explanation:
Solving (4a)
[tex]P (x_1,y_1) = (1,1)[/tex]
[tex]M (x,y) = (5,6)[/tex]
Required
Determine [tex]Q(x_2,y_2)[/tex]
To do this, we make use of the mid-point formula:
[tex]M(x,y) = (\frac{x_1+x_2}{2},\frac{y_1+y_2}{2})[/tex]
Substitute values for x, y, x1 and y1
[tex](5,6) = (\frac{1+x_2}{2},\frac{1+y_2}{2})[/tex]
Multiply through by 2
[tex]2 * (5,6) = (\frac{1+x_2}{2},\frac{1+y_2}{2}) * 2[/tex]
[tex](10,12) = (1+x_2,1+y_2)[/tex]
By comparison:
[tex]10 = 1 + x_2[/tex]
[tex]x_2 = 10 - 1[/tex]
[tex]x_2 = 9[/tex]
[tex]12 = 1 + y_2[/tex]
[tex]y_2 = 12 - 1[/tex]
[tex]y_2 = 11[/tex]
Hence, the coordinates of Q is (9,11)
HELLPP PLEASE MY TEACHER KEEPS ASKING FOR ANSWER LOL IM SORRY I DONT HAVE MUCH POINTS :(
Answer:
-1/6
Step-by-step explanation:
Multiply numerators and denominators
4*3
9*6
You get -12/72
Simplify
Divide 12 by both numbers
You get -1/6
Answer:
[tex] - \frac{1}{6} [/tex]
Step-by-step explanation:
Multiplying a positive and a negative equals a negative: ( + ) × ( - ) = ( - )
[tex] - \frac{4}{9} \times \frac{3}{8} [/tex]
Reduce the number with the greatest common factor 4:
[tex] - \frac{1}{9} \times \frac{3}{2} [/tex]
Reduce the number with the greatest common factor 3:
[tex] - \frac{1}{3} \times \frac{1}{2} [/tex]
Multiplying the fractions:
[tex]1 \times 1 = 1 \\ 3 \times 2 = 6[/tex]
Therefore, the answer will be - 1/6.
12. The math department at a small school has 5 teachers. The ages of these teachers are 23, 34, 37, 42, and 58.Suppose you selected a random sample of 3 teachers and calculated the sample median.(a) List all 10 possible samples of size 3. Calculate the sample median for each sample.
Answer:
[tex]\begin\\\begin{tabular}{|l|l|l|}No&Sample&Median\\01&58,42,37 &42\\02&58,42,34&42\\03&58,42,23& 42\\04&37,34,23 & 34\\05&58,37,34 & 37\\06&58,37,23 & 37\\07&58,34,23 &34\\08&42,37,34 &37\\09&42,37,23 &37\\10&42,34,23 &34\end[/tex]
Step-by-step explanation:
From the question we are told that
5 samples(23 34 37 42 58)
To select 3 random sample
a) Generally to find all 10 possible outcomes and there median we make a table
Therefore
L1
58,42,37 Median=42
L2
58,42,34 Median=42
L3
58,42,23 Median=42
L4
37,34,23 Median=34
L5
58,37,34 Median=37
L6
58,37,23 Median=37
L7
58,34,23 Median=34
L8
42,37,34 Median=37
L9
42,37,23 Median=37
L10
42,34,23 Median=34
[tex]\begin\\\begin{tabular}{|l|l|l|}No&Sample&Median\\01&58,42,37 &42\\02&58,42,34&42\\03&58,42,23& 42\\04&37,34,23 & 34\\05&58,37,34 & 37\\06&58,37,23 & 37\\07&58,34,23 &34\\08&42,37,34 &37\\09&42,37,23 &37\\10&42,34,23 &34\end[/tex]
Whoever answers it first gets the title brainliest.
Answer:
2.24
Explanation:
2.5 - 2.4 / 1.2(0.13) = 2.24
A rectangle has a width measuring 6 units and a length measuring x + 4 units. Write and solve an inequality to find the value of x so that the area of the rectangle is more than 36 square units. Justify your answer.
Answer:
its easy the anser is x=40
Step-by-step explanation:
you just add the numbers
an oil tank loses 32% of its contents every hour. Peter says the tank will have lost 95% of its original contents by the end of the sixth hour. Is peter correct
Answer:
it lose 90 percent in 3 hours so peter is incorrect
Step-by-step explanation:
What is the solution of the system of equations? (_,_)
Answer:
0, 5
Step-by-step explanation: