After simplifying each proportion or using cross-multiplication, it is clear that option B.) 18/48 = 20/50 is the false proportion.
For A.) 24/30 = 20/25, we can simplify by dividing both sides by 5. Simplified, it becomes 4/5 = 4/5, which is true.
For B.) 18/48 = 20/50, simplifying both sides by 2 gives us 9/24 = 10/25, which is not equivalent since 9x25 is not equal to 24x10. Therefore, this proportion is false.
For C.) 25/45 = 75/135, simplifying by dividing both numbers by 15 gives us 5/9 = 5/9, which is true.
For D.) 10/25 = 40/100, simplifying by dividing both sides by 10 gives us 1/2.5 = 4/10. If we further simplify 4/10 by dividing both the numerator and denominator by 2, we get 1/2.5 = 2/5, which is true after rounding the decimal in the first ratio to 2.5.
Comparing these results, option B is the false proportion.
The Big Burger recipe calls for 3 meat patties, 2 slices of cheese, and four pickles. How many meat patties are needed if 52 pickles are used?
Answer:
39 meat patties
Step-by-step explanation:
Given,
The Big Burger recipe calls for 3 meat patties, 2 slices of cheese and 4 pickles,
That is, the ratio of meat patties, cheese and pickles in the recipe = 3 : 2 : 4
Let in a recipe,
Meat patties = 3x, Cheese = 2x, pickles = 4x
Where, x is a positive real number,
If there are 52 pickles,
⇒ 4x = 52 ⇒ x = 13
Hence, meat patties = 3 (13) = 39
A rectangle's length is 4 feet more than its width. Write a quadratic function
that expresses the rectangle's area in terms of its width.
A. A(W) = w^2 – 4w
B. A(w)= w^2+4w
c. A(w)=w+4
D. A(w) = lw
Answer:
B. A(w)=w∧2+4w
Step-by-step explanation:
Let the width of the rectangle be w.
The length is 4 feet longer than the width= w+4
Area of a rectangle= length× width
A=L×W
=(w+4)×w
Opening the brackets gives the:
A=w²+4w
Therefore the expression for the area in terms of width is A(w)=w∧2+4w
Dante is standing at horizontal ground level with the base of the Empire State Building in New York City. The angle formed by the ground and the line segment from his position to the top of the building is 48.4°. The height of the Empire State Building is 1,472 feet. Find his distance from the Empire State Building to the nearest foot.
A. 7.65 ft
B. 1, 968 ft
C. 1,307 ft
D. 2, 217 ft
Answer:
C. 1307 ft
Step-by-step explanation:
Given:
Angle = 48.4 degrees
Height, opposite side= 1472 feet
his distance from the Empire State Building, base=x
Now as per the trigonometric ratios:
Tan∅= Opposite/base
tan(48.4)= 1472/x
x=1472/(1.13)
x=1302.65
his distance from the Empire State Building is 1302.65 feet!
Answer:
The correct answer is option C.
Step-by-step explanation:
Height of Empire State Building = 1,472 feet
Angle formed by the line segment from the point of ground on which Dante is positioned to the top of the building is 48.4°.
Distance of Dante from the Empire State Building =?
In the fig ,ΔABC
AB = 1,472 feet, BC = ? , θ= 48.4°
[tex]\tan\theta =\frac{Perpendicular}{base}[/tex]
[tex]\tan 48.4^o=\frac{AB}{BC}[/tex]
[tex]BC=\frac{AB}{\tan 48.4^o}=\frac{1,472 feet}{1.1263}=1,306.9 feet\approx 1,307 feet[/tex]
Distance of Dante from the Empire State Building is 1,307 feet.
Consider the function f(x)=|x+3|−5 and its graph, which follows.
An absolute value function with vertex (negative 3, negative 5). It passes through (negative 8, 0) & (2, 0).
Suppose the function is transformed by the function g(x) = −1/5f(x).
Please graph response
Answer:
The graph in the attached figure
Step-by-step explanation:
we have
[tex]f\left(x\right)=\left|x+3\right|-5[/tex]
Obtain the function g(x)
[tex]g(x)=-\frac{1}{5} f(x)[/tex]
substitute
[tex]g(x)=-\frac{1}{5} [\left|x+3\right|-5][/tex]
[tex]g(x)=-\frac{1}{5}\left|x+3\right|+1[/tex]
using a graphing tool
The graph in the attached figure
The vertex is the point (-3,1)
The x-intercepts are the points (-8,0) and (2,0)
The y-intercept is the point (0,0.4)
Answer:
n
Step-by-step explanation:
34 + 3 ⋅ 5 = ____. (Input only whole numbers.)
[tex]34+3\cdot5=34+15=49[/tex]
Hope this helps.
r3t40
Answer:
[tex]\huge \boxed{49}\checkmark[/tex]
Step-by-step explanation:
Order of operations
PEMDAS
Parenthesis
Exponent
Multiply
Divide
Add
Subtract
Do multiply first.
[tex]\displaystyle 34+3\times5[/tex]
[tex]\displaystyle 3\times5=15[/tex]
Add from left to right to find the answer.
[tex]\displaystyle 34+15=49[/tex]
[tex]\huge \boxed{49}[/tex], which is our answer.
Hope this helps!
A 3-digit numeral is formed by selecting digits at random from 2,4,6,7 without repetition. Find the probability that the number is formed greater than 600. P(greater than 600)
The probability that the number is formed greater than 600 is [tex]\frac{1}{2}[/tex].
What is probability?Probability is the chance that something will happen, or how likely it is that an event will occur.
What is the formula for the probability?The formula for the probability is
[tex]P(E) = \frac{number \ of \ favorable \ outcomes }{Total\ number\ of\ outcomes}[/tex]
Where,
P(E) is the probability of any event.
What is permutation?A permutation is a mathematical technique that determines the number of possible arrangements in a set when the order of the arrangements matters.
What is the formula for the permutation?The formula for the permutation is given by
[tex]^{n} P_{r} = \frac{n!}{(n-r)!}[/tex]
Where,
[tex]^{n} P_{r}[/tex] is the permutation
n is the total number of objects
r is the total number of objects to be selected
According to the given question.
We have total four numbers 2, 4, 6, 7.
So,
The total number of three digits can be formed using these four numbers = [tex]^{4} P_{3}[/tex] = [tex]\frac{4!}{(4-3)!} =\frac{4\times 3\times 2\times 1}{1}[/tex][tex]=24[/tex]
Now, for making three digits number which are greater than 600 by using 2, 4, 6, 7 without repetition is given by
Number of ways for filling hundred place is 2 (either 6 or 7).
Number of ways for filling tens place is 3 (if 6 is placed at hundred place then remaining numbers are 7, 2, 4 and if 7 is place at hundred place then remaining numbers are 6, 2, 4).
Number of ways for filling one place is 2(because only 2 number are left).
Therefore, the total numbers of three digits can be formed by using these numbers 2, 4, 6, and 7
[tex]= 2\times 3\times 2\\=12[/tex]
So,
the probability that the number is formed greater than 600
= [tex]\frac{total\ three\ digits\ numbers\ which\ are \ formed \ by\ using\ 1,\ 2, \ 3, \ and\ 4\ which\ are\ greater\ than\ 600 }{Total \ three\ digits\ numbers\ formed\ by \ using \ 1,\ 2,\ 3,\ and \ 4}[/tex]
[tex]= \frac{12}{24}[/tex]
[tex]= \frac{1}{2}[/tex]
Therefore, the probability that the number is formed greater than 600 is [tex]\frac{1}{2}[/tex].
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5,731÷34 show ur work
Answer:
168.558
Step-by-step explanation:
168.58
34√5731.0
-34 ↓3
23 3
-204 ↓1
28 1
- 272 ↓0
19 0
-170 ↓0
20 0
-170 ↓0
30 0
-272 ↓0
28 0
and just goes on..
8. Point O is the circumcenter of the triangle ABC shown below.
Which segment passes through point O for all lengths of sides of the triangle?
A. angle bisector of angle ABC
B. perpendicular bisector of side AB
C. a line segment drawn from vertex C to bisect side AB
D. a line segment drawn from vertex A to cut side BC at right angles
The circumcenter O, formed by the intersection of the perpendicular bisectors of sides AB and BC, is equidistant from all vertices, with OA = OB = OC. Here option B is correct.
In a triangle, the circumcenter is the point where the perpendicular bisectors of its sides intersect. In this case, the circumcenter O is formed by the intersection of the perpendicular bisector of side AB and the perpendicular bisector of side BC.
The circumcenter is equidistant from all three vertices, making OA, OB, and OC equal, and these distances represent the radius of the circumcircle.
This line not only bisects side AB but also intersects with the perpendicular bisector of side BC at the circumcenter O. The equality of OA, OB, and OC ensures that the circumcircle passes through all three vertices of the triangle, making it a significant point in the context of the triangle's geometry. Here option B is correct.
I need help putting this in corresponding factored form. I got two wrong but I’m not sure how to do it and show my work.
Answer:
x^2-16 goes with (x+4)(x-4)
x^2+10x+16 goes with (x+8)(x+2)
Step-by-step explanation:
The first one you got wrong is known as a difference of squares.
To factor a difference of squares, a^2-b^2, you just write it as (a-b)(a+b) or (a+b)(a-b) would work too.
So x^2-16=(x-4)(x+4) or (x+4)(x-4).
Let's check (x+4)(x-4) using foil!
First: x(x)=x^2
Outer: x(-4)=-4x
Inner: 4(x)=4x
Last: 4(-4)=-16
----------------------Add
x^2-16
Bingo! (x+4)(x-4) definitely corresponds to x^2-16.
Here are more examples of factoring a difference of squares:
Example 1: x^2-25 = (x+5)(x-5)
Example 2: x^2-81 = (x+9)(x-9)
Example 3: x^2-100 =(x+10)(x-10)
Onward to the next problem:
x^2+10x+16
When the coefficient of the leading term of a quadratic is 1, all you have to do is find two numbers that multiply to be c=16 and add up be b=10.
Those numbers would be 8 and 2
because 8(2)=16 and 8+2=10.
So the factored form of x^2+10x+16 is (x+2)(x+8) or (x+8)(x+2).
Here is another example of when the leading coefficient of a quadratic is 1:
Example 1: x^2+5x+6=(x+2)(x+3) since 3(2)=6 and 3+2=5.
Example 2: x^2-x-6=(x-3)(x+2) since -3(2)=-6 and -3+2=-1.
Algebra 2 help please ASAP
Answer:
The option A,D and E are correct.
Step-by-step explanation:
Given: 2x^3-250x^2
Factor : 2x^2(x-125)
So, GCF = 2x^2
Now a = 1 and b= 5
we know that a^3-b^3 = (a-b)(a^2+ab+b^2)
(x)^3 - (5)^3 = (x-5)(x^2+5x+25)
So, the option A,D and E are correct.
Please help!!!!!!!!!!!!!!!!!!
Answer:
1) 95
2) -12
3) 7
4) 1,700
5) 57
6) 3,070
What is the smallest positive integer that will make x^x > 500,000? What
is the largest negative integer that will make x^(-x) > 500,000?
Answer:
For [tex]x^x > 500,000[/tex] [tex]x=7[/tex]
For [tex]x^{(-x)} > 500,000[/tex] [tex]x=-7[/tex]
Step-by-step explanation:
We need to find the smallest positive whole number that satisfies the inequality:
[tex]x^x > 500,000[/tex]
We tested with x = 6
[tex]6^6=46,656\\\\46,656 > 500,000[/tex]
Inequality is not met because [tex]46,656 < 500,000[/tex]
We test with the following integer x = 7
Then we have that:
[tex]7^7=823,543\\\\823,543 > 500,000[/tex]
Then the smallest positive integer that will make [tex]x^x > 500,000[/tex] is 7 because Inequality is met.
In the same way the largest negative integer that will make [tex]x^{(-x)} >500000[/tex] is [tex]x=-7[/tex] Beacuse [tex]7^{-(-7)}=823,543>500,000[/tex]
Answer:
Smallest positive integer value for [tex]x^x>500000[/tex] is,
x = 7,
Largest negative integer value for [tex]x^{-x}>500000[/tex] is,
x = -8
Step-by-step explanation:
If [tex]x^x>500000[/tex]
By graphing calculator,
[tex]x>6.83[/tex]
Thus, the smallest possible positive integer value of x is 7,
Now,
[tex]x^{-x}>500000[/tex]
Possible negative integer values of x are -6, -7 and -8,
If x = -6, -7, and -8,
[tex](-6)^{6}=46656[/tex]
[tex](-7)^{7}=-823543[/tex]
[tex](-8)^{8}=16777216[/tex]
[tex]\because 16777216 > 500000[/tex]
Thus, the largest negative integer value of the inequality [tex]x^{-x}>500000[/tex] is,
x = -8.
Find the distance between (0,4) and (3,-1)
Answer:
see explanation
Step-by-step explanation:
Calculate the distance (d) using the distance formula
d = √ (x₂ - x₁ )² + (y₂ - y₁ )²
with (x₁, y₁ ) = (0, -4) and (x₂, y₂ ) = (3, - 1)
d = [tex]\sqrt{(3-0)^2+(-1+4)^2}[/tex]
= [tex]\sqrt{3^2+ 3^2}[/tex]
= [tex]\sqrt{9+9}[/tex]
= [tex]\sqrt{18}[/tex] = 3[tex]\sqrt{2}[/tex] ≈ 4.24 ( to 2 dec. places )
Simplify 7(x - 2) - 4x + 9
Answer:
3x - 5
Step-by-step explanation:
Perform the indicated multiplication. We get:
7x - 14 - 4x + 9.
Combining like terms, we get 3x - 5
The office manager at a small law firm has taken a survey on how many cups of coffee each person drinks per 5-day work week. A table of her results is below.
Employee Cups per Week
1 29
2 13
3 27
4 26
5 9
6 15
7 17
8 19
9 25
10 32
11 14
On average, how many cups of coffee does each person at the firm drink per hour, assuming a 10-hour work day?
Answer:
A person drinks 4.52 cups per hour
Step-by-step explanation:
No of work days = 5
No of hours in day = 10
No of hours in week = 10*5= 50 hours
Total cups consumed = 226
No of cups consumed per hour = Total no of cups/ Total week hours
= 226/50
= 4.52 cups/ hour
HELP ASAP!!!
Lena makes home deliveries of groceries for a supermarket. Her only stops after she leaves the supermarket are at traffic lights and the homes where
she makes the deliveries. The graph shows her distance from the store on her first trip for the day. What is the greatest possible number of stops she
made at traffic lights?
answers:
a) 9
b) 5
c) 3
d) 4
Answer:
3
Step-by-step explanation:
3 is the greatest possible number of stops she made at traffic lights.
What is a Distance-Time Graph?A distance-time graph suggests how the distance an item has traveled in a given time. it's far a simple line graph that denotes distance as opposed to time findings the graph. Distance is plotted on the Y-axis.
How to read a distance graph?In a distance-time graph, the gradient of the line is equal to the velocity of the object. The more the gradient (and the steeper the road) the faster the object is transferring.
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A retail shop accepts only cash or checks suppose that 45% of its customers carry cash 44% carry checks and 31% carry both cash and checks what is the probability that a randomly chosen customer at the top of the shop is carrying cash or checks are both
Answer:
Step-by-step explanation:
The number of customers carrying cash=45% = 0.45
The number of customers carrying checks= 44% =0.44
The number of customers carrying both = 31% = 0.31
So,
To find the probability we will write the expression:
cash+checks-cash or checks(both)=cash and checks
0.45+0.44-both=0.31
0.45+0.44-0.31=both
0.58=both....
Carl's Candies has determined that a candy bar measuring 3 inches long has a z-score of +1 and a candy bar measuring 3.75 inches long has a z-score of +2.
What is the standard deviation of the length of candy bars produced at Carl's Candies?
A 0.75
B 3
C 3.75
D 2
Answer:
A. d = 0.75.
Step-by-step explanation:
The z-score =( x - m) / d where m = the mean and d = the standard deviation.
So we have
(3 - m) / d = 1
3 - m = d.............(1)
and
(3.75 - m) / d = 2
3.75 - m = 2d....... (2)
Subtract (2) - (1):
0.75 = d.
Answer: A 0.75
Step-by-step explanation:
Formula for z-score :
[tex]z=\dfrac{x-\mu}{\sigma}[/tex]
, where x= random variable
[tex]\mu[/tex] = Population mean
[tex]\sigma[/tex] = Standard deviation
As per given , we have
[tex]+1=\dfrac{3-\mu}{\sigma}\\\\\Rightarrow\ \sigma=3-\mu\\\\\Rightarrow\ \mu=3-\sigma---(i)[/tex]
[tex]+2=\dfrac{3.75-\mu}{\sigma}\\\\\Rightarrow\ \mu=3.75-2\sigma---(ii)[/tex]
From (i) and (ii) , we have
[tex]3-\sigma=3.75-2\sigma\\\\\Rightarrow\ 2\sigma-\sigma=3.75-3\\\\\Rightarrow\ \sigma=0.75[/tex]
Hence, the standard deviation of the length of candy bars produced at Carl's Candies is 0.75.
Thus , the correct answer is A. 0.75.
A plane is taking off from Bangladesh headed to
New York City. At the same time, a plane from
New York City is headed to Bangladesh is also
taking off. The plane bound to New York City is
traveling at 600 mph, while the plane traveling
to Bangladesh is traveling at 400 mph. How far
from New York City will the two planes meet
if the distance between New York City and
Bangladesh is 8,000 miles?
Answer:
4800 miles
Step-by-step explanation:
What are the zeros of the polynomial function f(x) =x^2+5x-6
Answer:
The zeros are x=1, x=-6
Step-by-step explanation:
f(x) =x^2+5x-6
Factor: What two numbers multiply to -6 and add to 5
-1 *6 = -6
-1 +6 = 5
f(x) = (x-1) (x+6)
Using the zero product property
0 = (x-1) (x+6)
x-1 =0 x+6=0
x=1 x=-6
The zeros are x=1, x=-6
AB ll CD and EF ll GH use the figure above to find the value of each angle
Answer:
[tex]x=k=117\°[/tex]
Step-by-step explanation:
Givens
AB || CD
EF || GH
These parallels gives us certain pair of angles which are congruent.
The given angle 63° is congruent with its corresponding angles. The image attached shows all corresponding angles that are equal to 63° (red angles).
Basically,
[tex]x+63=180[/tex] and [tex]k+63=180[/tex], by suplementary angles and straight angle definition.
Then, solving for [tex]x[/tex] and [tex]k[/tex], we have
[tex]x=180-63\\x=117\°[/tex]
[tex]k=180-63\\k=117\°[/tex]
Therefore, both angles are equal to 117°.
The measure of each angle are:
<m = 63 <n = 117<k = 63 <x = 63 <w = 63 <p = 63Given:
AB || CD
EF || GH
As, the lines are parallel and parallel line have following properties
Corresponding angleAlternate Interior angleAlternate Exterior angleCo- interior angleSo, <m = 63 {Corresponding angle}
Now, <m + <n = 180 {Linear pair}
63 + <n =180
<n =180 - 63
<n = 117
and, <k = 63 {Vertical opposite angle}
and, <k = <x = 63 {Corresponding angle}
and, <w = <x = 63 (alternate Interior angle}
and, <p = <w = 63 {Vertical opposite angle}
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Roofing material costs $84.52 per square (10ft×10ft). The roofer charges $55.75 per square for labor, plus $9.65 per square for supplies. Find the total cost for 26.3 squares of installed roof. Round to the nearest cent.
Answer:
$1720.00
Step-by-step explanation:
55.75 + 9.65 = 65.40
65.40 x 26.3 = 1720.02
During the first four months of the year, Jack earned $1270, $1150, $870 and $1450 If Jack must have an average salary of at least $1150 in order to earn retirement benefits, what must Jack earn in the fifth month in order to qualify for benefits?
Answer:
1010
Step-by-step explanation:
There are a whole class of questions that rely on the method to this one.
First add up what you know
1270 + 1150 + 870 + 1450 = 4740
Now add on the 5th month (which you don't know. Call it x)
4740 + x
Divide by 5
(4740 + x)/5 = 1150 and that is your equation
Solution
Multiply both sides by 5
5*(4740 + x) / 5 = 1150 * 5
4740 + x = 5750
Subtract 4740 from both sides
4740 - 4740 + x = 5750 - 4740
x = 1010
Which seems kind of low, but that's what the numbers come to.
What is the lateral area of a regular pyramid with a square base which has a slant height of 9 units and base side lengths of 7 units?
Answer:
126 units
Step-by-step explanation:
the lateral area of a regular pyramid with a square base of 126 units has a slant height of 9 units and base side lengths of 7 units.
Given: DE || BC. Find the measure of AED in the triangle AED.
Answer:
<AED = 60
Step-by-step explanation:
Since triangle ADE is similar to triangle ABC
Angle ADE equals angle ADC
The angles of a triangle add to 180 degrees
<A + <ADE + <AED = 180
36 + 84+ <AED = 180
Combine like terms
120 + <AED = 180
Subtract 120 from each side
120-120 +<AED = 180-120
<AED = 60
Answer:
58°
Step-by-step explanation:
Since DE and BC are parallel, ADE will be equal to ABC.
So ADE is 84°.
And, the sum of the angles of a triangle is 180°. So AED will be equal to 180-(38 + 84),which is 58°
[Only if Angle A is 38°]
The pentagon on the left is a reflection of the pentagon on the right.
The pentagon is reflected over line ____.
Answer:
A
Step-by-step explanation:
If you reflect over line A both pentagons are equally spaced in proportion to the line
The pentagon is reflected over the line A.
What is Reflection?Reflection is a type of geometric transformation where the figure is flipped. In other words, a figure when undergoes reflection becomes it's mirror image.
Here given are two pentagons on left and right.
The pentagon on the left is a reflection of the pentagon on the right.
This means that both the pentagons should be proportionally spaced from the line.
If we consider the line of reflection as B, the the pentagon on the right is nearer to the line compared to that on the left.
If we consider line D as the line of reflection, then pentagon on the left is nearer to the line compared to that on the right.
So if line A is the line of reflection, the both pentagons are equally spaced from the line.
Hence line A is the line of reflection.
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is this a parallelogram? Just checking
Answer:
Step-by-step explanation:
Yes it’s parallel because the lines do not meet
Find the length of RW
Answer:
A 54
Step-by-step explanation:
RC = 126
WC = 72
RC = WC + RW
126 = 72+RW
Subtract 72 from each side
126-72 = 72-72+RW
54 = RW
Answer: First option
[tex]RW=54[/tex]
Step-by-step explanation:
Notice in the image that the distance from R to C is equal to 126.
Then we write:
[tex]RC = 126[/tex]
Also note that the distance between W and C is 72.
We know that:
[tex]RC=RW + WC[/tex]
In this case we want to find RW, so we solve the equation for RW
[tex]RC-WC=RW + WC-WC[/tex]
[tex]RC-WC=RW[/tex]
[tex]RW=RC-WC[/tex]
Now we substitute the values of RC and WC into the equation
[tex]RW=126-72[/tex]
[tex]RW=54[/tex]
Find the equation of the horizontal line that passes through the point (-5,6) using the point slope formula
Answer:
So first blank=6
Second blank=0
Third blank=-5
Fourth blank=6
Step-by-step explanation:
It is a horizontal line so the equation is just going to be
[tex]y=\text{ whatever y-coordinate they gave you in that one point}[/tex].
So it will be [tex]y=6[/tex]
We are going to do the point-slope form too as I requested.
Point-slope form is [tex]y-y_1=m(x-x_1)[/tex].
We have m=0 since you said the slope was 0 and since it said your have a horizontal line (which means the slope slope or m is 0).
You also that it contains the point [tex](-5,6)=(x_1,y_1)[/tex].
So our line in point-slope form is [tex]y-6=0(x-(-5))[/tex].
F(x)=x^2+9x-16
What is vertex
Axis of semetry
Answer:
axis of symmetry is [tex]x=\frac{-9}{2}[/tex].
The ordered pair of the vertex is [tex](\frac{-9}{2},\frac{-145}{4})[/tex].
Step-by-step explanation:
Your function is a quadratic.
Compare [tex]x^2+9x-16[/tex] to [tex]ax^2+bx+c[/tex].
You should see that [tex]a=1,b=9,c=-16[/tex].
The x-coordinate of the vertex or the axis of symmetry since the axis symmetry goes through the vertex can be found by computing [tex]\frac{-b}{2a}[/tex].
So here we go!
The axis of symmetry is [tex]x=\frac{-9}{2(1)}=\frac{-9}{2}[/tex].
When you write your axis of symmetry be sure to write it as an equation.
That is the axis of symmetry is [tex]x=\frac{-9}{2}[/tex].
Now that was also the x-coordinate of your vertex. To find the corresponding y-coordinate of the vertex, plug your value for [tex]x[/tex] into
[tex]y=x^2+9x-16[/tex].
[tex]y=(\frac{-9}{2})^2+9(\frac{-9}{2})-16[/tex]
Put into calculator:
[tex]y=\frac{-145}{4}[/tex] when [tex]x=\frac{-9}{2}[/tex]
The ordered pair of the vertex is [tex](\frac{-9}{2},\frac{-145}{4})[/tex].
Answer:
Vertex: [tex](h,k)\rightarrow(-4.5,-36.25)[/tex]
Axis of symmetry: [tex]x=-4.5[/tex]
Step-by-step explanation:
Finding the Axis of Symmetry:First I'll find the axis of symmetry. This formula lets us find the a.o.s: [tex]x=\frac{-b}{2a}[/tex].
In [tex]x^2+9x-16[/tex], the values of a, b, and c are:
a: 1b: 9c: -16We only need a and b to find the axis of symmetry. Substitute these values into the formula.
[tex]x=\frac{-(9)}{2(1)}[/tex]Simplify this fraction.
[tex]x=\frac{-9}{2} =-4.5[/tex]The axis of symmetry of this quadratic function is x = -4.5.
Finding the Vertex:Now to find the vertex, we have to take into account that this quadratic is in standard form, making it a little harder. We have to convert this function into vertex form.
Start by changing f(x) to 'y' and adding 16 to both sides.
[tex]y+16=x^2+9x[/tex]Use the completing the square formula: [tex](\frac{b}{2} )^2[/tex]
[tex](\frac{9}{2} )^2=20.25[/tex]Keep the balance by adding 20.25 on the left side and adding it on the right side of the equation.
[tex]y+16+20.25=x^2+9x+20.25[/tex]Combine like terms.
[tex]y+36.25=x^2+9x+20.25[/tex]Factor the right side of the equation. Ask yourself, "What two numbers multiply to 20.25 (c) and add up to 9 (b)?" These two numbers are 4.5 and 4.5. Rewrite the right side with factors.
[tex]y+36.25=(x+4.5)(x+4.5)[/tex][tex]y+36.25=(x+4.5)^2[/tex]Isolate y by subtracting 36.25 from both sides of the equation.
[tex]y=(x+4.5)^2-36.25[/tex]Now this quadratic function is in vertex form, making it super simple to find the vertex using [tex](h, k)[/tex].
Vertex form of a quadratic is:
[tex]y=a(x-h)^2+k[/tex]Compare [tex]y=(x+4.5)^2-36.25[/tex] with the original vertex form and find where h and k are. Those are the x (h) and y (k) values of the vertex.
Since the original vertex form has x - h, the h value in [tex]y=(x+4.5)^2-36.25[/tex] would be a negative since two negatives make a positive. The k value would stay "normal"---negative would mean it is a negative and positive would mean it is a positive number.
Therefore the h value is -4.5, and the k value is -36.25.
The ordered pair of the vertex is [tex](-4.5, -36.25)[/tex].