Given:
The limit problem is:
[tex]\lim_{x\to -\infty}(-2x^5-3x+1)[/tex]
To find:
The value of the given limit problem.
Solution:
We have,
[tex]\lim_{x\to -\infty}(-2x^5-3x+1)[/tex]
In the function [tex]-2x^5-3x+1[/tex], the degree of the polynomial is 5, which is an odd number and the leading coefficient is -2, which is a negative number.
So, the function approaches to positive infinity as x approaches to negative infinity.
[tex]\lim_{x\to -\infty}(-2x^5-3x+1)=\infty[/tex]
Therefore, [tex]\lim_{x\to -\infty}(-2x^5-3x+1)=\infty[/tex].
William has been contracted to paint a school classroom. The classroom is 20 m long, 15 m wide and 5 m high. There are four windows (2m by 3m) and a door (2m by 1m). Determine the cost of painting the ceiling at N$ 6.50/m²
Answer:
Step-by-step explanation:
l -> length
b -> width
h -> height
Find the area of four walls and ceiling. then subtract the area of four windows and a door form that area.
Area of four walls + ceiling = 2( lh + bh) +lb
= 2*(20*5 + 15*5) + 20*15
= 2( 100 + 75) + 300
= 2* 175 + 300
= 350 +300
= 650 sq m
Area of window = 2 *3 = 6 sq.m
Area of four windows = 4*6 = 24 sq.m
Area of door = 2 * 1 = 2 sq.m
Area of four walls excluding 4 windows and door = 650 - 24 - 2 = 624 sq.m
Cost of painting = 624 * 6.50
= $ 4056
Answer: 1950 dollars to paint the ceiling only (ignoring the walls)
The cost to paint the walls only is 2106 dollars.
The cost to paint the walls and ceiling is 4056 dollars.
==================================================
Explanation:
It seems a bit strange how your teacher mentions the windows and doors, but then asks about the ceiling only. Perhaps this is a red herring, but I'm not sure.
Anyway, to directly answer the question, we'll need to find the area of the ceiling first. The ceiling is a rectangle of dimensions 20 m by 15 m, so its area is 20*15 = 300 square meters.
Since paint costs 6.50 dollars per square meter, the total cost for the ceiling alone is 6.50*300 = 1950 dollars
If your teacher only cares about the ceiling, then you can stop here (and ignore the next section below).
---------------------------
If you wanted to find the cost to paint the walls, then we need to find the area of the walls.
For now, ignore the windows and door. Two opposite walls have area of 20*5 = 100 m^2 each. That accounts for 2*100 = 200 m^2 of wall area so far.
The other pair of opposite walls have area 15*5 = 75 m^2 each. That's another 2*75 = 150 m^2 of wall area.
In all, the total wall area without considering the windows or door is 200+150 = 350 m^2.
Now we consider the windows. Each window is 2 m by 3 m, yielding an area of 2*3 = 6 m^2. Four such windows have a total area of 4*6 = 24 m^2.
The door is 2 m by 1 m, so its area is 2*1 = 2 m^2
We'll subtract the wall area and the combined window+door areas to get
wallArea - windowArea - doorArea = 350-24-2 = 324
So after accounting for the windows and door, the amount of wall to paint is 324 m^2, which leads to a cost of 6.50*324 = 2106 dollars.
Therefore, painting the walls and ceiling gets us a total cost of 1950+2106 = 4056 dollars
This section is entirely optional if your teacher only cares about the ceiling.
A student has test scores of 75 and 82respectively. What is the student’s average score for a third test
Answer:
78.5 (I think 90% sure)
Step-by-step explanation:
sum of both scores
75+82 = 157
average for a third test
157÷2=78.5
using the 1 to 9 at the most time each, fill in the boxes to make a true statement
Answer:
2
Step-by-step explanation:
8*8 is 64
Since it looks like the empty box is an exponent, and there are 2 8s being multiplied, the answer is 2
A study is done to determine the average salary for all Santa Clara County teachers. If we were to randomly pick 15 schools in Santa Clara County and average together the salaries of all teachers, would this be a good sampling technique or a bad sampling technique
Answer:
Good sampling technique
Step-by-step explanation:
In other to determine the mean value for a population, it may be necessary to make inference from the a small subset of the population, called the sample, if it is impossible, difficult or inefficient to get all population data. Hence, in other to select a subset of the population, then the sample must be selected at random, such that all elements of the population have equal chances of being selected and hence, be representative of the population. Since, the sampling technique adopted above is done at random, hence, it is a good sampling technique.
4x-1,9x-1,7x-3 find the perimeter
20x-5
Answer:
Solution given;
perimeter=sum of all sides
=4x-1+9x-1+7x-3=20x-5
The perimeter of the line segments 4x - 1, 9x - 1, and 7x - 3 is 20x - 5.
To find the perimeter of the given line segments, you need to add up the lengths of all the line segments.
The lengths of the line segments are:
4x - 1,
9x - 1,
7x - 3.
To find the perimeter, add these lengths together:
Perimeter = (4x - 1) + (9x - 1) + (7x - 3)
= 4x + 9x + 7x - 1 - 1 - 3
= 20x - 5.
Therefore, the perimeter of the line segments 4x - 1, 9x - 1, and 7x - 3 is 20x - 5.
To learn more on Perimeter click:
https://brainly.com/question/7486523
#SPJ6
Find in the triangle. Round to the nearest degree.
Answer:
D. 34
Step-by-step explanation:
Because this is a right triangle we can use sin, cos, tan.
Use cosine because the values of the adjacent side and hypotenuse are already given.
cos(θ) = 72/87
Because we are solving for the angle measure (and not the measure of the side) we need to use inverse cos.
cos⁻¹ = 72/87
put into a calculator and answer is approximatelyn34 degrees.
The following is a scatterplot of the percent of children under age 18 who are not in school or in the labor force vs. the number of juvenile violent crime arrests for each of the 50 states. The least-squares regression line has been drawn in on the plot. We would like to predict what the number of juvenile violent crime arrests would be in a state if 25% of children are not in school or in the labor force. This is called
Answer:
Extrapolation
Step-by-step explanation:
From the linear regression plot created in the picture given, se could see that Tha percentage of student covered by the the plot is just above 16%. Therefore, to predict the percentage of the number of juvenile violent crime arrests would be in a state if 25% of children are not in school or in the labor force will require us to assume that the current trend continues into the future. Hence, we use the information and indications we have at present to make prediction into the future based on the assumption that we the current trend will remain relevant and applicable. This assumption into the future based on current trend is called EXTRAPOLATION.
Badluckville had been expecting a $50 million investment for a new golf resort, but the contract was cancelled.
Badluckville's mayor estimates that this will cost the city $300 million in total economic activity. What is the mayor's
estimate of the multiplier and marginal propensity to consume?
Answer:
multiplier = 6
marginal propensity to consume = .83
Step-by-step explanation:
/ means divided by
1 .
300,000,000 / 50,000,000 =
multiplier =
6
2.
300,000,000 - 50,000,000 =
250,000,000
250,000,000/300,000,000 =
marginal propensity to consume =
0.83333333333
or
.83
quizlet
marginal propensity to consume is equal to ΔC / ΔY, where ΔC is the change in consumption, and ΔY is the change in income
investopedia
A chemical engineer must report the average volume of a certain pollutant produced by the plants under her supervision. Here are the data she has been given by each plant:plantvolume of pollutantPittCross CreekSusquehannaWhat average volume should the chemical engineer report
Answer:
Please find the complete question in the attached file.
Step-by-step explanation:
Total quantities of plant-produced pollutants:
[tex]=(10.88+15.82+0.92) \ L\\\\=27.62\ L[/tex]
We are three medicinal plants here, Pinecrest, Macon, and Ogala. The average number of contaminants produced by plants would be
[tex]\to 27.62\div 3 \\\\\to \frac{27.62}{3} \\\\ \to 9.206 \ L[/tex]
When f(x) is divided by x + 4 the quotient is x2+5x−3+2x+4. What is f(−4)?
Javier volunteers in community events each month. He does not do more than five events in a month. He attends exactly five events 25% of the time, four events 30% of the time, three events 20% of the time, two events 15% of the time, one event 5% of the time, and no events 5% of the time. Find the probability that Javier volunteers for less than three events each month. P (x < 3) = 2 Find the expected number of events Javier volunteers in a month. 3.6 It is given that x must be below a certain value, which limits the rows to use in the PDF table. What is the sum of the probabilities of those rows?
Answer:
[tex]P(x < 3) = 25\%[/tex]
[tex]E(x) = 3[/tex]
Step-by-step explanation:
The given parameters can be represented as:
[tex]\begin{array}{ccccccc}x & {5} & {4} & {3} & {2} & {1}& {0} & P(x) & {25\%} & {30\%} & {20\%} & {15\%} & {5\%} & {5\%} \ \end{array}[/tex]
Solving (a): P(x < 3)
This is calculated as:
[tex]P(x < 3) = P(x = 0) + P(x = 1) + P(x =2)[/tex] ----- i.e. all probabilities less than 3
So, we have:
[tex]P(x < 3) = 5\% + 5\% + 15\%[/tex]
[tex]P(x < 3) = 25\%[/tex]
Solving (b): Expected number of events
This is calculated as:
[tex]E(x) = \sum x * P(x)[/tex]
So, we have:
[tex]E(x) = 5 * 25\% + 4 * 30\% + 3 * 20\% + 2 * 15\% + 1 * 5\% + 0 * 5\%[/tex]
[tex]E(x) = 125\% + 120\% + 60\% + 30\% + 5\% + 0\%[/tex]
[tex]E(x) = 340\%[/tex]
Express as decimal
[tex]E(x) = 3.40[/tex]
Approximate to the nearest integer
[tex]E(x) = 3[/tex]
Find the direction cosines and direction angles of the vector. (Give the direction angles correct to the nearest degree.) c, c, c , where c > 0
Answer:
cos(∝) = 1/√3
cos(β) = 1/√3
cos(γ) = 1/√3
∝ = 55°
β = 55°
γ = 55°
Step-by-step explanation:
Given the data in the question;
vector is z = < c,c,c >
the direction cosines and direction angles of the vector = ?
Cosines are the angle made with the respect to the axes.
cos(∝) = z < 1,0,0 > / |z|
so
cos(∝) = < c,c,c > < 1,0,0 > / √[c² + c² + c²] = ( c + 0 + 0 ) / √[ 3c² ]
cos(∝) = c / √[ 3c² ] = c / c√3 = 1/√3
∝ = cos⁻¹( 1/√3 ) = 54.7356° ≈ 55°
cos(β) = < c,c,c > < 0,1,0 > / √[c² + c² + c²] = ( 0 + c + 0 ) / √[ 3c² ]
cos(β) = c / √[ 3c² ] = c / c√3 = 1/√3
β = cos⁻¹( 1/√3 ) = 54.7356° ≈ 55°
cos(γ) = < c,c,c > < 0,0,1 > / √[c² + c² + c²] = ( 0 + 0 + c ) / √[ 3c² ]
cos(γ) = c / √[ 3c² ] = c / c√3 = 1/√3
γ = cos⁻¹( 1/√3 ) = 54.7356° ≈ 55°
Therefore;
cos(∝) = 1/√3
cos(β) = 1/√3
cos(γ) = 1/√3
∝ = 55°
β = 55°
γ = 55°
You work for a parts manufacturing company and are tasked with exploring the wear lifetime of a certain bearing. You gather data on oil viscosity used and load. You see the regression output given below.
Predictor Coef Stdev t-ratio P
Constant -147.973 41.972 -3.53 0.004181
viscosity 6.262 0.474 13.21 <0.0001
load 0.298 0.04 7.43 <0.0001
s = 13.507 R² = 95.73% R² (adj = 95.02%
Analysis of Variance
Source DF SS MS F
Regression 2 49131.93 24565.96 134.65
Error 12 2189.38 182.45
Total 14 51321.3
Required:
What is the correct conclusion about the regression slopes based solely on the F-test
Answer:
We reject the Null and conclude that There is significant evidence that the slope values are greater than 0.
Step-by-step explanation:
Based on the ANOVA output given :
The F critical value can be obtained thus ;
F(df regression, df error)
Using an α-value of 0.01
F(2, 12) at α = 0.01 is 6.927
The F statistic as obtained from the ANOVA table = 134.65
Since, F statistic > F critical we reject the Null and conclude that slope values are significantly > 0
Similarly,
Using the Pvalue :
The Pvalue of the slope are extremely small :
Viscosity <0.0001
Load <0.0001
At α = 0.01, 0.025
The Pvalue < α ; The null will be rejected.
Use Cramer's Rule to solve (if possible) the system of linear equations.
x1 + 2x2 =8
- x1 + x2 = 1
Required:
Find the coefficient matrix.
Answer:
x1 = 2
x2 = 3
Step-by-step explanation:
[tex]x_1=\frac{D_{x1}}{D}\\\\x_2=\frac{D_{x2}}{D}[/tex]
Here D is the coefficient matrix.
Hence
[tex]x_1=\frac{6}{3}\\x_1=2[/tex]
&
[tex]x_2=\frac{9}{3}\\x_2=3[/tex]
A hotel manager believes that 23% of the hotel rooms are booked. If the manager is right, what is the probability that the proportion of rooms booked in a sample of 610 rooms would differ from the population proportion by less than 3%
Answer:
0.9216 = 92.16% probability that the proportion of rooms booked in a sample of 610 rooms would differ from the population proportion by less than 3%
Step-by-step explanation:
To solve this question, we need to understand the normal probability distribution and the central limit theorem.
Normal Probability Distribution
Problems of normal distributions can be solved using the z-score formula.
In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the z-score of a measure X is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the p-value, we get the probability that the value of the measure is greater than X.
Central Limit Theorem
The Central Limit Theorem establishes that, for a normally distributed random variable X, with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]s = \frac{\sigma}{\sqrt{n}}[/tex].
For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.
For a proportion p in a sample of size n, the sampling distribution of the sample proportion will be approximately normal with mean [tex]\mu = p[/tex] and standard deviation [tex]s = \sqrt{\frac{p(1-p)}{n}}[/tex]
A hotel manager believes that 23% of the hotel rooms are booked.
This means that [tex]p = 0.23[/tex]
Sample of 610 rooms
This means that [tex]n = 610[/tex]
Mean and standard deviation:
[tex]\mu = p = 0.23[/tex]
[tex]s = \sqrt{\frac{p(1-p)}{n}} = \sqrt{\frac{0.23*0.77}{610}} = 0.017[/tex]
What is the probability that the proportion of rooms booked in a sample of 610 rooms would differ from the population proportion by less than 3%?
p-value of Z when X = 0.23 + 0.03 = 0.26 subtracted by the p-value of Z when X = 0.23 - 0.03 = 0.2. So
X = 0.26
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
By the Central Limit Theorem
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{0.26 - 0.23}{0.017}[/tex]
[tex]Z = 1.76[/tex]
[tex]Z = 1.76[/tex] has a p-value of 0.9608
X = 0.2
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{0.2 - 0.23}{0.017}[/tex]
[tex]Z = -1.76[/tex]
[tex]Z = -1.76[/tex] has a p-value of 0.0392
0.9608 - 0.0392 = 0.9216
0.9216 = 92.16% probability that the proportion of rooms booked in a sample of 610 rooms would differ from the population proportion by less than 3%
if cars A and B are traveling at the speed of 55km/hr and 75km/hr respectively. What is their average speed?
Answer:
65 km/hr
Step-by-step explanation:
The average of numbers can be calculated by adding them up and dividing that by how many numbers there are.
Here, we have two numbers. Therefore, we first add them (55+75 = 130) and then divide by 2 because there are 2 numbers, so 130/2 = 65
Solve the equation
tan^2 thetha-3 tan thetha+2=0 for 0
Step-by-step explanation:
[tex]\tan^2 \theta - 3\tan \theta + 2 = 0[/tex]
Let [tex]x = \tan \theta[/tex]
We can then write
[tex]x^2 -3x + 2 = 0\:\:\Rightarrow\:\:(x - 2)(x - 1) = 0[/tex]
or
[tex](\tan \theta - 2)(\tan \theta - 1) = 0[/tex]
The zeros occur when
[tex]\tan \theta = 2\:\:\:\text{or}\:\:\:\tan \theta = 1[/tex]
or when [tex]\theta = 63.4°[/tex] or [tex]\theta = 45°[/tex].
11 Emilio makes metal fences.
He is making a fence using this design.
1.44 m
DO NOT WRITE IN THIS AREA
1.8 m
.
The fence will need
3 horizontal metal pieces of length 1.8m
2 tall metal pieces of length 1.44 m
5 medium metal pieces
6 short metal pieces as shown on the diagram.
The heights of the tall, medium and short metal pieces are in the ratio 9:8:7
.
How many metres of metal in total does Emilio need to make the fence?
Answer:
Step-by-step explanation: 7 3 13 31
3 × 1.8 long pieces Calculate the ratios
2 × 1.44 9 tall vertical piece 1.44 / 9 = x / 9 x = 1.44
5 × ___ 8 medium vertical pieces 1.44 / 9 = x / 8 x = 1.28
6 × ___ 7 short vertical pieces 1.44 / 9 = x / 7 x = 1.12
3 × 1.8 long pieces
2 × 1.44 9 tall vertical piece 1.44 / 9 = x / 9 x = 1.44
5 × 1.28 8 medium vertical pieces 1.44 / 9 = x / 8 x = 1.28
6 × 1.12 7 short vertical pieces 1.44 / 9 = x / 7 x = 1.12
3 × 1.8 long pieces = 5.4 m
2 × 1.44 9 tall vertical piece = 2.88 m
5 × 1.28 8 medium vertical pieces = 6.4 m
6 × 1.12 7 short vertical pieces = 6.72 m
Total = 21.4 m
Find the probability of selecting 3 cards from a standard deck without
replacing them and all 3 are hearts.
Answer:
B
Step-by-step explanation:
THere are 52/4= 13 hearts we can draw
The number of ways to draw 3 hearts from 13 is
13C3= 286
The number of ways to draw 3 cards from 52 is
52C3= 22100
divide these
286/22100= 11/850
This is answer B
In this diagram, ABAC – AEDF. If the
area of ABAC = 6 in?, what is the
area of AEDF?
Answer:
2.7 in²
Step-by-step explanation:
similar triangles have the same angles, and all side lengths (or other distances) of one triangle have the same scaling factor to the side lengths of the other triangle.
so, we know the relation between the 2 baselines is 2/3, as this is the factor to turn the baseline of the large triangle into the baseline of the smaller triangle.
in other words
EF = BC × 2/3
2 = 3 × 2/3
correct
how do we calculate the area of a triangle ?
Area = baseline × height / 2
from BAC we know
Area = 6
baseline = 3
height = ?
6 = 3 × height / 2
12 = 3 × height
height = 4
aha !
now, EDF has a height too that we need to calculate is Area. and this height has the same scaling factor compared to the larger triangle as the side lengths : 2/3
so, for EDF we know
Area = ?
baseline = 2
height = 4 × 2/3 = 8/3
therefore, the area is
Area = (2 × 8/3) / 2 = (16/3) / 2 = 8/3 = 2.66666... ≈ 2.7
the shirt answer would be :
we know from the 2 baselines that the scaling factor for each distance is 2/3.
for the area we need to multiply 2 distances, so that means we have to multiply both by 2/3. and so on the formula for the area we have to use 2/3 × 2/3.
2/3 × 2/3 = 4/9
=>
Area small = Area large × 4/9 = 6 × 4/9 = 24/9 = 8/3 ≈ 2.7
Plz help ASAP problem down below
Explanation:
This is known as a cyclic quadrilateral since all four points are on the circle's edge, and the quadrilateral is entirely inside the circle (no parts of the quadrilateral spill outside the circle). Another term is "inscribed quadrilateral"
Since we have an inscribed quadrilateral, this means the opposite angles of the quadrilateral are supplementary.
B+D = 180
120+x = 180
x = 180-120
x = 60
What is the range of g?
y
--
.
9
7.
6+
5+
4+
3+
2+
1+
.
3
{4}}
-7 -6 -5 -4 -3 -2
+++
2 3 4 5 6 7
-2
-3
-4+
-5
-6+
7
7
Answer:
y 9 5 473648
Step-by-step explanation:
You return from a trip with 480 Canadian dollars. How much are your Canadian dollars worth in U.S. dollars? Use the exchange rate shown below. Currency U.S. dollars per Canadian dollar Canadian dollars per U.S. dollar Canadian dollar 0.5823 1.717 The 480 Canadian dollars are equivalent to about $ (Round to the nearest cent as needed.)
591 Dollars 42 Cents (591 Dollars when rounded)
If 6 playes cost 54$ how much do 30 plates cost
Answer:
270 plates
Step-by-step explanation:
First, you need to find how much one plate costs.
6x = 54
---- ----
6 6
x = 9
Now, multiply 30 plates with x, which is 9.
30(9) = 270
The answer is 270.
Answer:
270
Step-by-step explanation:
54($)÷6= 9 then 9×30=270
Sofia bought a clothes iron that was discounted 15% off of the original price of $35. What was the sale price of the clothes iron?
Answer:
35 - 0.15 * 35 so it is $29.75
Step-by-step explanation:
I got u
Answer:
$29.75
Step-by-step explanation:
15% = .15
.15 x 35 = 5.25
35 - 5.25 = 29.75
how do you find the slope of -2
What are the intercepts of the graphed function?
3
-2
Х
O x-intercept = (-1,0)
y-intercept = (-3,0)
O x-intercept = (0, -1)
y-intercept = (0, -3)
O x-intercept = (0, -1)
y-intercept = (-3,0)
x-intercept = (-1,0)
y-intercept = (0, -3)
6
Step-by-step explanation:
x intercept=(-1,0) because the graph is passing this point on the x axis
y intercept=(0,-3)
Answer:
4th option
Step-by-step explanation:
The x- intercept is where the graph crosses the x- axis.
This is at (- 1, 0 )
The y- intercept is where the graph crosses the y- axis.
This is at (0, - 3 )
Zach read a book for 10 minutes every weekend in the first month, 20 minutes in the second month, 40 minutes in the third month, and 80 minutes in the fourth month.
Victoria read a book for 35 minutes every weekend in the first month, 50 minutes in the second month, 65 minutes in the third month, and 80 minutes in the fourth month.
Which statement best describes the methods used by Zach and Victoria to increase the time they spent reading a book? (1 point)
Select one:
a. Zach's method is linear because the number of minutes increased by an equal factor every month.
b. Victoria's method is linear because the number of minutes increased by an equal number every month.
c. Both Victoria's and Zach's methods are exponential because the number of minutes increased by an equal factor every month.
d. Both Victoria's and Zach's methods are exponential because the number of minutes increased by an equal number every month.
9514 1404 393
Answer:
b. Victoria's method is linear because the number of minutes increased by an equal number every month
Step-by-step explanation:
Zach's reading doubled each month, a characteristic of an exponential function.
Victoria's reading increased by 15 minutes each month, characteristic of a linear function.
The only reasonable description among those offered is the one shown above: choice B.
HELP! AAHHHHH SOMEBODY HELP!
If each square of the grid below is $0.5\text{ cm}$ by $0.5\text{ cm}$, how many square centimeters are in the area of the blue figure?
Answer:
8.50 cm²
Step-by-step explanation:
The dimension of each square is given as 0.5cm by 0.5cm
The area of the a square is, a²
Where, a = side length
Area of each square = 0.5² = 0.25cm
The number of blue colored squares = 34
The total area of the blue colored squares is :
34 * 0.25 = 8.50cm²
math help plz
how to solve parabola and its vertex, how to understand easily and step by step with an example provided please
Answer:
The general equation for a parabola is:
y = f(x) = a*x^2 + b*x + c
And the vertex of the parabola will be a point (h, k)
Now, let's find the values of h and k in terms of a, b, and c.
First, we have that the vertex will be either at a critical point of the function.
Remember that the critical points are the zeros of the first derivate of the function.
So the critical points are when:
f'(x) = 2*a*x + b = 0
let's solve that for x:
2*a*x = -b
x = -b/(2*a)
this will be the x-value of the vertex, then we have:
h = -b/(2*a)
Now to find the y-value of the vertex, we just evaluate the function in this:
k = f(h) = a*(-b/(2*a))^2 + b*(-b/(2*a)) + c
k = -b/(4*a) - b^2/(2a) + c
So we just found the two components of the vertex in terms of the coefficients of the quadratic function.
Now an example, for:
f(x) = 2*x^2 + 3*x + 4
The values of the vertex are:
h = -b/(2*a) = -3/(2*2) = -3/4
k = -b/(4*a) - b^2/(2a) + c
= -3/(4*2) - (3)^2/(2*2) + 4 = -3/8 - 9/4 + 4 = (-3 - 18 + 32)/8 = 11/8
