1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
Genrish500 [490]
3 years ago
15

What would y=x^2 +x+ 2 be in vertex form

Mathematics
1 answer:
balu736 [363]3 years ago
7 0

Answer:

y = (x +  \frac{1}{2} )^{2}  +  \frac{7}{4}

Step-by-step explanation:

y =  {x}^{2}  + x + 2

We can covert the standard form into the vertex form by either using the formula, completing the square or with calculus.

y = a(x - h)^{2}  + k

The following equation above is the vertex form of Quadratic Function.

<u>Vertex</u><u> </u><u>—</u><u> </u><u>Formula</u>

h =  -  \frac{b}{2a}  \\ k =  \frac{4ac -  {b}^{2} }{4a}

We substitute the value of these terms from the standard form.

y = a {x}^{2}  + bx + c

h =  -  \frac{1}{2(1)}  \\ h =  -  \frac{ 1}{2}

Our h is - 1/2

k =  \frac{4(1)(2) - ( {1})^{2} }{4(1)}  \\ k =  \frac{8 - 1}{4}  \\ k =  \frac{7}{4}

Our k is 7/4.

<u>Vertex</u><u> </u><u>—</u><u> </u><u>Calculus</u>

We can use differential or derivative to find the vertex as well.

f(x) = a {x}^{n}

Therefore our derivative of f(x) —

f'(x) = n \times a {x}^{n - 1}

From the standard form of the given equation.

y =  {x}^{2}  +  x + 2

Differentiate the following equation. We can use the dy/dx symbol instead of f'(x) or y'

f'(x) = (2 \times 1 {x}^{2 - 1} ) + (1 \times  {x}^{1 - 1} ) + 0

Any constants that are differentiated will automatically become 0.

f'(x) = 2 {x}+ 1

Then we substitute f'(x) = 0

0 =2x + 1 \\ 2x + 1 = 0 \\ 2x =  - 1 \\x =  -  \frac{1}{2}

Because x = h. Therefore, h = - 1/2

Then substitute x = -1/2 in the function (not differentiated function)

y =  {x}^{2}  + x + 2

y = ( -  \frac{1}{2} )^{2}  + ( -  \frac{1}{2} ) + 2 \\ y =  \frac{1}{4}  -  \frac{1}{2}  + 2 \\ y =  \frac{1}{4}  -  \frac{2}{4}  +  \frac{8}{4}  \\ y =  \frac{7}{4}

Because y = k. Our k is 7/4.

From the vertex form, our vertex is at (h,k)

Therefore, substitute h = -1/2 and k = 7/4 in the equation.

y = a {(x - h)}^{2}  + k \\ y = (x - ( -  \frac{1}{2} ))^{2}  +  \frac{7}{4}  \\ y = (x +  \frac{1}{2} )^{2}  +  \frac{7}{4}

You might be interested in
In 2011 and 2015, the study was held to determine the proportion of people who read books. 948 people of 1200 said they read at
GuDViN [60]

Answer:

(0.79-0.72) - 1.96 \sqrt{\frac{0.72(1-0.72)}{1200} +\frac{0.79(1-0.79)}{1500}}=0.0373  

(0.79-0.72) + 1.96 \sqrt{\frac{0.72(1-0.72)}{1200} +\frac{0.79(1-0.79)}{1500}}=0.103  

And the 95% confidence interval would be given (0.0373;0.103).  

We are confident at 95% that the difference between the two proportions is between 0.0373 \leq p_A -p_B \leq 0.103

Step-by-step explanation:

Previous concepts

A confidence interval is "a range of values that’s likely to include a population value with a certain degree of confidence. It is often expressed a % whereby a population means lies between an upper and lower interval".  

The margin of error is the range of values below and above the sample statistic in a confidence interval.  

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".  

p_A represent the real population proportion for A  

\hat p_A =\frac{948}{1200}=0.79 represent the estimated proportion for  A

n_A=1200 is the sample size required for  A

p_B represent the real population proportion for B  

\hat p_B =\frac{1080}{1500}=0.72 represent the estimated proportion for B

n_B=1500 is the sample size required for B

z represent the critical value for the margin of error  

The population proportion have the following distribution  

p \sim N(p,\sqrt{\frac{p(1-p)}{n}})  

Solution to the problem

The confidence interval for the difference of two proportions would be given by this formula  

(\hat p_A -\hat p_B) \pm z_{\alpha/2} \sqrt{\frac{\hat p_A(1-\hat p_A)}{n_A} +\frac{\hat p_B (1-\hat p_B)}{n_B}}  

For the 95% confidence interval the value of \alpha=1-0.95=0.05 and \alpha/2=0.025, with that value we can find the quantile required for the interval in the normal standard distribution.  

z_{\alpha/2}=1.96  

And replacing into the confidence interval formula we got:  

(0.79-0.72) - 1.96 \sqrt{\frac{0.72(1-0.72)}{1200} +\frac{0.79(1-0.79)}{1500}}=0.0373  

(0.79-0.72) + 1.96 \sqrt{\frac{0.72(1-0.72)}{1200} +\frac{0.79(1-0.79)}{1500}}=0.103  

And the 95% confidence interval would be given (0.0373;0.103).  

We are confident at 95% that the difference between the two proportions is between 0.0373 \leq p_A -p_B \leq 0.103

4 0
2 years ago
9/16÷7/10 help please
horsena [70]
90/112 because all you have to do is 9/16 times 10/7. You just have to flip the 2nd fraction and make it multiplication.
3 0
3 years ago
Read 2 more answers
Will give brainlist to who finds this out!
castortr0y [4]

Answer:

1

2

4

Step-by-step explanation:

8 0
2 years ago
Read 2 more answers
(07.01 MC)
sweet-ann [11.9K]

Answer:

C

Step-by-step explanation:

5 0
2 years ago
4 value of 546 210 in words
FrozenT [24]
Five hinder and fourty six and two hundred and ten
3 0
2 years ago
Other questions:
  • Is -12, -10, -8, and -6 linear, exponential or neither
    15·2 answers
  • Write a word problem that can be solved by finding the number that have 4 as a factor
    11·1 answer
  • 61/28 + a = 19/4a - 11/7
    8·2 answers
  • Find the
    10·1 answer
  • There are 90 students, 10 leave. What percentage stayed?
    7·2 answers
  • Cant do word problems. ​
    10·1 answer
  • 6.71/(1.15)^4 doesn't equal 5.73
    7·2 answers
  • Factor the polynomial.
    10·1 answer
  • If ABCD is dilated by a factor of 2 the coordinate of b would be? Will mark barinlyist.
    14·1 answer
  • Please help me solve this! 2 1/3 x 1 1/2
    13·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!