The point-slope forms of the equation from the information given is 1. y + 2 = ï½(x-1), 2.y - 3 = 2(x+1) , 3.y - 5 = 1(x-8), 4. y + 7 = -5/7(x+1) and 5.y - 3/2 = -1(x-1/2).
1.
slope m = ï½ and (h, k) = (1, -2)
The point slope is y + 2 = ï½(x-1)
2.
slope = 2 , point (-1, 3)
y - 3 = 2(x+1)
3.
The line goes through the points (8,5) and (9, 6).
slope m = 6-5 / 9-8
= 1/1
= 1
y - 5 = 1(x-8)
4.
The line goes through the points (-1, -7) and (-8, -2)
slope m = -2+7 / -8+1
= 5/-7
= -5/7
y + 7 = -5/7(x+1)
5.
The line goes through the points (1/2, 3/2) and (-1/4, 5/4)
slope m = 5/4 - 3/2 / -1/4 - 1/2
= 5 - 6 / 4 / -1 + 2 / 4
= -1/1
= -1
y - 3/2 = -1(x-1/2).
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To find the local extrema of a polynomial function f(x) = anxn + an-1xn-1 + … + a2x2 + a1x + a0 (or any other function), you would apply the equation f ’(x) = 0. Consider the basic form of a quadratic equation: f(x) = ax2 + bx + c. Apply the equation above to this quadratic function and solve for x. The resulting equation should be familiar from algebra/pre-calculus. What did you call this equation and how is it related to the location of a local extrema on a quadratic function?
The resulting equation is the x-coordinate of the point of local extrema of a quadratic function.
To find the local extrema of a polynomial function f(x), we apply the equation f'(x) = 0. We are given the basic form of a quadratic equation. The equation is given below.
f(x) = ax² + bx + c
We need to find the differentiation of the above equation.
f'(x) = 2ax + b
Now, we will equate the obtained equation to zero and solve for the value of the variable "x".
f'(x) = 0
2ax + b = 0
2ax = -b
x = -b/2a
This resulting equation is familiar from algebra/pre-calculus. This is the x-coordinate of the point of local extrema of a quadratic function.
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A rectangular sheet of metal measures 8 inches by 10 inches. The metal is worth $2.00 per square inch. How much is the sheet of metal worth?
The cost of the 80 square inches metal sheet is $160
The length of the rectangular sheet = 8 inches
The width of the rectangular sheet = 10 inches
The area of the rectangular sheet = The length of the rectangular sheet × The width of the rectangular sheet
Substitute the values in the equation
= 8 × 10
= 80 square inches
The cost of metal per square inch = $2
Then the cost of metal sheet = The area of the rectangular sheet × The cost of metal per square inch
Substitute the values in the equation
= 80 × 2
= $160
Hence, the cost of the 80 square inches metal sheet is $160
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A communications satellite is directly above the extension of a line between receiving towers A and B. It is determined
from radio signals that the angle of elevation from tower A is 88.9° and the angle of elevation from tower B is 85.8°. If
A and B are 658 km apart, how far is the satellite from A?
(Ignore the curvature of the earth.)
The distance of the satellite from tower A, using the law of sines, is given by:
11987.216 km.
What is the law of sines?Suppose we have a triangle in which the sides and angles can be seen as follows:
Side with a length of a is opposite to the angle A.Side with a length of b is opposite to the angle B.Side with a length of c is opposite to the angle C.The lengths and the sine of the angles are related by the equality shown as follows:
[tex]\frac{\sin{A}}{a} = \frac{\sin{B}}{b} = \frac{\sin{C}}{c}[/tex]
The internal angles of the triangle in this problem are given as follows:
85.8º -> given.91.1º -> as an angle is supplementary with it's exterior angle, that is, the sum of their measures is of 180º.3.1º -> as the sum of the internal angles of a triangle is of 180º.The distance of the satellite from the tower A is opposite to the angle of 85.8º, while the distance between the towers of 658 km is opposite from the angle of 3.1º, hence the following relation is established:
sin(85.8º)/d = sin(3.1º)/658.
Applying cross multiplication, the distance is obtained as follows:
d = 658 x sin(85.8º)/sin(3.1º)
d = 11987.216 km.
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The function
h
(
x
)
=
(
x
+
3
)
5
can be expressed in the form
f
(
g
(
x
)
)
where
f
(
x
)
=
x
5
, and
g
(
x
)
is defined below:
The auxiliary function used in the composition between the two functions f(x) and g(x) is equal to linear function g(x) = x + 3.
How to derive the auxiliary function in a composition function
Herein we find the result of a composition between two functions, where the variable x of the parent function f(x) is substituted by the auxiliary function g(x). The composition between the two functions is defined below:
h(x) = f ° g (x) = f [g (x)]
If we know that the parent function f(x) = x⁵, then the auxiliary function within the composition is the linear function g(x) = x + 3.
Remark
The statement is poorly formatted, the correct form is shown below: The function h(x) = (x + 3)⁵ can be expressed in the form f[g(x)] where f(x) = x⁵ and g(x) is defined below.
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Write a linear function $f$ with $f\left(0\right)=7$ and $f\left(3\right)=1$ .
A linear function will have the form f(x) = mx + b.
f(0)=7 means the point (0,7) is on this line.
f(3)=1 means the point (3,1) is on the line as well.
Using those two points, we can find the slope:
m = (1-7)/(3-0) = -6/3 = -2
Since we were given f(0)=7, we also know the y-intercept.
So our function is f(x) = -2x+7
NO LINKS!! Please help me with this sequence Part 1x
Answer:
2, 3, 4, 9, 32, 279, 8896, 2481705, 22077238784 2,490,930 neither 121,800Step-by-step explanation:
1. Recursively-defined sequenceYou want the first 9 terms of the recursive sequence defined by ...
[tex]\begin{cases}s_1=2\\s_2=3\\s_n=s_{n-2}(s_{n-1}-1)\end{cases}[/tex]
The attached spreadsheet uses the given formula for the next term. It shows the terms to be ...
2, 3, 4, 9, 32, 279, 8896, 2481705, 22077238784
2. Sequence sumThe attached spreadsheet sum function has been used to find the sum of the first 8 terms. That sum is ...
2490930
3. Sequence typeThe sequence of problem 1 has neither a common difference nor a common ratio between successive terms. It is neither arithmetic nor geometric.
4. Sum of arithmetic sequenceThe sum of the first n terms of an arithmetic sequence with first term a1 and common difference d is given by ...
Sn = (2a1 +d(n -1))(n/2)
You have a sequence with a1 = 12 and d = (18-12) = 6. You want the sum of the first 200 terms.
S200 = (2·12 +6(200 -1))(200/2) = 121,800
The sum is 121,800.
__
Additional comment
A spreadsheet is a nice tool for finding terms of a recursively-defined sequence. The formula can include as many terms as necessary, and it can be replicated thousands of times, if necessary. The limitation is that arithmetic is generally limited to 16 significant figures, or so.
Answer:
1.
s₁ = 2
s₂ = 3
s₃ = 4
s₄ = 9
s₅ = 32
s₆ = 279
s₇ = 8,896
s₈ = 2,481,705
s₉ = 22,077,238,784
2. 2,490,930
3. Neither.
4. 121,800
Step-by-step explanation:
Question 1A recursive rule for a sequence allows you to find the nth term of the sequence provided you know the value of the previous term in the sequence.
Given recursive rule:
[tex]\begin{cases}s_n=s_{n-2} \cdot (s_{n-1}-1)\\s_1=2\\s_2=3\end{cases}[/tex]
Therefore, the first 9 terms of the sequence are:
[tex]s_1=2[/tex]
[tex]s_2=3[/tex]
[tex]\begin{aligned}s_3&=s_{3-2} \cdot (s_{3-1}-1)\\&=s_{1} \cdot (s_{2}-1)\\&=2 \cdot (3-1)\\&=2 \cdot 2\\&=4 \end{aligned}[/tex]
[tex]\begin{aligned}s_4&=s_{4-2} \cdot (s_{4-1}-1)\\&=s_{2} \cdot (s_{3}-1)\\&=3 \cdot (4-1)\\&=3 \cdot 3\\&=9 \end{aligned}[/tex]
[tex]\begin{aligned}s_5&=s_{5-2} \cdot (s_{5-1}-1)\\&=s_{3} \cdot (s_{4}-1)\\&=4 \cdot (9-1)\\&=4 \cdot 8\\&=32 \end{aligned}[/tex]
[tex]\begin{aligned}s_6&=s_{6-2} \cdot (s_{6-1}-1)\\&=s_{4} \cdot (s_{5}-1)\\&=9 \cdot (32-1)\\&=9 \cdot 31\\&=279\end{aligned}[/tex]
[tex]\begin{aligned}s_7&=s_{7-2} \cdot (s_{7-1}-1)\\&=s_{5} \cdot (s_{6}-1)\\&=32 \cdot (279-1)\\&=32 \cdot 278\\&=8896\end{aligned}[/tex]
[tex]\begin{aligned}s_8&=s_{8-2} \cdot (s_{8-1}-1)\\&=s_{6} \cdot (s_{5}-1)\\&=279 \cdot (8896-1)\\&=279\cdot 8895\\&=2481705\end{aligned}[/tex]
[tex]\begin{aligned}s_9&=s_{9-2} \cdot (s_{9-1}-1)\\&=s_{7} \cdot (s_{8}-1)\\&=8896\cdot (2481705-1)\\&=8896\cdot 2481704\\&=22077238784\end{aligned}[/tex]
Question 2Given series:
[tex]\displaystyle \sum^8_{k=1} s_k[/tex]
The sum notation asks to find the sum of the first 8 terms of the sequence from question 1.
Therefore:
[tex]\begin{aligned}\displaystyle \sum^8_{k=1} s_k&=s_1+s_2+s_3+s_4+s_5+s_6+s_7+s_8\\&=2+3+4+9+32+279+8896+2481705\\&=2490930\end{aligned}[/tex]
Question 3If a sequence is arithmetic, the difference between consecutive terms is the same (this is called the common difference).
If a sequence is geometric, the ratio between consecutive terms is the same (this is called the common ratio).
As the difference between consecutive terms it not the same, the sequence is not arithmetic.
As the ratio between consecutive terms is not the same, the sequence is not geometric.
Therefore, the sequence is neither arithmetic nor geometric.
Question 4[tex]\boxed{\begin{minipage}{7.3 cm}\underline{Sum of the first $n$ terms of an arithmetic series}\\\\$S_n=\dfrac{1}{2}n[2a+(n-1)d]$\\\\where:\\\phantom{ww}$\bullet$ $a$ is the first term. \\ \phantom{ww}$\bullet$ $d$ is the common difference.\\ \phantom{ww}$\bullet$ $n$ is the position of the term.\\\end{minipage}}[/tex]
Given arithmetic sequence:
12, 18, 24, ...Therefore:
a = 12d = 18 - 12 = 6To find the sum of the first 200 terms, substitute the found values of a and d into the formula along with n = 200:
[tex]\begin{aligned}S_{200}&=\dfrac{1}{2}(200)[2(12)+(200-1)(6)]\\&=100[24+(199)(6)]\\&=100[24+1194]\\&=100[1218]\\&=121800\end{aligned}[/tex]
Shayla is tracking the weight of the smallest puppy in a liter. use the table to write a linear function that model’s the puppy’s growth
The linear function that models the puppy's growth is written as: y = 3x + 3.
How to Write the Equation of a Linear Function?The values we need to find before we can write he linear function that models a situation, are:
The slope = mThe y-intercept = bThe linear function would be expressed as y = mx + b, which is in the slope-intercept form.
The table of the puppy's growth that Shayla is tracking is shown below. Using two points on the table, say, (0, 3) and (1, 6):
Slope (m) = change in y / change in x = (6 - 3)/(1 - 0)
Slope (m) = 3/1 = 3
Substitute m = 3 and (0, 3) into y = mx + b to find b:
3 = 3(0) + b
3 = 0 + b
3 = b
b = 3
To write the linear function, substitute m = 3 and b = 3 into the equation, y = mx + b:
y = 3x + 3
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Answer: A
Step-by-step explanation: FLVS question the answer is A
Judging by the appearance, name a acute angle, an obtuse angle, and a right angle
Answer:
acute angle - UVW
obtuse angle - VWX
right angle - UYX
1. Find the y-intercept of the line passing through the points A (3, -2) and B (-1, 3). (5 pts)
Answer:
-1.25
Step-by-step explanation:
m=
[tex]y2 - y1 \div x2 - x1[/tex]
3- (-2)÷ -1-3
5/-4
m=-1.25
HELP PLS AND HURRY!!!
Answer:
1. 5√7
2. (√15)/6
Step-by-step explanation:
1. √175
= √(25) * √(7)
= 5√7
2. √(5/12)
= (√5)/(√12)
= (√5)/(√(4) * √(3))
= (√5)/(2√3)
Radicals can't be in the denominator:
(√5)/(2√3)
(√5 * 2√3)/(2√3 * 2√3)
(2√15)/12
(√15)/6
3 to the second power times 3 to the third power
please help me ..............................!!!
Answer:
I hope this will be helpful for you.
0 (zero)
Step-by-step explanation:
[tex]{ \blue{ \sf{ {5}^{ - 3} \times {2}^{4} \times {5}^{3} \times {2}^{ - 5}}}} [/tex]
Arrange the like terms in order
[tex]{ \blue{ \sf{ {5}^{ - 3} \times {5}^{3} \times {2}^{4} \times {2}^{ - 5}}}} [/tex]
This is in the form of [tex]{ \red{ \sf{ {a}^{m} \times {a}^{n} = {a}^{m + n}}}} [/tex]
[tex]{ \blue{ \sf{ {5}^{ - 3 + 3} \times {2}^{4 + ( - 5)}}}} [/tex]
[tex]{ \blue{ \sf{ {5}^{0} \times {2}^{4 - 5}}}} [/tex]
[tex] { \blue{ \sf{0 \times {2}^{ - 1}}}} [/tex]
[tex]{ = \blue{ \sf{0}}}[/tex]
Any number which is multiplied with zero is zero itself.
The length of one parallel side of a trapezium measures 24 m and the distance between parallel sides measures 28 m. The area of the trapezium is 756 m2. What is the measure of the other parallel side?
Answer:
30 m
Step-by-step explanation:
Given a trapezium with area 756 m², height 28 m, and one base of length 24 m, you want to know the length of the other base.
Area formulaThe formula for the area of a trapezium is ...
A = 1/2(b1 +b2)h
ApplicationUsing the given values, we have ...
756 = 1/2(24 +b2)(28)
54 = 24 +b2 . . . . . . . . . . . divide by 14
30 = b2 . . . . . . . . . . . . subtract 24
The measure of the other parallel side is 30 m.
Solve the equation 32 = v + 27 for v.
−5
5
−59
59
Answer:
Step-by-step explanation: 32=v + 27
V= 5
how:
32-27=5
v=5
I don"t know what you meant for at the bottom, if that is included please explain what you mean.
Answer:
5
Step-by-step explanation:
32 = v + 27
Subtract 27 from both sides and make v the subject.
32 - 27 = v
5 = v
which expressions are equivalent to 3^4/9/3^2/9? select all that apply
Answer:
Two correct answers: subtracting exponents and
3^ 2/9
Step-by-step explanation:
To divide, subtract exponents. To subtract fractions keep the bottom number the same and subtract the top number. See image.
i need help what is the ancer to 7+7
Answer: 14 is the ancer
Step-by-step explanation:
Answer:
its 14
Step-by-step explanation:
Given the equation F=95C+32 where C is the temperature in degrees Celsius and F is the corresponding temperature in degrees Fahrenheit, and the following ordered pairs:
(25,F1), (−30,F2)
The corresponding values of F1 and F2 in the ordered pairs are 77 degrees Fahrenheit and -22 degrees Fahrenheit
How to use the equation to convert from degrees Celsius to degrees Fahrenheit?
Given: the equation F=9/5C+32 and ordered pairs: (25, F1), (−30, F2)
In order to convert the given values to degrees Fahrenheit, substitute the given values into the equation to get F1 and F2 respectively:
For (25, F1):
F = 9/5 C+32 (put C= 25 in the equation)
F1 = 9/5(25) + 32
F1 = 45 +22
F1 = 77 degrees Fahrenheit
For (−30, F2):
F = 9/5 C+32 (put C= -30 in the equation)
F2 = 9/5(-30) + 32
F2 = -54 + 32
F2 = -22 degrees Fahrenheit
Therefore, the values of F1 and F2 are 77 degrees Fahrenheit and -22 degrees Fahrenheit respectively
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PLEASE HELP MEEE!!!
Which point is located on the line represented by the equation y+8=3/7(x-5)
a. (3/7, 5)
b. (-5, 8)
c. (8, -5)
d. (5, -8)
e. (-8, 5)
Answer:
D, (5,-8)
Step-by-step explanation:
On the graph in the image, you can see option D's only point on the line.
Answer: D
Step-by-step explanation:
You will plug every value into the equation; if they are true, that is the correct option.
A.
5+8=3/7(3/7-5)
13=3/7(-32/7)
13=(-96/49)
A is INCORRECT
B.
8+8=3/7(-5-5)
16=3/7(-10)
16=-30/7
B is INCORRECT
C.
-5+8=3/7(8-5)
3=3/7(3)
3=9/7
C is INCORRECT
D.
-8+8=3/7(5-5)
0=3/7(0)
0=0
D IS CORRECT
Darren planted sunflowers in the community garden. During his last visit to the garden, the plants were 60 inches tall. Today, Darren found them to be 10% taller. How tall are the sunflowers now?
please due tommroe
Answer:
66 inches
Step-by-step explanation:
60 + .1(60)
60 + 6
66
Answer:
66 inches
Step-by-step explanation:
60 x .10 = 6
60 +6 =66
3, 12, 27, 64, 75, 108
a) give the next number in this pattern
b) what is the rule to find the nth term?
c) what is the 20th term?
Please help + explain!
The third term from the sequence should be 48 so that (a) the next term is 147 (b) the rule to find the nth term is 3×n² and (c) the 20th term is 1200.
How to find the next term of the sequenceFrom the question, 3, 12, 27, 64, 75, 108 the third term 64 is supposed to be 48. hence the terms are derived as follows:
1st term; 3 × 1² = 3
2nd term;3 × 2² = 12
3rd term; 3 × 3² = 27
4th term; 3 × 4² = 48
5th term; 3 × 5² = 75
6th term; 3 × 6² = 108
Therefore, the next term which the 7th term is 3 × 7² = 147, the rule for the sequence is 3×n² and the 20th term is 3 × 20² = 1200.
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A square with sides length x are cut out of each corner of a rectangular piece of cardboard measuring 5ft by 8ft. The resulting piece of cardboard is then folded into a box without a lid. Find the volume of the largest box that can be formed this way.
The volume of the largest box that can be made from the rectangular cardboard obtained by using the value of the derivative of the function for the volume at the extremum point is 18 cubic feet
What is an extremum point in mathematics?An extremum of a function is the point where the value of the function is a maximum or a minimum.
The side length of the cut out squares = x
The width of the rectangular cardboard = 5 feet
The length of the rectangular cardboard = 8 feet
The length of the side of the cut square = 8 - 2·x
Width of the side of the cut square = 5 - 2·x
Height of the cut square = x
Volume of the cut square, V = (8 - 2·x) × (5 - 2·x) × x = 4·x³ - 26·x² + 40·x
When the volume of the box is largest, we have;[tex]V' = \dfrac{d}{dx} \left(4\cdot x^3 - 26\cdot x^2 + 40\cdot x\right) = 12\cdot x^2 - 52\cdot x + 40=0[/tex]
12·x² - 52·x + 40 = 0
3·x² - 13·x + 10 = 0
Which gives;
(3·x - 10)·(x - 1) = 0
The dimensions of the cut square that gives the maximum volume are therefore;
x = 1 or x = 10/3
When x = 10/3, we have;
V = (8 - 2×(10/3)) × (5 - 2×(10/3)) × (10/3) ≈ -200/27 < 0
When x = 1, we have;
V = (8 - 2×1) × (5 - 2×1) × 1 = 18
The value of x that gives the maximum volume is therefore, x = 1
The dimensions of the box that give the maximum volume are therefore;
Height = x = 1
Length = 8 - 2×1 = 6
Width = 5 - 2 × 1 = 3
The dimensions of the box that gives the maximum volume are;
Height = 1 ft
Length = 6 ft
Width = 3 ft
The volume of the largest box that can be formed is 18 cubic feet
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it Question 3 Sandra draws a graph representing the relationship between time and distance on her morning run. She wants to show her coach that her rate in minutes per mile was the same for the entire run. Sandra chose two points on the line and drew vertical line segments from the points to the x-axis. How can she use these segments to prove that her speed was constant? Explain your reasoning.
Using the two points on the graph, Sandra can prove that her speed was constant using: Slope (m) = (18 - 6) / (3 - 1).
What is the Slope of a Graph?The slope of a graph is the ratio of the vertical distance along the line to the horizontal distance across the line, which is given as, m = change in y / change in x.
For a proportional graph, the slope is always constant, for whichever points we choose to use on the line.
Given two points on the line that Sandra chose, (3, 18) and (1, 6), therefore:
Slope (m) = change in y / change in x
Slope (m) = (18 - 6) / (3 - 1)
Slope (m) = 12/2
Slope (m) = 6
Therefore, to show her speed, which is the slope of the line, is constant, she can use the equation below:
Slope (m) = (18 - 6) / (3 - 1).
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Assume the radius of a certain planet is 4790 km and the planet is a sphere. What is its surface area?
Answer:
A ≈ 2.88×108km²
Step-by-step explanation:
A= 4 π r² = 4·π· 47902 ≈ 2.88324×10 power 8 km²
A giant tortoise moves at a slow but steady
pace. It takes the giant tortoise 1 hour to travel
2 miles. ¿How many miles (mi) a tortoise
travels in ¼ hour? Use the bar diagram to help
you.
According to the information, it can be inferred that the turtle travels 0.5 miles in a quarter of an hour (1/4 hours).
How to find the distance that the tortoise travels in a quarter of an hour (1/4 hours)?To find the distance that the tortoise travels in a quarter of an hour we must perform the following operations and take into account the information provided.
1 hour = 2 miles1/2 hour = 1 mile1/4 hour = 0.5 miles1/8 hour = 0.25 milesTo verify that the turtle takes 1/4 of an hour to travel 0.5 miles, we must divide the total number of miles it travels in one hour by 4, because we want to identify how much distance it travels in a quarter of the time, as shown below:
2 miles / 4 = 0.5 miles
Based on the above, we can infer that the tortoise travels 0.5 miles in 1/4 hour.
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How many more white squares are there than black squares in the 47th term? Explain how you know.
Expression (White Squares):
Expression: (Black Squares):
There are 9216 more white squares than black squares in the 47th term with it's an odd number of 97
How to calculate the 47th termThe terms are generated with the square of odd numbers with the black squares being the addition of the odd number and one subtracted from it.
For the 47th term:
Given that the first term is consists of 25 squares with 9 black and 16 white squares
first term has odd number 5
the square of 5; 5² = 25
black squares = 5 + (5-1) = 9
white squares = 25 - 9 = 16
Therefore, for the 47th term:
47th term has odd number 97
the square of 97; 97² = 9409
black squares = 97 + (97-1) = 193
white squares = 9409 - 193 = 9216
Hence, the 47th term has an odd number 97 with 9216 more black squares than the white squares,
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subtract -2x^2+5x+10 and -7x-3
The difference of the given algebraic expressions is -2x²+12x+13.
The given algebraic expressions are -2x²+5x+10 and -7x-3.
What is the subtraction of expressions?To subtract an algebraic expression from another, we should change the signs (from '+' to '-' or from '-' to '+') of all the terms of the expression which is to be subtracted and then the two expressions are added.
The subtraction of given expressions is as follows
-2x²+5x+10-(-7x-3)
= -2x²+5x+10+7x+3
= -2x²+12x+13
Hence, the difference of the given algebraic expressions is -2x²+12x+13.
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I need help in answering this question
Answer:
2 6 12 20,000
6 18 36 60,000
Step-by-step explanation:
The bottom number is 3 times the top number.
compare 5.9 and 152 using <, > or =
Answer:
Step-by-step explanation:
Rectangle A measures 12 inches by 8 inches. Rectangle B is a scaled copy of Rectangle A. Select all of the
measurement pairs that could be the dimensions of Rectangle B.
6 inches by 2 inches
8 inches by 6 inches
120 inches by 80 inches
60 inches by 36 inches
3 inches by 2 inches
Answer:
120 inches by 80 inches
3 inches by 2 inches
Step-by-step explanation:
So the ratio of 12/8 is 1.5 so we divide all of the numbers to find ratios that equal that
120/80=1.5
3/2=1.5
but we dont put say 6 by 2 because 6/2 is 3
What is the standard form of the quadratic function y=ax^2+bx+c shown in the graph below? a=_ b=_ c=_
The standard form of the quadratic function y=ax² + bx + c = a (x - h)² + k
How to find the standard for of a quadratic function?By definition, quadratic function is a function in the form f(x) = ax² + bx + c, where a, b, and c are numbers with a not equal to zero.
ax² + bx + c = a (x² - 2xh + h²) + kax² + bx + c
= ax² - 2ah x + (ah² + k)
Comparing the coefficients of x on both sides,b = -2ah ⇒ h = -b/2a .............................................(i)
Comparing the constants on both sides,c = ah² + kc = a (-b/2a)² + k (From (1))
c = b²/(4a) + kk = c - (b²/4a)k = (4ac - b²) / (4a)
Therefore, we can use the formulas h = -b/2a and k = (4ac - b²) / (4a) to convert the standard form to its vertex form.
the vertex form of the parabola is y = a (x - h)² + k
The standard form of the parabola is y = ax² + bx + c
The standard form of the quadratic function y=ax²+bx+c is x=-b± [tex]\frac{\sqrt{b^{2}- } 4ac}{2a}[/tex]
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