Answer:
A dozen chocolate cookies cost $6
and a dozen oatmeal cookies cost $7
Step-by-step explanation:
Let's assume that cost of 1 dozen chocolate cookies = x
Let's assume that cost of 1 dozen oatmeal cookies = y
then we get equations:
9x+8y=110...(i),
and 9x+5y=89...(ii)
Solve equation (i) for x
9x+8y=110
9x=110-8y
[tex]x=\frac{110-8y}{9}[/tex]...(iii)
plug (iii) into (ii)
9x+5y=89
[tex]9\left(\frac{110-8y}{9}\right)+5y=89[/tex]
[tex]110-8y+5y=89[/tex]
[tex]9\left(\frac{110-8y}{9}\right)+5y=89[/tex]
[tex]110-3y=89[/tex]
[tex]-3y=89-110[/tex]
[tex]-3y=-21[/tex]
[tex]y=7[/tex]
plug y=7 into (iii)
[tex]x=\frac{110-8y}{9}=\frac{110-8(7)}{9}=6[/tex]
So the final answer is given by:
A dozen chocolate cookies cost $6
and a dozen oatmeal cookies cost $7
Susana bought a laptop for $500. It was marked $50 off because it was out of the box and slightlly scratched. She also got 20% student discount, which was taken off the original price. What was the original peice of the laptop? Write and solve an equation to answer the question
Answer: 500*20%=100-50=50
Step-by-step explanation:
500*20%=100-50=50
PLZ HELP!! What is the average rate of change of f(x) from -2 to 2?
Answer:
-4 units per unit
Step-by-step explanation:
As x increases from a = -2 to b = 2 (a change of 4 units), f(x) changes from f(-2) = 25 to f(2) = 9, a decrease of 16. Thus, the average rate of change of f(x) on the interval [-2, 2] is:
-16
---------- = -4 units per unit.
4
Please help!! 10 points
See the attached picture:
what is the value of
tan-1(100/41)
The value of [tex]\(\tan^{-1}\left(\frac{100}{41}\right)\)[/tex] is approximately [tex]\(65.386^\circ\)[/tex] or [tex]\(1.141\)[/tex] radians.
To find the value of [tex]\(\tan^{-1}\left(\frac{100}{41}\right)\)[/tex], you can use a calculator or trigonometric tables. However, if you want to understand how to solve it step by step, here's how you can do it:
1. First, calculate the ratio [tex]\( \frac{100}{41} \)[/tex]. This gives you approximately [tex]\(2.439\)[/tex].
2. Now, you need to find the angle whose tangent is [tex]\(2.439\)[/tex]. You can do this using a calculator or by using trigonometric identities.
Typically, this is done using an inverse tangent function, often denoted as [tex]\(\tan^{-1}\)[/tex], [tex]\(\arctan\)[/tex], or [tex]\(\text{atan}\)[/tex] depending on the notation of your calculator or programming language.
So, [tex]\(\tan^{-1}\left(\frac{100}{41}\right) \approx 65.386^\circ\)[/tex].
Therefore, the value of [tex]\(\tan^{-1}\left(\frac{100}{41}\right)\)[/tex] is approximately [tex]\(65.386^\circ\)[/tex].
12w-27=-34+14w whats the answer
12w-27=-34+14w
Add 34 to each side:
12w + 7 = 14w
Subtract 12w from each side:
7 = 2w
Divide both sides by 2:
w = 7/2 ( depending on how you need to write the answer it could also be written as 3 1/2 or 3.5)
Answer:
w=7/2
Step-by-step explanation:
12w-27=-34+14w
12w-27+27=-34+14w+27
12w=14w-7
12w-14w=14w-7-14w
-2w=-7
w=7/2
If you need a decimal divide 7 by 2
hope this helps mark me brainliest
y=4x^{2 -4x+9 Find the x-intercepts for the parabola
Answer: The parabola does not intercept the x-axis.
Step-by-step explanation:
The parabola intercepts the x-axis when [tex]y=0[/tex], then, you need to substitute [tex]y=0[/tex] into the equation:
[tex]y=4x^2 -4x+9\\0=4x^2 -4x+9[/tex]
Now, use the Quadratic formula:
[tex]x=\frac{-b\±\sqrt{b^2-4ac}}{2a}[/tex]
In this case:
[tex]a=4\\b=-4\\c=9[/tex]
Substituting these values and evaluating, you get:
[tex]x=\frac{-(-4)\±\sqrt{(-4)^2-4(4)(9)}}{2(4)}\\\\x=\frac{4\±\sqrt{-128}}{8}[/tex]
Remeber that:
[tex]\sqrt{-1}=i[/tex]
Then, rewriting:
[tex]x=\frac{4\±8i\sqrt{2}}{8}[/tex]
Simplifying:
[tex]x=\frac{4(1\±2i\sqrt{2}}{4(2)}\\\\x=\frac{1\±2i\sqrt{2}}{2}\\\\x=\frac{1}{2}\±\frac{2i\sqrt{2}}{2}\\\\x=\frac{1}{2}\±i\sqrt{2}[/tex]
Then:
[tex]x_1=\frac{1}{2}+i\sqrt{2}[/tex]
[tex]x_2=\frac{1}{2}-i\sqrt{2}[/tex]
The roots are complex, therefore, the parabola does not intercept the x-axis.
Which statement best describes the association between variable X and variable Y?
Moderate negative association
Perfect positive association
Moderate positive association
Perfect negative association
Answer:
Perfect positive association
Step-by-step explanation:
Definition: A perfect positive association means that a relationship appears to exist between two variables, and that relationship is positive 100% of the time. Two variables have a positive association when the values of one variable tend to increase as the values of the other variable increase. (+1 indicates a perfect positive linear relationship)
Definition: A perfect negative association means that a relationship appears to exist between two variables, and that relationship is negative 100% of the time. Two variables have negative association when the values of one variable tend to decrease as the values of the other variable increase. (-1 indicates a perfect negative linear relationship)
Values between 0.3 and 0.7 (-0.3 and -0.7) indicate a moderate positive (negative) linear relationship.
From the graph we can see that this relationship shows perfect positive association (both variables increase and we can plot the straight line which will include all points)
Answer:
perfect positive
Step-by-step explanation:
HELP!
LOOK AT IMAGES BELOW
Answer:
D)y-intercept of f(x) is greater than the y-intercept of g(x)
Step-by-step explanation:
Y-intercept is the point y-coordinate value on graph where graph line intersects the y-axis and the value of x coordinate on x-axis is zero at that point.
From the given graph of f(x), it can be seen that
y-intercept of f(x) is 2
For the given function g(x)
finding y-intercept
when x=0
y=-3
because g(x) = 3 for 1<x≤3
As 2>-3
hence y-intercept of f(x)>g(x) !
Given: circle k(O), m FH = (31x+3°) m GH = (33x+3°) m∠FHG = (28x−3°) Find: m∠FGH
Answer:
The measure of angle FGH is m∠FGH=48°
Step-by-step explanation:
step 1
Find the measure of arc FG in terms of x
we know that
The inscribed angle measures half that of the arc comprising
m∠FHG=(1/2)[arc FG]
substitute the given values
(28x−3°)=(1/2)[arc FG]
arc FG=(56x−6°)
step 2
Find the value of x
we know that
arc FG+arc FH+arc GH=360° -----> by complete circle
substitute the values
(56x−6°)+(31x+3°)+ (33x+3°)=360°
120x=360°
x=3°
step 3
Find the measure of angle m∠FGH
we know that
The inscribed angle measures half that of the arc comprising
m∠FGH=(1/2)[arc FH]
substitute the given values
m∠FGH=(1/2)[31x+3°]
m∠FGH=(1/2)[31(3)+3°]
m∠FGH=48°
40 points answer asap!!!!!What is the surface area of the cone? (Use 3.14 for pie and round to the nearest hundredth.)
Answer:
216[tex]\pi[/tex]
Step-by-step explanation:
Plug the given values into the equation:
SA = ([tex]\pi[/tex])(9)(15) + [tex](\pi)[/tex] [tex](9^{2})[/tex]
Condense the equation:
216[tex]\pi[/tex]
Answer:
216 is going o be your answer
Step-by-step explanation:
construction worker needs to put a rectangular window and the side of the building he knows from the measure at the top and bottom of the window has a width of 5 ft and decides otherwise of 12 ft he also measures 1 diagonal to be 13 ft which is the length of the other diagonal
17
12
13
5
13 because each diagonal in a rectangle are the same
which of the function below could possibly have created this graph ?
ANSWER
D.
[tex]f(x) = - {x}^{4} + {x}^{3} + 3 {x}^{2} [/tex]
EXPLANATION
The given graph rises on the left and falls on the right.
This implies that the degree of the function is even.
Since the graph opens downward, the coefficient of the leading term must be negative.
The graph graph of the function also has 3 intercepts with one of them having an even multiplicity.
This means that the degree must be 4.
Therefore the last choice is correct.
Answer:
D
Step-by-step explanation:
The measure of angle zero is 600°. The point XY corresponding to zero on the unit circle is
Answer:
[tex]x= -\frac{1}{2}\\\\ y= -\frac{\sqrt{3}}{2}[/tex]
Step-by-step explanation:
We are given that the measure of angle θ is 600° and we are to fin the point (x, y) corresponding to angle θ on the unit circle,
We know that the x,y coordinates in the unit circle are the cosine and sine ratios, respectively.
θ = 600
1 unit circle= 360°
It means that 600° corresponds to 600° - 360° = 240°.
So, θ = 360° + 240°
180° < 240° < 270° (this tells that the point is in the third quadrant and its coordinates are negative)
240° - 180° = 60° (using the supplementary angles to find the sin and cos)[tex]\sin 60^\circ = \frac{\sqrt{3}}{2}\\\\\cos 60^\circ= \frac{1}{2}[/tex]
Therefore, the x-coordinate is the cos of the given angle while y-coordinate is the sine of the given angle and the coordinates are:
[tex]x= -\frac{1}{2}\\\\ y= -\frac{\sqrt{3}}{2}[/tex]
What is 40% of 39 ?
Answer:
15.6
Step-by-step explanation:
40% × 39 = 15.6
Hope this helps
What is the sum of the series?
For this case we must find the sum of the given series. For this we must expand the series for each value of k.
[tex](-2 (3) +5) + (- 2 (4) +5) + (- 2 (5) +5) + (- 2 (6) +5) =\\(-6 + 5) + (- 8 + 5) + (- 10 + 5) + (- 12 + 5) =[/tex]
Different signs are subtracted and the sign of the major is placed, while equal signs of sum and the same sign is placed.
[tex]-1-3-5-7 =\\-16[/tex]
The value of the series is -16
ANswer:
-16
The value of c is equal to _____.
For this case we have that by definition, the Pythagorean theorem states that:
[tex]c = \sqrt {a ^ 2 + b ^ 2}[/tex]
Where:
a, b: Are the legs
c: It is the hypotenuse
According to the figure we have to:
[tex]a = 20\\b = 21[/tex]
Substituting in the formula we have:
[tex]c = \sqrt {(20) ^ 2 + (21) ^ 2}\\c = \sqrt {400 + 441}\\c = \sqrt {841}\\c = 29[/tex]
Answer:
Option D
Answer:
841
Step-by-step explanation:
hope this helps
How do you get to 2,982 using 42 + something?
Answer:
2940
Step-by-step explanation:
Hello There!
To Solve This Equation, I Would Not Use Addition Till The Very End.
Remember, 42 + Something Equals 2,982.
That is The Same As Subtracting 42 From 2,982
2,940 Is Your Answer.
Select all of the situations that represent a linear relationship between x and y.
the total area, in square meters, y, of a square garden if the side length is x meters
the height, y, of a 6-foot tree that grows 0.2 feet per month for x months
the total cost, y, of a ski rental from a company that charges $15 plus an additional $8 per day rented for x days
the total miles, y, traveled by a car going 62 miles per hour in x hours
Answer:
We love the rebel in you, but please don't violate our Terms and Conditions. Your answer has been removed, but please review our posting guidelines and repost your answer. Thanks! https://brainly.com/pages/terms_of_use
Step-by-step explanation:
We love the rebel in you, but please don't violate our Terms and Conditions. Your answer has been removed, but please review our posting guidelines and repost your answer. Thanks! https://brainly.com/pages/terms_of_useWe love the rebel in you, but please don't violate our Terms and Conditions. Your answer has been removed, but please review our posting guidelines and repost your answer. Thanks! https://brainly.com/pages/terms_of_useWe love the rebel in you, but please don't violate our Terms and Conditions. Your answer has been removed, but please review our posting guidelines and repost your answer. Thanks! https://brainly.com/pages/terms_of_useWe love the rebel in you, but please don't violate our Terms and Conditions. Your answer has been removed, but please review our posting guidelines and repost your answer. Thanks! https://brainly.com/pages/terms_of_useWe love the rebel in you, but please don't violate our Terms and Conditions. Your answer has been removed, but please review our posting guidelines and repost your answer. Thanks! https://brainly.com/pages/terms_of_useWe love the rebel in you, but please don't violate our Terms and Conditions. Your answer has been removed, but please review our posting guidelines and repost your answer. Thanks! https://brainly.com/pages/terms_of_useWe love the rebel in you, but please don't violate our Terms and Conditions. Your answer has been removed, but please review our posting guidelines and repost your answer. Thanks! https://brainly.com/pages/terms_of_useWe love the rebel in you, but please don't violate our Terms and Conditions. Your answer has been removed, but please review our posting guidelines and repost your answer. Thanks! https://brainly.com/pages/terms_of_useWe love the rebel in you, but please don't violate our Terms and Conditions. Your answer has been removed, but please review our posting guidelines and repost your answer. Thanks! https://brainly.com/pages/terms_of_useWe love the rebel in you, but please don't violate our Terms and Conditions. Your answer has been removed, but please review our posting guidelines and repost your answer. Thanks! https://brainly.com/pages/terms_of_useWe love the rebel in you, but please don't violate our Terms and Conditions. Your answer has been removed, but please review our posting guidelines and repost your answer. Thanks! https://brainly.com/pages/terms_of_useWe love the rebel in you, but please don't violate our Terms and Conditions. Your answer has been removed, but please review our posting guidelines and repost your answer. Thanks! https://brainly.com/pages/terms_of_useWe love the rebel in you, but please don't violate our Terms and Conditions. Your answer has been removed, but please review our posting guidelines and repost your answer. Thanks! https://brainly.com/pages/terms_of_useWe love the rebel in you, but please don't violate our Terms and Conditions. Your answer has been removed, but please review our posting guidelines and repost your answer. Thanks! https://brainly.com/pages/terms_of_useWe love the rebel in you, but please don't violate our Terms and Conditions. Your answer has been removed, but please review our posting guidelines and repost your answer. Thanks! https://brainly.com/pages/terms_of_useWe love the rebel in you, but please don't violate our Terms and Conditions. Your answer has been removed, but please review our posting guidelines and repost your answer. Thanks! https://brainly.com/pages/terms_of_useWe love the rebel in you, but please don't violate our Terms and Conditions. Your answer has been removed, but please review our posting guidelines and repost your answer. Thanks! https://brainly.com/pages/terms_of_useWe love the rebel in you, but please don't violate our Terms and Conditions. Your answer has been removed, but please review our posting guidelines and repost your answer. Thanks! https://brainly.com/pages/terms_of_useWe love the rebel in you, but please don't violate our Terms and Conditions. Your answer has been removed, but please review our posting guidelines and repost your answer. Thanks!
Linear relationship always produces a straight line and are without exponential.
Lets check all the given options:
1. the total area, in square meters, y, of a square garden if the side length is x meters.
The area of a square is [tex](side)^{2}[/tex]
Here, side is 'x' and area 'y' will be [tex]x^{2}[/tex].
This is not linear.
2. the height, y, of a 6-foot tree that grows 0.2 feet per month for x months
The equation forms:
[tex]y=6+0.2x[/tex]
When x = 1
[tex]y=6.2[/tex]
When x = 2
[tex]y=6+0.2(2)[/tex]
=> [tex]y=6.4[/tex]
When x = 3
[tex]y=6+0.2(3)[/tex]
=> [tex]y=6.6[/tex]
This is a linear relation.
3. the total cost, y, of a ski rental from a company that charges $15 plus an additional $8 per day rented for x days
The equation forms:
[tex]y=15+8x[/tex]
When x = 1
[tex]y=15+8=23[/tex]
When x = 2
[tex]y=15+8(2)[/tex]
=> [tex]y=31[/tex]
When x = 3
[tex]y=15+8(3)[/tex]
=> [tex]y=39[/tex]
This is also linear.
4. the total miles, y, traveled by a car going 62 miles per hour in x hours
The equation forms:
[tex]y=62x[/tex]
When x = 1
[tex]y=62[/tex]
When x = 2
[tex]y=62(2)[/tex]
=> [tex]y=124[/tex]
When y = 3
[tex]y=62(3)[/tex]
=> [tex]y=186[/tex]
This is also linear in nature.
Question is shown below.
ANSWER
D. 4 minutes 40 seconds
EXPLANATION
The given data set showing the number of seconds of each song the playlist is:
283,204,198,229,173,230,168,236,215,204
Let the length of the 11th song be x.
Because he wants the mean to be 3 minutes 40 seconds, which is equal to 220 seconds, we write and solve the following equation for x.
[tex] \frac{283 + 204 + 198 + 229 + 173 + 230 + 168 + 236 + 215 + 204 + x}{11} = 220[/tex]
[tex] \frac{2140 + x}{11} = 220[/tex]
[tex]2140 + x = 2420[/tex]
[tex]x = 2420 - 2140[/tex]
[tex]x = 280[/tex]
We convert 280 seconds to minutes.
[tex] \frac{280}{60} = 4 \frac{2}{3} = 4mins : 40 sec[/tex]
The correct choice is D
In how many ways can a committee of 4 be chosen from a group of 9 people? 36 126 3,024
Answer:
the answer is 126!
Step-by-step explanation:
i worked it out myself, and got it correct! have a great day :)
There are 126 ways can a committee of 4 be chosen from a group of 9 people, the correct option is B.
What is the combination?The solution is obtained through a combination where we choose "r of n", where r is the amount of things we choose and n the total number of things that can be chosen.
The combinations form uses factorial numbers. This is the formula:
[tex]\rm ^nC_r=\dfrac{n!}{(n-r)!r!}[/tex]
Where the value of n is 9 and r is 4.
The number of ways can a committee of 4 be chosen from a group of 9 people is;
[tex]\rm ^nC_r=\dfrac{n!}{(n-r)!r!}\\\\\rm ^9C_4=\dfrac{9!}{(9-4)!4!}\\\\ ^9C_4=\dfrac{9!}{5!4!}\\\\ ^9C_4=\dfrac{9\times 8 \times 7 \times 6 \times 5!}{5!\times 4\times 3\times 2\times 1}\\\\ ^9C_4=\dfrac{3024}{24}\\\\ ^9C_4=126[/tex]
Hence, there are 126 ways can a committee of 4 be chosen from a group of 9 people, the correct option is B.
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How do i solve this with steps
5x+5(x−3y)+12(4x−6y)
Expand
5x + 5x - 15y + 12 (4x - 6y)
Expand
5x + 5x - 15y + 48x - 72y
Collect like terms
(5x + 5x + 48x) + (-15y - 72y)
Simplify
= 58x - 87y
In the rho.bus, m angle 1 =4y-2. Find the value of y.
Answer:
y = 23
Step-by-step explanation:
The diagonals of a rhombus are perpendicular bisectors of each other, thus
∠1 = 90° and ∠1 = 4y - 2, so equating gives
4y - 2 = 90 ( add 2 to both sides )
4y = 92 ( divide both sides by 4 )
y = 23
Data set B: 87, 90, 77, 83, 99, 94, 93, 90, 85, 83 Find the MEDIAN
Answer: 88.5
Step-by-step explanation:
First you need to order the data set from least to greatest:
So,
77,83,83,85,87,90,90,93,94,99
Then you find the middle number, in this case it's 87 and 90.
Because there is two middle numbers in the data set, you need to find the average between those numbers.
The average between 87 and 90 is 88.5.
Hope this helps :)
The median of the Data set B: {87, 90, 77, 83, 99, 94, 93, 90, 85, 83} is 88.5.
What is Median?Median is the mid-value of the given data set when the set of the data is properly arranged.
The value of the data set is {87, 90, 77, 83, 99, 94, 93, 90, 85, 83}, now rearranging the set we will get,
77, 83, 83, 85, 87, 90, 90, 93, 94, 99
As we can see that the number of data points in the set is 10, therefore, the value of the median is the arithmetic average of the 5th and the 6th term,
[tex]\text{The average of the 5^{th} and the 6^{th} value}=\dfrac{87 + 90}{2} = 88.5[/tex]The average of the 5th and 6th = (87+90)/2 = 88.5
hence, the median of the Data set B: {87, 90, 77, 83, 99, 94, 93, 90, 85, 83} is 88.5.
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Suppose Henry is laying on the ground looking up at the sun that appears just over the top of a tree at
an angle of elevation of 32°. He is located 20 meters away from the base of the tree. How tall is the
tree? Round to the nearest meter.
Answer:
Good day! Use trigonometry to define the formula.
Step-by-step explanation:
The height of the tree that the sun appears just over it top is 12 meters
The situation forms a right angle triangle.
The height of the tree is the opposite side of the triangle.
His distance away from the base of the tree is the adjacent side of the right angle triangle.
Therefore,
Using trigonometric ratio,
tan 32 = opposite / adjacent
tan 32 = h / 20
cross multiply
height of the tree = 20 tan 32
height of the tree = 20 × 0.6248693519
height of the tree = 12.4973870382
height of the tree ≈ 12 meters
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what is the answer for 40÷32/4
Answer:
5
Step-by-step explanation:
40/(32/4)
40/8
5
Answer:
[tex]{\huge \boxed{5}}[/tex]
Step-by-step explanation:
Parenthesis
Exponent
Multiply
Divide
Add
Subtract
Do divide from left to right, to find the answer.
[tex]40\div32\div4[/tex]
[tex]32\div4=8[/tex]
[tex]40\div8=5[/tex]
5 is the correct answer.
What is the volume of each solid figure made of 1-inch cubes? What is the correct unit of measure?
Answer:
a is 12 inches cubed because there are four hidden blocks
b is 9 inches cubed because you can see there are 9 cubes
Please help me with this!
Answer:
The graph is a parabola with vertex (0 , -2)
Equation: y = x² -2
For x = -2, y = (-2)² -2 ⇒ y = 4 - 2 ⇒ y = 2
For x = -1, y = (-1)² -2 ⇒ y = 1 - 2 ⇒ y = -1
For x = 0, y = (0)² -2 ⇒ y = 0 - 2 ⇒ y = -2
For x = 1, y = (1)² -2 ⇒ y = 1 - 2 ⇒ y = -1
For x = 2, y = (2)² -2 ⇒ y = 4 - 2 ⇒ y = 2
What is the measure in radians for the central angle of a circle whose radius is 5 and intercepted arc length is 7.2? Round your answer to the earnest hundredth. (THERE ARE NO ANSWER CHOICES. I HAVE TO TYPE THIS ANSWER OUT
Answer:
[tex]0.46\pi\ radians[/tex]
Step-by-step explanation:
step 1
Find the circumference
The circumference of a circle is equal to
[tex]C=2\pi r[/tex]
we have
[tex]r=5\ units[/tex]
assume
[tex]\pi=3.14[/tex]
substitute
[tex]C=2(3.14)(5)=31.4\ units[/tex]
step 2
Find the measure in radians for the central angle of a circle whose radius is 5 and intercepted arc length is 7.2 units
Remember that
The circumference of [tex]31.4\ units[/tex] subtends a central angle of [tex]2\pi \ radians[/tex]
so
using proportion
Let
x-----> the central angle in radians
[tex]\frac{2\pi}{31.4}=\frac{x}{7.2} \\ \\x=2\pi (7.2)/31.4\\ \\x=0.46\pi\ radians[/tex]
Which is the area of a parallelogram with a base of 13 centimeters and a height of 3 centimeters?
Step-by-step explanation:
Area of a parallelogram is the base times the height.
A = bh
A = (13 cm) (3 cm)
A = 39 cm²
Answer:
The area is 39 square cm.
Step-by-step explanation:
The area of a parallelogram is given as :
[tex]A=base\times height[/tex]
Given is:
base = 13 cm
height = 3 cm
So, area of the parallelogram becomes:
[tex]A=13\times3[/tex]
[tex]A=39[/tex] square cm.
Therefore, the area of the parallelogram is 39 square centimeters.
1. Consider the function f(x) = x2 + 2x - 8.
(a) What are the x-intercepts of the graph of the function?
(b) What is the equation of the axis of symmetry? Explain how you know.
(c) What is the vertex of the function? Show your work.
(d) What is the domain of the function? Use set notation (i.e. { x | x > 1} )
(e) What is the range of the function? Use set notation (i.e. { y | y ≥ 1} )
(f) Graph the function.
ANSWER
a)The x-intercepts are:
(-4,0) and (2,0)
b) The axis of symmetry is x=-1
c) The vertex of this function is (-1,-9)
d)Domain: { [tex]x|x \in \: R[/tex]}
e) Range : {[tex]y |y \geqslant - 9[/tex]}
EXPLANATION
The given function is:
[tex]f(x) = {x}^{2} + 2x - 8[/tex]
We complete the square to write this function in the form:
[tex]f(x)=a{(x - h)}^{2} + k[/tex]
We add and subtract the square of half the coefficient of x.
[tex]f(x) = {x}^{2} + 2x + {1}^{2} - {1}^{2} - 8[/tex]
[tex]f(x) = {(x + 1)}^{2} - 9[/tex]
The vertex of this function is (h,k) which is (-1,-9)
The equation of axis of symmetry is x=h
But h=-1, hence the axis of symmetry isx=-1
To find the x-intercepts, we put f(x)=0
[tex]{(x + 1)}^{2} - 9 = 0[/tex]
[tex]{(x + 1)}^{2} = 9[/tex]
[tex]x + 1= \pm \sqrt{9} [/tex]
[tex]x= - 1 \pm3[/tex]
x=-4, 2
The x-intercepts are:
(-4,0) and (2,0)
The given function is a polynomial function, the domain is all real numbers.
[tex]x|x \in \: R[/tex]
e) The function has a minimum value of y=-9.
Therefore the range is
[tex]y |y \geqslant - 9[/tex]
Using the intercepts and vertex we can now draw this graph easily.
The graph of this function is shown in the attachment.
Final answer:
To address the student's question about the quadratic function, x-intercepts are found by setting the equation to zero, the axis of symmetry is calculated using -b/(2a), and the vertex is derived from these elements. The domain is all real numbers, and the range is all y-values greater than or equal to the y-value of the vertex. Finally, the function is graphically represented by plotting key points and drawing a parabola.
Explanation:
The student has inquired about the quadratic function f(x) = x2 + 2x - 8.
The x-intercepts of a quadratic function are the values of x where f(x) = 0. To find the x-intercepts, we set the equation to zero and solve for x: x2 + 2x - 8 = 0. In this case, the x-intercepts can be found using factorization or the quadratic formula.
The axis of symmetry is a vertical line that divides the parabola into two mirror images. For a quadratic function in the form of f(x) = ax2 + bx + c, the axis of symmetry equation is x = -b/(2a). We can use this formula to find the axis of symmetry for the given function.
The vertex of the function is the highest or lowest point on the graph, depending on whether the parabola opens upwards or downwards. To find the vertex, we use the axis of symmetry and plug that x-value into the function to find the corresponding y-value.
The domain of any quadratic function is all real numbers, which is denoted as {x | x is a real number}.
The range of a quadratic function depends on whether the parabola opens upwards or downwards. For the given function f(x) = x2 + 2x - 8, which opens upwards, the range is {y | y is greater than or equal to the y-value of the vertex}.
To graph the function, plot the vertex, axis of symmetry, and x-intercepts on a coordinate plane, then draw a smooth curve through these points to complete the parabola.