A doghouse is to be built in the shape of a right trapezoid, as shown below. What is the area of the doghouse?
Answers
Area of the doghouse =- 32.5 sq. ft
Solution:
Given height = 5 ft, base1, [tex]b_1[/tex] = 5ft
Base2, [tex]b_2[/tex] = [tex]b_1+3=5+3=8[/tex] ft
Area of the trapezoid = [tex]\frac{1}{2}\times height\times (base1 +base2)[/tex]
[tex]=\frac{1}{2}\times 5\times (5+8)[/tex]
[tex]=\frac{1}{2}\times 5\times 13[/tex]
[tex]=\frac{1}{2}\times 65[/tex]
[tex]=32.5[/tex] sq. ft
Area of the trapezoid = 32.5 sq.ft
Hence the area of the doghouse is 32.5 sq. ft.
Related Questions
how do you graph the trigonometric function y= -sin(2x)+1?
Answers
Step-by-step explanation:
[tex]\displaystyle \boxed{y = -cos\:(2x - \frac{\pi}{2}) + 1} \\ \\ y = Acos(Bx - C) + D \\ \\ Vertical\:Shift \hookrightarrow D \\ Horisontal\:[Phase]\:Shift \hookrightarrow \frac{C}{B} \\ Wavelength\:[Period] \hookrightarrow \frac{2}{B}\pi \\ Amplitude \hookrightarrow |A| \\ \\ Vertical\:Shift \hookrightarrow 1 \\ Horisontal\:[Phase]\:Shift \hookrightarrow \frac{C}{B} \hookrightarrow \boxed{\frac{\pi}{4}} \hookrightarrow \frac{\frac{\pi}{2}}{2} \\ Wavelength\:[Period] \hookrightarrow \frac{2}{B}\pi \hookrightarrow \boxed{\pi} \hookrightarrow \frac{2}{2}\pi \\ Amplitude \hookrightarrow 1[/tex]
OR
[tex]\displaystyle y = Asin(Bx - C) + D \\ \\ Vertical\:Shift \hookrightarrow D \\ Horisontal\:[Phase]\:Shift \hookrightarrow \frac{C}{B} \\ Wavelength\:[Period] \hookrightarrow \frac{2}{B}\pi \\ Amplitude \hookrightarrow |A| \\ \\ Vertical\:Shift \hookrightarrow 1 \\ Horisontal\:[Phase]\:Shift \hookrightarrow 0 \\ Wavelength\:[Period] \hookrightarrow \frac{2}{B}\pi \hookrightarrow \boxed{\pi} \hookrightarrow \frac{2}{2}\pi \\ Amplitude \hookrightarrow 1[/tex]
You will need the above information to help you interpret the graph. First off, keep in mind that although this looks EXACTLY like the sine graph, if you plan on writing your equation as a function of cosine, then there WILL be a horisontal shift, meaning that a C-term will be involved. As you can see, the centre photograph displays the trigonometric graph of [tex]\displaystyle y = -cos\:2x + 1,[/tex]in which you need to replase "sine" with "cosine", then figure out the appropriate C-term that will make the graph horisontally shift and map onto the sine graph [photograph on the left], accourding to the horisontal shift formula above. Also keep in mind that the −C gives you the OPPOCITE TERMS OF WHAT THEY REALLY ARE, so you must be careful with your calculations. So, between the two photographs, we can tell that the cosine graph [centre photograph] is shifted [tex]\displaystyle \frac{\pi}{4}\:unit[/tex]to the left, which means that in order to match the sine graph [photograph on the left], we need to shift the graph FORWARD [tex]\displaystyle \frac{\pi}{4}\:unit,[/tex]which means the C-term will be positive, and by perfourming your calculations, you will arrive at [tex]\displaystyle \boxed{\frac{\pi}{4}} = \frac{\frac{\pi}{2}}{2}.[/tex]So, the cosine graph of the sine graph, accourding to the horisontal shift, is [tex]\displaystyle y = -cos\:(2x - \frac{\pi}{2}) + 1.[/tex]Now, with all that being said, in this case, sinse you ONLY have a graph to wourk with, you MUST figure the period out by using wavelengths. So, looking at where the graph hits [tex]\displaystyle [-\pi, 1],[/tex]from there to [tex]\displaystyle [-2\pi, 1],[/tex]they are obviously [tex]\displaystyle \pi\:units[/tex]apart, telling you that the period of the graph is [tex]\displaystyle \pi.[/tex]Now, the amplitude is obvious to figure out because it is the A-term, but of cource, if you want to be certain it is the amplitude, look at the graph to see how low and high each crest extends beyond the midline. The midline is the centre of your graph, also known as the vertical shift, which in this case the centre is at [tex]\displaystyle y = 1,[/tex]in which each crest is extended one unit beyond the midline, hence, your amplitude. Now, there is one more piese of information you should know -- the sine graph in the photograph farthest to the right is the OPPOCITE of the sine graph in the photograph farthest to the left, and the reason for this is because of the negative inserted in front of the amplitude value. Whenever you insert a negative in front of the amplitude value of any trigonometric equation, the whole graph reflects over the midline. Keep this in mind moving forward. Now, with all that being said, no matter how far the graph shifts vertically, the midline will ALWAYS follow.
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write the slope intercept form of the equation of the line with a slope of -2/5 that passes through 15, -9/2
Answers
The slope-intercept form of the equation of the line with a slope of -2/5 that passes through (15, -9/2) is y = -2/5x + 3/2.
Explanation:To write the slope-intercept form of the equation of a line, we use the formula y = mx + b, where m represents the slope and b represents the y-intercept. In this case, the slope is -2/5. To find the y-intercept, we substitute the coordinates of the point (15, -9/2) into the equation. Thus, we have:
y = -2/5x + b
-9/2 = -2/5(15) + b
-9/2 = -6 + b
b = -9/2 + 6 = 3/2
Therefore, the equation of the line with a slope of -2/5 that passes through (15, -9/2) is y = -2/5x + 3/2.
Final answer:
The equation of the line with a slope of -2/5 that passes through (15, -9/2) in slope-intercept form is y = (-2/5)x + 3.
Explanation:
The student is asking for the slope-intercept form of an equation of a line with a given slope and a point through which it passes. The slope-intercept form is given by y = mx + b, where m represents the slope and b represents the y-intercept. Given that the slope (m) is -2/5 and the line passes through the point (15, -9/2), we can substitute the slope and the point's coordinates into the slope-intercept formula to find b. Doing so, we get -9/2 = (-2/5)(15) + b. Solving for b, we find that the intercept is 3. Finally, the equation of the line in slope-intercept form is y = (-2/5)x + 3.
Stacey has a square piece of cloth. She cuts 3 inches off of the length of the square and 3 inches off of the width. The area of the smaller square is 1/4the area of the original square. What was the side length of the original square?
Answers
Answer:
The side length of the original square was 6 inches
Step-by-step explanation:
we know that
The area of a square is
[tex]A=b^2[/tex]
where
b is the length side of the square
Let
x ---> the length of the original square
The area of the original square is
[tex]A=x^{2}\ in^2[/tex]
The length of the smaller square is
[tex]b=(x-3)\ in[/tex]
The area of the smaller square is
[tex]A=(x-3)^2\ in^2[/tex]
The area of the smaller square is 1/4 the area of the original square
so
[tex](x-3)^2=\frac{1}{4} x^{2}[/tex]
solve for x
[tex]x^2-6x+9=\frac{1}{4} x^{2}[/tex]
Multiply by 4 both sides
[tex]4x^2-24x+36=x^{2}[/tex]
[tex]4x^2-x^2-24x+36=0\\3x^2-24x+36=0[/tex]
Solve the quadratic equation by graphing
using a graphing tool
x=2, x=6
see the attached figure
The solution is x=6 in
Remember that the solution must be greater than 3 inches (because Stacey cuts 3 inches off of the length of the square and 3 inches off of the width)
therefore
The side length of the original square was 6 inches
write the following decimals in order from smallest to largest: .021, .12, .2, .02
Answers
Item 17
Are the expressions 8x2+3(x2+y) and 7x2+7y+4x2−4y equivalent? Which best explains your reasoning?
Answers
Answer: yes
Step-by-step explanation: the expressions are equivalent because they both simplify to 11x^2 +3y
Yes, the expressions [tex]8x^2 + 3(x^2 + y) and 7x^2 + 7y + 4x^2-4y[/tex]- are equivalent.
To see the equivalence, we can simplify both expressions.
Start with the first expression: [tex]8x^2 + 3(x^2 + y)[/tex].
Distribute the 3 inside the parentheses: [tex]8x^2 + 3x^2 + 3y[/tex].
Now, combine like terms: [tex]11x^2 + 3y[/tex].
Now, let's simplify the second expression: [tex]7x^2 + 7y + 4x^2 - 4y[/tex].
Combine like terms: [tex]7x^2 + 4x^2 + 7y - 4y[/tex], which also results in [tex]11x^2 + 3y[/tex].
Both expressions simplify to the same form, [tex]11x^2 + 3y[/tex], which demonstrates their equivalence.
The coefficients and variables are the same in both expressions, just arranged differently, but they yield the same result when simplified.
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what is the measure of angle A, in degrees, in the figure shown?
Answers
Answer:
12.7 degrees
Step-by-step explanation:
there are 180 degrees in a straight line, and since the angle opposite a is 167.3, we subtract that from 180 and we get 12.7
Find the next three terms of the geometric sequence -8, 12, -18,27
Answers
Answer:
-40.5, 60.75, -91.125
Step-by-step explanation:
first you want to find the common difference. So just divide 12 by -8, which is -1.5 and then multiply 27 by -1.5, which is -40.5 and continue two more times.
What is the solution of the system?
10x – 2y = 24
6x+2y = 8
Answers
Answer: x = 2 , y = 5
Step-by-step explanation:
10x - 2y = 24 ..................... equation 1
6x + 2y = 8 ........................ equation 2
solving the system of linear equation by elimination method , add equation 1 and 2
16x = 32
divide through by 16
x = 2
substitute x = 2 into equation 1 to find the value of y
10(2) - 2y = 2y
20 - 2y = 2y
20 = 4y
y = 5
Describe Bob’s data in terms of center, spread, and shape.
Answers
Step-by-step explanation:
Bob's points per game are
5, 7, 7, 7, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 12, 12, 12, 12, 13, 13, 13, 14, 14, 14, 15, 15
Calculating Median:
As the median is the middle number in a sorted list of numbers when there is an odd number of terms.
As the the total number of terms = 37
Therefore, the median is the center term which is 10.
5, 7, 7, 7, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 12, 12, 12, 12, 13, 13, 13, 14, 14, 14, 15, 15
Calculating Range:
As the given data
5, 7, 7, 7, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 12, 12, 12, 12, 13, 13, 13, 14, 14, 14, 15, 15
The range is the difference between the highest and lowest values in the data set.
The lowest value is 5
The highest value is 15
The range = 15 - 5 = 10
Calculating the interquartile range (IQR)
The interquartile range is the difference between the third and first quartiles.
The first quartile (or lower quartile or 25th percentile) is the median of the bottom half of the numbers.So, the bottom half is
5 7 7 7 8 8 8 8 8 9 9 9 9 9 10 10 10 10
The median of these numbers is 8.5
The third quartile (or upper quartile or 75th percentile) is the median of the upper half of the numbers.So, the upper half is
10 11 11 11 11 11 12 12 12 12 13 13 13 14 14 14 15 15
The median of these numbers is 12
As
The third quartile is 12
The first quartile is 8.5
Therefore,
The interquartile range = 12 - 8.5 = 3.5
Finding Mode
As the given data
5, 7, 7, 7, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 12, 12, 12, 12, 13, 13, 13, 14, 14, 14, 15, 15
The mode of a set of data is the value in the set that occurs most often.
So, It is bimodal.
Therefore, the mode is 10.
Finding Mean
The mean of a data set is the sum of the terms divided by the total number of terms. Using math notation we have:
[tex]Mean=Sum\:of\:terms\:\div Number\:of\:terms[/tex]
[tex]Sum\:of\:terms\:=\:385[/tex]
[tex]Number\:of\:terms\:=\:37[/tex]
[tex]Mean\:=\:\frac{385}{37}=10.4[/tex]
Determining whether the data is symmetrical or non-symmetrical
The data is non-symmetric, they do not have about the same shape on either side of the middle. In other words, if you fold the histogram in half, it does not look about the same on both sides. Please check the histogram attached below.
Calculating Mean Absolute Deviation
The mean deviation is a measure of dispersion, A measure of by how much the values in the data set are likely to differ from their mean.
As the given data
5, 7, 7, 7, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 12, 12, 12, 12, 13, 13, 13, 14, 14, 14, 15, 15
Population size = 37
[tex]Mean=10.4[/tex]
Mean Absolute Deviation (MAD): 2.0
Keywords: mode, median, mean, non-symmetrical data, range, Interquartile Range (IQR)
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After school, Isaac skateboards directly from school to an ice cream parlor and then from the
ice cream parlor to a candy store. The ice cream parlor is 3 miles south of the school and the
candy store is 4 miles east of the ice cream parlor. What is the straight line distance between
the school and the candy store?
Answers
Answer:
The straight line distance between the school and the candy store is 5.
Step-by-step explanation:
Use Pythagorean theorem
a²+b²=c²
3²+4²=c²
9+16=c²
25=c²
5=c
The straight line distance between the school and the candy store will be 5 miles.
What is a right-angle triangle?It is a type of triangle in which one angle is 90 degrees and it follows the Pythagoras theorem and we can use the trigonometry function. The Pythagoras is the sum of the square of two sides is equal to the square of the longest side.
After school, Isaac skateboards directly from school to an ice cream parlor and then from the ice cream parlor to a candy store.
The ice cream parlor is 3 miles south of the school and the candy store is 4 miles east of the ice cream parlor.
Then the straight line distance between the school and the candy store will be
[tex]\rightarrow \sqrt{4^2 + 3^2}\\\\\rightarrow \sqrt{16 + 9}\\\\\rightarrow \sqrt{25}\\\\\rightarrow 5[/tex]
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Find the value of x
Answers
Answer:
85
Step-by-step explanation:
All triangles equal 180
Answer:
85
Step-by-step explanation:
(A triangle is 180 degrees)
52 + 43 = 95
180 - 95 = 85
Simplify the algebraic expression by combining like terms 0.8x + 19 - 0.7x
Answers
Answer:
Let's simplify step-by-step.
0.8x+19−0.7x
=0.8x+19+−0.7x
Combine Like Terms:
=0.8x+19+−0.7x
=(0.8x+−0.7x)+(19)
=0.1x+19
Answer:
=0.1x+19
Step-by-step explanation:
I hooe it helps plz Mark me brainliest
Please help me with this!
Answers
Answer:
Step-by-step explanation:
i don't know I am a forth grader
9e-7=7e-11 the answer
Answers
Answer:
e=-2
Step-by-step explanation:
9e-7=7e-11
9e-7e-7=-11
2e-7=-11
2e=-11+7
2e=-4
e=-4/2
e=-2
Answer:
e=2
Step-by-step explanation:
The solution set for -18 < 5 x - 3 is _____.
a -3 < x
b 3 < x
c -3 > x
d 3 > x
Answers
-18 < 5x - 3 Isolate/get x by itself, first add 3 to both sides of the equation
-18 + 3 < 5x - 3 + 3
-15 < 5x Divide 5 on both sides
-3 < x Your answer is A
You flip the sign [</>] if you multiply or divide a negative number to both sides of the equation.
What is the domain of this relation?
Answers
Answer:
The domain is the set of all first elements of ordered pairs (x-coordinates). The range is the set of all second elements of ordered pairs (y-coordinates). Only the elements "used" by the relation or function constitute the range.
Step-by-step explanation:
what dose 5x+3-48 mean because im strugling in math at south county middle school and im doing this on horizon
Answers
Answer:the answer is 40
Step-by-step explanation:
Because you do 5 times x
Answer:-9
Step-by-step explanation:5x+3-48 you do 3-48 you get -45 then you divide 5x by -45 and you get the final answer of -9.
ryan has a mix of action and sports games for a total of 39 video games his action collection is three more than half his sports collection how many action games and how many sports games does He have
Answers
Answer:
He have 15 action games and 24 sports games.
Step-by-step explanation:
Given:
Ryan has a mix of action and sports games for a total of 39 video games his action collection is three more than half his sports collection.
Now, to find the number of action games and sports games.
Let the sports games be [tex]x.[/tex]
As, given his action collection is three more than half his sports collection.
Thus, the action games = [tex]3+\frac{x}{2}[/tex] .
Total number of video games = 39.
So, we set an equation to get the number of action games and sports games:
[tex]x+(3+\frac{x}{2})=39[/tex]
[tex]x+3+\frac{x}{2} =39[/tex]
Subtracting both sides by 3 we get:
[tex]x+\frac{x}{2} =36[/tex]
[tex]\frac{2x+x}{2}=36[/tex]
[tex]\frac{3x}{2} =36[/tex]
Multiplying both sides by 2 we get:
[tex]3x=72[/tex]
Dividing both sides by 3 we get:
[tex]x=24.[/tex]
The sports games he have = 24.
Now, to get the action games we substitute the value of [tex]x[/tex]:
[tex]3+\frac{x}{2}[/tex]
= [tex]3+\frac{24}{2}[/tex]
= [tex]3+12[/tex]
= [tex]15.[/tex]
The action games he have = 15.
Therefore, he have 15 action games and 24 sports games.
The problem involves a basic algebraic system of equations. By defining variables, setting up the equations based on the problem statement, and solving the system, it is found that Ryan has 24 sports games and 15 action games.
Explanation:This is a problem related to the field of algebra in mathematics. To find the answer, we first need to assign variables: let's denote the number of sports games as 's' and the number of action games as 'a'.
From the problem, we know that the total number of sports and action games is 39, which can be written in algebra as: a + s = 39.
Also, the problem states that the number of action games is three more than half of the number of sports games. That means, a = s/2 + 3.
Now we need to solve this system of equations. Substituting the second equation into the first (replacing 'a' in the equation 'a + s = 39' with 's/2 + 3'), we get s/2 + 3 + s = 39. Solving this equation for 's', we find that Ryan has 24 sports games.
Substitute s = 24 into the first equation we got 'a + 24 = 39' . Solving this for 'a', we find that Ryan has 15 action games.
Therefore, Ryan has 24 sports games and 15 action games.
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.10.A cone with a height of 15 yards has a volume of 457.17 yd3 . Find the diameter of the cone
Answers
[tex]\bf \textit{volume of a cone}\\\\ V=\cfrac{\pi r^2 h}{3}~~ \begin{cases} r=radius\\ h=height\\[-0.5em] \hrulefill\\ V=457.17\\ h = 15 \end{cases}\implies 457.17=\cfrac{\pi r^2(15)}{3}\implies 457.17=5\pi r^2 \\\\\\ \cfrac{457.17}{5\pi }=r^2\implies \sqrt{\cfrac{457.17}{5\pi }}=r\implies 5.39 \approx r~\hfill \boxed{\stackrel{diameter = 2r}{2(5.39) = 10.78}}[/tex]
F(x)=3x+5
G(x)=4x^2-2
H(x)=x^2-3x+1
Find f(x) + g(x)-h(x)
A.6x^2+6x-1
B.5x^2+4
C.3x^2+2
D.3x^2+6x+2
Answers
Answer:
Step-by-step explanation:
F(x) = 3x + 5
G(x) = 4x^2 - 2
H(x) = x^2 - 3x + 1
F(x) + G(x) - H(x) =
3x + 5 + 4x^2 - 2 - (x^2 - 3x + 1) =
3x + 5 + 4x^2 - 2 - x^2 + 3x - 1 =
3x^2 + 6x + 2 <===== ur answer
Answer:
it's a D
Step-by-step explanation:
3x^2+6x+2
3. PENTAGON The Pentagon is a five-sided
building that houses the U.S.
Department of Defense in Arlington,
Virginia. If each side of the Pentagon
measures 921 feet, what is the walking
distance around its exterior?
Answers
The Pentagon has five sides, each measuring 921 feet. To find the walking distance around its exterior, multiply the length of one side by five. The total distance is 4605 feet.
The student is asking about the walking distance around the exterior of the Pentagon, a five-sided building. Given that each side of the Pentagon measures 921 feet, we would calculate the perimeter for the distance by multiplying the length of one side by the number of sides.
To find the walking distance around the Pentagon we perform this calculation:
Perimeter of Pentagon = Side Length × Number of Sides
Perimeter of Pentagon = 921 feet× 5
Perimeter of Pentagon = 4605 feet
Therefore, the walking distance around the Pentagon is 4605 feet.
What is x in -3x-8y=20, -5x+y=19
Answers
Answer:
x = -4
Step-by-step explanation:
solving simultaneously
What is the slope of y-3=-4(x-5)
Answers
Answer:
-4
Step-by-step explanation:
y-3=-4(x-5)
y=-4x+20+3
y=-4x+23
y=mx+b where m=slope and b=y-intercept
suppose a 9 minute overseas call costs $6.48 and a 19 minute call costs $13.68. What is the cost, c, of a call of m minutes duration? write a equation to model the cost.
Answers
9m = 6.48
Divide both sides by 9:
m = 0.72
19m = 13.68
Divide both sides by 19:
m = 0.72
So a one minute call costs 72 cents. We can now set up an equation like this:
c = 0.72m
8 1/3 subtract 2 2/3
Answers
Answer:
5 2/3........................
Solve each system of equations by substitution
Y=4x+13 y=6x+19
Answers
Answer:
I believe the answer to this question is: (-3,1) X=-3, Y=1.
Will mark brainliest once it becomes available
Answers
Answer:
m=47
Step-by-step explanation:
Answer:
m=47
Step-by-step explanation:
Each marble bag sold by Latoya’s Marble Company contains 7 purple marbles for every 4 red marbles. If a bad has 32 red marbles, how many purple marbles does it contain?
Answers
Answer:
Step-by-step explanation:
7 purple to 4 red = x purple to 32 red
7 / 4 = x / 32....cross multiply because this is a proportion
(4)(x) = (7)(32)
4x = 224
x = 224/4
x = 56 purple <==
Final answer:
To find the number of purple marbles in a bag that has 32 red marbles, given the ratio of 7 purple marbles for every 4 red marbles, simply divide 32 by 4 to find the number of sets and multiply by 7. The bag will contain 56 purple marbles.
Explanation:
Given the ratio of 7 purple marbles to 4 red marbles in a bag sold by Latoya's Marble Company, we can determine the number of purple marbles if a bag has 32 red marbles. To do this, we need to use the ratio to find out how many times the number of red marbles fits into the given amount and then multiply that number by the amount of purple marbles in the ratio.
Step-by-Step Calculation:
Start with the given ratio: 7 purple marbles for every 4 red marbles.
Determine how many sets of 4 red marbles there are in 32 red marbles by dividing 32 by 4, which is 8.
Since there are 7 purple marbles for every set, multiply 7 by the number of sets, which is 8.
The result is 56 purple marbles.
Therefore, a bag with 32 red marbles will contain 56 purple marbles.
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Answers
Knowing this:
2x + (x - 12) = 180
(2x and (x - 12) are same side angles)
3x - 12 = 180
3x = 192
x = 64
Do the same for the other pair of same-side angles:
3y + (y + 20) = 180
4y + 20 = 180
4y = 160
y = 40
x = 64 and y = 40
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-T.B.
Consider the following piece-wise function. Which of the below correctly describes the graph shown?
Answers
Answer:
C.
[tex]f(x)=\left\{\begin{array}{l}\dfrac{1}{2}x-3,\ \ x<2\\ \\3x-8,\ \ x\ge 2\end{array}\right.[/tex]
Step-by-step explanation:
The graph of the piecewise function consists of two linear functions which graphs meets at point with x-coordinate x = 2.
1. The left linear function is determined for all x < 2 and is passing through the points (0,-3) and (-2,-4). So, its slope is
[tex]\dfrac{-4-(-3)}{-2-0}=\dfrac{-1}{-2}=\dfrac{1}{2}[/tex]
and its equation is
[tex]y-(-3)=\dfrac{1}{2}(x-0)\\ \\y=\dfrac{1}{2}x-3[/tex]
2. The right linear function is determined for all x ≥ 2 and is passing through the points (2,-2) and (3,1). So, its slope is
[tex]\dfrac{1-(-2)}{3-2}=\dfrac{3}{1}=3[/tex]
and its equation is
[tex]y-1=3(x-3)\\ \\y=3x-8[/tex]
3. Hence, the expression for the piece-wise function is
[tex]f(x)=\left\{\begin{array}{l}\dfrac{1}{2}x-3,\ \ x<2\\ \\3x-8,\ \ x\ge 2\end{array}\right.[/tex]
The slope of a line is the rate of change. The correct option is the first option.
What is the slope of a line?The slope of a line is the rate of change, also it can be described as the gradient of a line is a number that helps to know both the direction and the steepness of the line.
In order to know the correct option of function that describes the graph, we will find the function of the graph, therefore,
Let's find the equation of the first function that is on the left side of the graph.
We know that the slope of the line is given by the formula,
[tex]\text{Slope of the line} = \dfrac{y_2-y_1}{x_2-x_1}[/tex]
Considering the two of the points of the slope,
[tex](x_1, y_1) = (2,-2)\\(x_2, y_2) = (-4,-5)\\[/tex]
Substituting the values to know that slope of the line, therefore,
[tex]\text{Slope of the line}, m= \dfrac{-5-(-2)}{-4-2} = \dfrac{-3}{-6} = \dfrac{1}{2}[/tex]
Now, using one of the points to know the value of the constant,
[tex]y=mx+c\\y_2 = mx_2+c\\-5=(\dfrac{1}{2}\times -4)+c\\-5=-2+c\\-3=c[/tex]
Hence, the equation that will represent the graph for the values of x less than -2 is [tex]y=\dfrac{1}{2}x-3[/tex].
Since, there are only two options that match this condition, the first option and the third option, now other functions will represent the rest of the graph, therefore, for the value of x greater than -2 (x≥-2).
Hence, the correct option is the first option.
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