Slope-intercept form is often expressed using the formula y = mx + b, where m denotes the line's slope and b denotes the y-value of the line's y-intercept.
Define the slope of a line?The ratio of the change in the y coordinate to the slope of a line in mathematicsThe formula for the change in coordinate with respect to the changeAny three numbers that are proportionate to the line's direction cosines can be used to represent a line's direction ratios. In addition to direction ratios, direction numbers are also used.The steepness and direction of a line are frequently indicated by its slope.It is necessary to compute the ratio of the coordinate differences between the two points that make up the line. The result is a value that is determined by the line's slope. When compared to its, it exhibits.To learn more about Slope of Line refers to:
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What is the 4th equivalent fraction to 3/5
Answer:
If you are refering to the 4th in a sequence being 3/5 the first.
3/5, 6/10, 12/20, 24/40
24/40 would be your answer.
If you are looking for 3/5 in 4ths
2.4/4
7. A radio station is going to construct a 24-foot tower for a new antenna. The tower will be supported by three cables, each attached to the top of the tower and to points on the roof of the building that are 10 feet from the base of the tower. Find the total length of the three cables. Drawing it will help.
78 ft is the total length of the 3 cables.
How do you explain the Geometry?Geometry is the study of various mathematical or actual-world forms, figures, and sizes. In geometry, we study various angles, transformations, and figure-to-figure similarities. The fundamental concepts of geometry primarily depend on point, line, angle, and plane.Geometry, which derives from the Ancient Greek words "gemetra" (meaning "land measurement"); "gê" (meaning "earth, land," and "v" (metron) "a measure") Along with arithmetic, [citation needed] is one of the earliest subfields of mathematics. It is concerned with spatial characteristics including the separation, form, size, and relative placement of objects. A geometer is a mathematician who specialises in geometry.Euclidean geometry, which incorporates the ideas of point, line, plane, distance, angle, surface, and curve as fundamental concepts, was nearly entirely the focus of geometry up until the 19th century.Given data :
[tex]24^{2}[/tex]+[tex]10^{2}[/tex] = [tex]d^{2}[/tex]
576 +100 = [tex]d^{2}[/tex]
676 = [tex]d^{2}[/tex]
d = 26
d = 26 is the length of the cable.
3 x 26 = 78 ft is the total length of the 3 cables.
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Zach scores a ringer 40\%40%40, percent of the time that he throws a horseshoe. Let rrr be the number of throws it takes zach to score his first ringer in a game. Assume the results of each throw are independent. Find the probability that it takes zach 555 or more throws to score his first ringer.
There is about a 7.3% probability that it will take Zach 555 or more throws to score his first ringer.
To solve this problem, we can use a geometric distribution, which models the number of trials it takes to get a success in a series of independent Bernoulli trials (trials with two outcomes: success or failure) with a fixed probability of success.
In this case, the probability of success is 40% (the probability of scoring a ringer), and the probability of failure is 60% (the probability of not scoring a ringer). We can use the formula for the probability mass function of a geometric distribution to calculate the probability that it takes Zach 555 or more throws to score his first ringer:
P(X ≥ 555) = 1 - P(X < 555)
= 1 - (P(X = 1) + P(X = 2) + ... + P(X = 554))
= 1 - (0.4 + (0.4 * 0.6) + (0.4 * 0.6^2) + ... + (0.4 * 0.6^554))
To calculate this probability, we can use a computer to sum up the series, or we can use a calculator with a geometric distribution function. Using a calculator, we get P(X ≥ 555) ≈ 0.073, or about 7.3%.
This means that there is about a 7.3% probability that it will take Zach 555 or more throws to score his first ringer.
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In this case, the probability of success is 40% (the probability of scoring a ringer), and the probability of failure is 60% (the probability of not scoring a ringer). We can use the formula for the probability mass function of a geometric distribution to calculate the probability that it takes Zach 555 or more throws to score his first ringer:
P(X ≥ 555) = 1 - P(X < 555)
= 1 - (P(X = 1) + P(X = 2) + ... + P(X = 554))
= 1 - (0.4 + (0.4 * 0.6) + (0.4 * 0.6^2) + ... + (0.4 * 0.6^554))
To calculate this probability, we can use a computer to sum up the series, or we can use a calculator with a geometric distribution function. Using a calculator, we get P(X ≥ 555) ≈ 0.073, or about 7.3%.
The plane intersects the paraboloid in an ellipse. Find the points on this ellipse that are nearest to and farthest from the origin. Point farthest away occurs at.
The point on the ellipse that is farthest from the origin will be the point that is closest to one of the foci.
To find the points on the ellipse that are nearest to and farthest from the origin, we need to consider the equation of the paraboloid and the equation of the ellipse.
The equation of a paraboloid is given by:
[tex]z = ax^2 + by^2 + c[/tex]
where a, b, and c are constants.
The equation of an ellipse is given by:
[tex](\frac{x}{a})^2 + (\frac{y}{b})^2 = 1[/tex]
where a and b are the semi-major and semi-minor axes of the ellipse, respectively.
If the plane intersects the paraboloid in an ellipse, then the equations of the paraboloid and the ellipse must be equal at the points of intersection. Therefore, we can set the equation of the paraboloid equal to the equation of the ellipse and solve for the x and y values.
[tex]z = ax^2 + by^2 + c[/tex]
[tex](\frac{x}{a})^2 + (\frac{y}{b})^2 = 1[/tex]
Substituting the equation of the paraboloid into the equation of the ellipse, we get:
[tex](\frac{x}{a})^2 + (\frac{y}{b})^2+c = 1[/tex]
We can then solve this equation for the values of x and y that correspond to the points on the ellipse that are nearest to and farthest from the origin.
The point farthest away from the origin occurs at the point on the ellipse that is farthest from the center, which is called the "foci" of the ellipse. The distance from the center of the ellipse to the foci is called the "eccentricity" of the ellipse. The eccentricity is denoted by the letter "e" and is calculated using the following formula:
[tex]e = \frac{\sqrt{(a^2 - b^2)}}{a}[/tex]
where a and b are the semi-major and semi-minor axes of the ellipse, respectively.
Therefore, to find the point on the ellipse that is farthest from the origin, we need to calculate the eccentricity of the ellipse and then use this value to determine the location of the foci. The point on the ellipse that is farthest from the origin will be the point that is closest to one of the foci.
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Calculate the mean for the discrete probability distribution shown here. Use the formula: Hx = E[x - P(x)] х 2 6 10 14 P(x) 0.2 0.3 0.3 0.2 M = 4x • P(x) = 2(0.2) +66(0.3) + 1020.3+420.2 = 18 1. Using the information in the above problem, calculate the standard deviation using your calculator.
The mean for the discrete probability distribution is 3.73
What is a discrete probability distribution?Generally, To calculate the mean of a discrete probability distribution, you can use the formula:
Mean = ∑x * P(x)
Using this formula, the mean of the given distribution is:
Mean = (2 * 0.2) + (6 * 0.3) + (10 * 0.3) + (14 * 0.2) = 4
To calculate the standard deviation of a discrete probability distribution, you can use the formula:
Standard Deviation = √∑(x - mean)^2 * P(x)
Using this formula and the mean that we calculated above, the standard deviation of the given distribution is:
Standard Deviation = √( (2 - 4)^2 * 0.2 + (6 - 4)^2 * 0.3 + (10 - 4)^2 * 0.3 + (14 - 4)^2 * 0.2 )
= √( (-2)^2 * 0.2 + (2)^2 * 0.3 + (6)^2 * 0.3 + (10)^2 * 0.2 )
= √( 4 * 0.2 + 4 * 0.3 + 36 * 0.3 + 100 * 0.2 )
= √( 0.8 + 1.2 + 10.8 + 20 ) = √( 13.8 )
= approximately 3.73
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Marissa's mom purchased coffee from the local coffee shop. The first cup of coffee was $3.95. Each refill is $1.50. She also decided to get a bagel for $3.00. If Marissa's mom had 2 refills and a $0.50 discount as part of the rewards program, how much did she spend at the coffee shop?
$9.95
$9.45
$8.65
$7.95
Answer: $9.45
Step-by-step explanation:
The first cup is $3.95
2 refills are $1.50 times 2 = $3
A bagel is $3
3.95+3+3 - 0.50 = $ 9.45
Answer:
$9.45
Step-by-step explanation:
add 3.95, 1.50, 1.50, and 3.00, then subtract the $.50 and you get 9.45
What theorems apply to equilateral triangles?
Theorems apply for equilateral triangles are theorem 1 and 2, In theorem 1 each angle of an equilateral triangle is the same and measures 60 degrees each while in theorem 2 a triangle is said to be equilateral if and only if it is equiangular.
What is theorems?Theorem can be defined as a statement which has been proved true by a special kind of logical argument called a rigorous proof.
There are different types of theorem such as:
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What angle is bigger than right?
Answer:
look this up next time..with this said, obtuse
Step-by-step explanation:
What is nature of roots formula?
The nature of the roots for the equation ax² + bx + c is
α + β = -b/a and αβ = c/a
where α and β are the two roots of the equation.
What is an expression?An expression is a way of writing a statement with more than two variables or numbers with operations such as addition, subtraction, multiplication, and division.
Example: 2 + 3x + 4y = 7 is an expression.
We have,
A quadratic equation ax² + bx + c.
Now,
There will be two roots of the equation α and β,
Now,
The nature of the roots are
α + β = -b/a
αβ = c/a
Thus,
The roots are α and β and α + β = -b/a and αβ = c/a.
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The nature of roots of quadratic depends on its discriminant, formula is D=√(b²-4ac).
What is root of quadratic equation?It is the value of the unknown variable for which the quadratic equation holds true. Its nature can be real, imaginary, rational, irrational, equal and unequal as well that depends on the value of the discriminant.
For equation ax²+ bx + c =0 , expression b²- 4ac is called discriminant and denoted by D.
Given equation is , ax²+ bx + c =0
then, according to quadratic formula for the roots of quadratic equation
x = {-b±√(b²-4ac)}/4ac
Let us take a real number
k > 0
√k can be defined and is a positive quantity but √-k can not be defined.
The quantity which is under square root in the expression for roots is
b²-4ac, which is called discriminant of the quadratic equation. It is represented by D.
If D>0 and is perfect square then the equation will have two real, rational and distinct roots.
If D>0 and is not a perfect square then the equation will have two real, irrational and distinct roots.
If D=0 the equation will have two real and equal roots.
If D<0, the equation will have no real roots.
Hence, nature of roots formula for quadratic equation depends on the value of √b²-4ac.
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What are the 3 ways to prove similarity?
The 3 ways to prove similarity are Angle - Angle (AA), Side - Angle - Side (SAS), and Side - Side - Side (SSS).
The three theorems are as follows :
Two triangles are said to be compared according to the Angle-Angle (AA) principle if they have two pairs of corresponding angles that are congruent.The Angle-Side-Side (SAS) According to the theorem, two triangles are similar if two of their sides are proportionate to two of their corresponding sides in another triangle and their corresponding included angles are congruent.The Side-Side-Side (SSS) theorem states that two triangles are comparable if their sides are proportionate.To learn more about similarity here:
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The amount P of pollution varies directly with the population N of people. City A has a population of 468,000 and produces 260,000 tons of pollutants. Find how many tons of pollution we should expect City B to produce, if we know that its population is 350,000.
Answer: you should expect 218,750 tons of pollutants from City B.
Step-by-step explanation:
When two variables vary "directly", it means that they fit into the equation y=mx, where y varies directly with x, with the multiplier m to be determined by the information you are given.
For this problem, P (pollutants) varies directly with N (people) so the equation is P=mN for their relation.
You are given that City A has P = 260,000 tons of pollutants and N = 416,000 people. Plug these into the equation above (P=mN) and you get 260,000=(m)(416,000). If we divide each side by 416,000, you get that m=0.625. So now you have the multiplier for your equation: P=(0.625)N.
Use the complete equation now to find out how many P (pollutants) City B will produce.
P=(0.625)N
P=(0.625)(350,000) [[350,000 was the given "N" or amount of people in City B]]
P=218,750 tons of pollutants
What are the vertical asymptotes of f/x 10 /( x 2 1?
Taking the function given in the question [tex]f(x) = \frac{10}{x^{2} +1}[/tex] , there are no vertical asymptotes since there no any real values for x , which makes the function undefined.
What is vertical asympotote?A vertical line that directs the function's graph but is not actually a part of it is called a vertical asymptote. Because it appears at an x-value outside of the function's scope, the graph can never cross it. There may be several vertical asymptotes for a given function.
How to calculate vertical asymptote of a function?To calculate the vetical asymptotes of a function simply equate the denominator of the equation to zero , which when solved gives the values of x at which the function is undefined. Thus the obtained values of x are the vertical asymptotes to the given function.
In the given function f(x), let's equate the denominator to zero to find the vertical asymptotes,
[tex]x^{2} +1 =0[/tex]
Solving the above equation, we get
x = [tex]\sqrt{-1}[/tex]
which is imaginary, hence there is no vertical asymptote for the given function.
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What is the graph of system of linear inequalities in two variables?
the graphing system of linear inequalities of two variables are bounded by solid line or dashed line.
what are the linear inequalities?The linear inequalities of two variables are the mathematical expressions denoted by greater than > or less than< or combination of ≤ or ≥ between two algebraic terms.
How to solve and plot linear inequalities?the way we solve linear inequalities are same as the way in how we solve linear equation. The graph of linear inequalities of two variables are bounded by solid line or dashed line depending on the unequal sign presented.
hence, there is a difference between the graph of linear equation and linear inequalities.
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What are the ways to prove two triangles are similar?
Using the SAS(Side-Angle-Side) Theorem, ASA(Angle-Side-Angle) Theorem, SSS(Side-Side-Side) Theorem and AA(Angle-Angle) Theorem we can easily prove that two triangles are similar.
When the sides and angles of two triangles match, the triangles are said to be congruent. When the angles of two triangles match, the triangles are said to be similar. Thus, two triangles can be superimposed side to side and angle to angle.
Hence, when the sides and angles of two triangles match that triangle is congruent and When the angles of two triangles match that triangle is similar.
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Fransisco gana
*650,16 mensuales.
¿Cuánto ganará Francisco en un año?
If Francisco wins 650.16 per month, he will earn a total of 7,801.92 per annum. Note that this is a multiplication exercise.
What is Multiplication?It is to be noted that Multiplication is a mathematical operation that involves multiplying two or more numbers together. It is often represented using the symbol "x" or "*".
Since there are 12 months in a year, and Francisco earns 650.16 per annum,
Hence,
12 x 650.16 = 7,801.92
Thus, it is correct to state that where Francisco gets 650.16 per month, by the year's end, he will have earned a total of 7,801.92.
Note that Per Year and Per Annum are synonyms and can be used interchangeably.
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How do you work out 15x15?
The product is 225.
What is multiplication?Augmentation is an activity that addresses the fundamental thought of rehashed expansion of a similar number. The numbers that are multiplied are referred to as the factors, and the result that is produced when two or more numbers are multiplied together is referred to as the product of those numbers. Repeated addition of the same number can be made easier by using multiplication.
Given 15 x 15
the product can be done in many ways,
like add 15times 15 we get the answer,
or 15 is written as 10 + 5
so (10 + 5) x 15
10 x 15 + 5 x 15
150 + 75 = 225
Therefore, 15 x 15 is 225
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Why we use mean method?
The assumed mean method is used to determine the mean or arithmetic mean of grouped data. If there is a large amount of data available, this method is suggested over calculating the mean directly.
What does the mode mean?
By summing all the numbers in a data collection and dividing by the total number of values in the set, the mean (average) of the data set can be determined.
A data collection is ordered from least to largest, and the median is the midpoint number. A data set's mode is the number that appears the most frequently.
What are the mean, median, and example?
The most frequent number, or the one that happens the most frequently, is known as the mode.
Example: Since the number 2 appears three times, more than any other number, it is the mode of the numbers 4, 2, 4, 3, and 2.
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In a company, 60% of the workers are men. If 1,880 people work for the company who aren't men, how many workers are there in all? Use pencil and paper. Show two different ways that you can solve this problem.
Answer:
4700 workers
Step-by-step explanation:
We know that 60% of the worker are men
1880 people work for the company who aren't men
100% - 60% = 40%
40% of workers = 1880
40% = 0.4
If 0.4 = 1880
100% of the workers x
100% = 1
We have equations
0.4 = 1880
1 = x
Cross multiply
0.4x = 1880
Divided both sides by 0.4
x = 4700 workers total
To find the number of men workers, we take
4700 - 1880 = 2820 men workers
So, there are a total of 4700 workers!
1880 women workers & 2820 men workers!
If you can, please give me a Brainliest; thank you!
Answer:
FIRST SOLUTION: (proportion)60% are men. 40% women
x - amount of men, 1 880 amount of women
60:x = 40:1 880
40x = 60(1 880)
40x = 112 800
x = 2 820
First solution said that there are 2 820 men and 1 880 womenthere are 4 700 workers all in allSECOND SOLUTION: base rate percentagePercentage = Rate / Base
= 1 880 / (1 - 0.60) (1-0.60 to find out the percent of not men)
= 1 880 / 0.40
= 4 700
There are total of 4 700 workers because there are 1 880 women meaning 4 700 - 1 880 = 2 820There are 2 820 womenCHECKING:Women: 4 700 * 0.4 = 1 880 (this statement is true)
Men: 4 700 * 0.6 = 2 820 (therefore, it is too)
Step-by-step explanation:
heart and star pls <3 brainliest will be appreciated <3(っ◔◡◔)っ -{ elyna s }-what is the lowest term of 2/30
Answer:
1/15
step by step
divide both numerator and denominator by 2
2÷2=1
30÷2=15
Is range and domain same?
No, the range and the domain are very different concepts because the domain is just a collection of natural numbers, whereas the output values are referred to as the range.
what is domain ?A function's authorized range of values is known as its domain. The x values in a function like f make up this set (x). A function's range is the collection of values that it can take as input. When we enter an x value, the function outputs this set of values.
here
The set of all feasible function inputs is known as a function's domain. The function f(x) = 2x can be represented by this box.
The domain is the set of all natural numbers when x = 11, 12, 13, 14, etc., the values that are returned are referred to as the range.
No, the range and the domain are very different concepts .
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The mapping diagram shows a functional relationship.
A mapping diagram shows a relation, using arrows, between domain and range for the following ordered pairs: (negative 1, 0), (0, 3), (3, negative 1), (9, 9).
Complete the statement.
f(3) is
The value of function f(3) is -1
What is a Mapping Diagram?
A mapping diagram is an image of two circles which shows two values corresponding to each other, such that one is the input value to the function and the other is the output value.
Given ordered pairs :
(- 1, 0), (0, 3), (3, - 1), (9, 9)
Domain and range from given ordered pairs.
Domain - means x - axis
Range - means y-axis
Now, take out domain and range like that,
Domain of the function : (-1, 0, 3, 9) [ input values]
Range of the function : (0, 3, -1, 9) [ output values]
Now, we need to determine the value of function f(3).
It means 3 is input value in f(3) so the output value at 3 is,
f(3) = -1
Hence, the value of function f(3) is -1.
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The lengths of the sides of a triangle are three consecutive integers. The perimeter of the triangle is 48 in. What is the length of the longest side of the triangle?.
The required lengths of side of triangle are 15 in, 16 in and 17 in.
Given :
The sum of the angles of a triangle is always 180°
Given that :
Sides of a triangle are consecutive integers.
Also given that,
Perimeter of triangle is 48.
Let the sides of triangle are x, x + 1 and x + 2.
x + x + 1 + x + 2 = 48
3x = 48 - 3
3x = 45
divide by 3 on both sides
x = 15
The length of the triangles
x + 1 = 15 + 1 = 16
x + 2 = 15 + 4 = 17
The side lengths of the triangle are 15, 16 and 17.
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p
118⁰
q
r
s
Which angle is equal to 118°?
s
q
r
p
Professor Delta was shooting hoops in the gym.
The victim was not assaulted with a fallen
object.
Chemical poisoning was not the cause of the
injury.
Dr. Alpha was drinking pop in the movie theater.
S angle is equal to 118°,Professor Delta was shooting hoops in the gym.
How big is each angle measured?An angle's measure is 118 degrees less than its supplemental angle's measurement.Pairs of angles that add up to 180 degrees in total are known as supplementary angles.A protractor is used to measure angles in degrees (°). With the aid of a protractor, one may measure and calculate angles in terms of degrees. For instance, using a protractor, the black arrow in the illustration below points to 100°, crossing 90°. As a result, the angle is 100° in size.Since we always know the total of the two angles' measurements, we can always find one angle if we know the other angle.Let x be the additional angle's degree measurement.The other angle's measurement will now be x degree 118 degree.The 118° angle can also be expressed in terms of its complementary angle, which is 62°. The complementary angle is the angle that, when added to the 118To learn more about How big is each angle measured refer to:
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Mateo has a piece of fabric that is 2 1/4 yards long. He cuts it into pieces that are each 3/8 yard long. How many pieces of fabric does Mateo cut?
Answer:6
Step-by-step explanation: 2 1/4 is a mixed fraction and needs to be changed into an improper fraction:
The whole number 2 is multiplied by the denominator 4 which gives 8 and we add 1 from the numerator this gives 9 and we put this over the denominator 9/4
9/4÷3/8=
The first fraction stays the same and we flip over the second fraction and use the multiplication sign in between
9/4 x 8/3=72/12=6
The tops of the fractions are multiplied together to give 72
The bottoms of the fractions are multiplied together to give 12
What is the square root of 2 as a power of 2?
Answer:
[tex]2^{\frac{1}{2}}[/tex]
Step-by-step explanation:
Taking the square root of a number is equivalent to putting that number to the power of 1/2.
Step-by-step explanation:
x^(a/b)
means
[tex] \sqrt[b]{ {x}^{a} } [/tex]
the numerator of the exponent is the actual "to the power of" factor.
the denominator defines the degree of the root to be taken.
as any integer number is also a fraction like
6 = 6/1
this applies to any form of rational numbers like
x^0.25 = x^(25/100) = x^(1/4) =
[tex] \sqrt[4]{ {x}^{1} } = \sqrt[4]{x} [/tex]
with irrational numbers as exponents this gets trickier, as with infinite numbers without a pattern after the decimal point we have a problem to precisely specify the degree of the root, as we also have a problem to specify the irrational number itself.
we try to find the origin of the irrational number (like a square root)
[tex] {x}^{ \sqrt{3} } [/tex]
or
[tex] \sqrt[ \sqrt{3} ]{ {x}^{2} } = {x}^{2 \div \sqrt{3} } [/tex]
so,
[tex] \sqrt{2} = \sqrt[2]{2} = {2}^{1 \div 2} [/tex]
In Lucy's coin collection, 15% of the coins are quarters.
If there are 12 quarters, find the total number of coins in Lucy's coin collection
The required total number of Lucy's coins,when we have 15% of quarter coins are 80.
What is percent?percentage, a general worth demonstrating 100th pieces of any amount. One percent (represented 1%) is a hundredth part; in this way, 100% addresses the total and 200 percent determines two times the given amount. rate.
According to question:Quarters coins = 12
Let x be total number of coins Lucy has,then
15% of x = 12
15x/100 = 12
x = 1200/15
x = 80 coins
Thus, there are 80 coins in total.
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Leyla and her sister Maryam are excited to begin saving for a new bike they can share. Their aunt said that if they saved up enough money, she would split the cost with them and pay for half. The new bike will cost $110 total.
The girls concluded that they would have just enough if they saved the money from 2 days’ worth of Leyla’s paper route and 3 days’ worth of Maryam’s dog walking job. Maryam has to work a little more because her dog walking job pays $5 less than Leyla’s paper route. How much do the paper route and dog walking jobs each earn per day?
Answer:
paper route: $14 per daydog walking: $9 per dayStep-by-step explanation:
You want to know the pay for a paper route and dog walking if 2 days' paper route pay and 3 days' dog walking pay will total half of $110, while dog walking pays $5 less than the paper route.
SetupLet p represent the pay for a day's pay for the paper route. Then (p-5) is the day's pay for dog walking. The total pay is ...
2p +3(p -5) = 110/2
SolutionSimplifying the equation, we have ...
5p -15 = 55
p -3 = 11 . . . . . . . . divide by 5
p = 14 . . . . . . . . add 3
p -5 = 14 -5 = 9 . . . . . pay for dog walking
The paper route and dog walking pay 14 and 9 dollars per day, respectively.
__
Additional comment
We could write two equations, and solve by substitution. We would get the same single equation by doing that. In this solution method, it works well to let the variable represent the largest contributor (highest pay).
We could also solve the two equations by any of several other methods. The attached graph shows the solution by graphing.
someone pls help me answer this question!!!! look through the directions and give a thorough explanation. thank you!
The sailboat and the medical aid should meet at the points
(0, 7) and (5, 12).
What is a quadratic equaton?A quadratic equation is an algebraic expression in the form of variables and constants.
A quadratic equation has two roots as its degree is two.
A straight line on the coordinate plane is represented by a linear function. As an illustration, the equation y = 3x – 2 depicts a linear function because it is a straight line in the coordinate plane.
The intersection point between the two graphs y = - x² + 6x + 7 and y = x + 7 is the point where they meet each other.
Now, For the equation y = - x² + 6x + 7,
When x = 0, y = 7 ⇒ (0, 7).
when x = 1, y = 12 ⇒ (1, 12).
Now for y = x + 7,
when x = 1, y = 8 ⇒ (1, 8).
when x = 2, y =9 ⇒ (2, 9).
Now by taking the graph paper with each box of 1 square unit will plot these points and we obtain that at (0, 7) and (5, 12) these graphs intersect each other.
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How do you know if the y-intercept is positive or negative?
A positive y-intercept means that the line crosses the y-axis above the origin, and a negative y-intercept means that the line crosses below the origin. Arbitrary lines can be defined simply by changing the values of m and b.
Slope Intercept:
The slope intercept of the straight line is used to find the equation of the straight line. The slope intercept formula requires knowing the slope of the line and the intersection of the line with the y-axis. Consider a line with slope 'm' and y-intercept 'b'. The slope-intercept shape formula for a straight line with slopes 'm' and 'b' as y-intercept is given as y = mx + b.
Positive Intercept:
A positive slope refers to a line that is slanted and slopes upward when viewed from left to right. The positive slope of a straight line can be calculated using the formula m = (y2 - y1)/(x2 - x1) = Tanθ = f'(x) = dy/dx. A positive slope means that the two quantities represented along the two axes of the coordinate system increase or decrease simultaneously. A line with a positive slope is a positive value for the ratio of climb to run. A straight line with a positive slope makes an acute angle with the positive x-axis.
Positive slope (m) = +slope/run = +Δy/Δx
Negative Intercept:
Negative slope means that the coordinates of the two given points are negatively related. A line with a negative slope has m < 0, and the angle θ it makes with the positive x-axis is obtuse θ, such that 90° < θ < 180°. The slope-to-run ratio of the negative slope line is negative. where slope is the change in y value, denoted by Δy, and run is the change in x value, denoted by Δx.
Negative slope = -rise/run = -Δy/Δx
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For each graph below, state whether it represents a function. Graph 1 Graph 2 Graph 3 24 . 864 7 21 6 & -2 4- -6- Function? Yes O No Graph 4 Yes ONo Graph 5 Yes O No Graph 6 60 Function? Yes O No Yes ONo Yes O No
Using the vertical line test, there are only two graphs that represent functions (graph function): graph, 1 and graph 6.
A graph function is a graph that represents a function on a coordinate plane, which means that every point on the graph fulfills the function equation. In a graph function, an x corresponds to only one y.
If we want to know whether a graph represents a function or not, we can perform a vertical line test. In this test, we can drag a vertical line from left to right on the graph. If the vertical line intersects with the graph at one point at a time, then it's a function.
Otherwise, if the vertical line intersects with the graph at two or more points at a time, then it's not a function. It could happen when an x value gives two or more y values (1 input results in multiple outputs), which means the graph is not a function.
Now we take a look at the given picture.
Graph 1: when we drag a vertical line from left to right on the graph, it intersects with the graph at one point at a time. So yes, the graph represents a function.
Graph 2: when we drag a vertical line from left to right on the graph, it intersects with the graph at two points. So no, the graph doesn't represent a function.
Graph 3: when we drag a vertical line from left to right on the graph, it intersects with the graph at one point at first. But later on, it crosses two points in the graph. So no, the graph doesn't represent a function.
Graph 4: when we drag a vertical line from left to right on the graph, it intersects with the graph at two points. So no, the graph doesn't represent a function.
Graph 5: when we drag a vertical line from left to right on the graph, it also intersects with the graph at two points. So no, the graph doesn't represent a function.
Graph 6: when we drag a vertical line from left to right on the graph, it intersects with the graph at one point at a time. So yes, the graph represents a function.
So there are only two graphs that represent functions: graph, 1 and graph 6 since the other graphs fail the vertical line test.
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