The equation that models the total amount Anna saves, y, in terms of the number of days she adds to her savings, x, can be represented as y = 2x + 125, where 2x represents the total amount she saves from her allowance and 125 represents the initial amount she saved.
This is a linear equation in slope-intercept form, where the slope of the line is 2, indicating that she saves $2 per day, and the y-intercept is 125, indicating that she had already saved $125 before starting to save from her allowance. To graph this equation, we can plot the y-intercept at (0, 125) and use the slope to find other points on the line. For example, after 5 days, she would have saved y = 2(5) + 125 = 135 dollars, giving us the point (5, 135). Plotting this point and connecting it with the y-intercept will give us a straight-line graph.
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PLEASE HELP!!! MIDDLE SCHOOL MATH!!!!!!!!!!!!!!!!!!!!!
Which of the following represent angle measures in the figure shown? select all that apply
PLEASE LOOK AT PICTURE BELOW!!!!!! MUST SHOW WORK!!!!!!!!!!!!!!
The angle measures in the figure are 45°, 55° and 145° respectively.
How to determine the measure of angles?Angles are measured in degrees (°) using a protractor, which is a measuring device used to calculate or draw angles in terms of degrees
From the figure,
7x + 5 + 35 = 180 (angle on a straight line)
7x = 180-35-5
7x = 180-40
7x = 140
Making x the subject of the relation by dividing both sides by 7
7x/7 = 140/7
x = 20
Now, 7(20) +5 = 140+5 = 145°
Also, 90 -35 = 55
Therefore the angles that are measures of angles are 45°, 55°,145°
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What is the solution to this system of equations X=12-y and 2x+3y=so
Answer:
x = 7, y = 5
Step-by-step explanation:
Given the 2 equations
x = 12 - y → (1)
2x + 3y = 29 → (2)
Substitute x = 12 - y into (2)
2(12 - y) + 3y = 29 ← distribute and simplify left side
24 - 2y + 3y = 29
24 + y = 29 ( subtract 24 from both sides )
y = 5
Substitute y = 5 into (1)
x = 12 - 5 = 7
Solution is x = 7, y = 5
pls help solve this unseen therom of circle
(n.24)
The length of line EF is proved to be parallel to the length of line EF.
What is the prove that EF// BD?To prove that EF is parallel to BD, we need to show that the opposite interior angles are equal.
Let's denote the center of the circle as O, and the intersection point of BD and AE as X. Since ABCD is a square, we have:
AD = AB, andBD is a diagonal, so BD passes through the center of the circle O.From the above information, we can deduce that:
∠ABD = 45°, and
∠OBD = 90°.
Since line_AF = line_AE, we can also deduce that:
∠FAE = ∠FEA.
Now, let's consider the triangle AFE. We have:
∠AFE + ∠FAE + ∠FEA = 180° (sum of angles in a triangle)
∠AFE + 2∠FAE = 180° (substituting ∠FEA with ∠FAE)
∠AFE = 180° - 2∠FAE
Also, in triangle AXε, we have:
∠AXε + ∠AEX + ∠XAE = 180° (sum of angles in a triangle)
∠AXε + ∠AEX + ∠FAE = 180° (substituting ∠XAE with ∠FAE)
∠AXε + ∠FAE + ∠FAE = 180° (rearranging terms)
∠AXε + 2∠FAE = 180°
Now, let's consider the quadrilateral ABXE. We have:
∠ABX + ∠AXε + ∠AEB = 360° (sum of angles in a quadrilateral)
∠ABX + ∠AEX + ∠AEB = 360° (rearranging terms)
∠ABX + ∠FAE + ∠AEB = 360° (substituting ∠AEX with ∠FAE)
∠ABX + 2∠FAE = 360° (substituting ∠AEB with ∠FAE)
∠ABX = 360° - 2∠FAE
Finally, let's consider the triangle BDE. We have:
∠BDE + ∠BED + ∠EBD = 180° (sum of angles in a triangle)
∠BDE + ∠AEB + ∠EBD = 180° (substituting ∠BED with ∠AEB)
∠BDE + ∠FAE + ∠EBD = 180° (substituting ∠AEB with ∠FAE)
∠BDE + 2∠FAE = 180° (since ∠FAE = ∠FEA)
∠BDE = 180° - 2∠FAE
From the above equations, we can see that:
∠ABX = ∠BDE (since both are equal to 360° - 2∠FAE)
∠ABD = ∠EBD (since both are equal to 45°)
Therefore, by the angle-angle (AA) criterion for similarity, we have:
triangle ABD is similar to triangle EXD.
This implies that:
BD/AD = XD/ED (by the property of similar triangles)
Since AD = BD (since ABCD is a square), we have:
BD/BD = XD/ED
1 = XD/ED
XD = ED
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Use the V n of the function y=f(x) below to answer the questions.
12
BEN
2011
Py
T
5
(a) Is f(1) negative?
O Yes No
(b) For which value(s) of x is f(x) > 0?
Write your answer using interval notation.
I
(c) For which value(s) of x is f(x)=0?
If there is more than one value, separate them with commas.
0
Using the graph, the answers to the questions are
a. Nob. The values of x in f(x) is greater than 0 at (-3 2)c. The function of x is zero at -3, 2What is f(1) in the graph
Using the graph f(1) is the value of y when x is zero.
This is solved by tracing point x = 1 to the graph then tracing the corresponding value of y
Doing this give f(1) to be equal to 2 and this is not negative
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A wind chime in the shape of a cylinder measures 16 inches long.
Will the wind chime fit inside the box?
Match the statement to the phrase that correctly completes it. A phrase can be used more than once or not used.
A rectangular prism with a base that measures 8 inches by 10 inches, and a height of 6 inches. A dashed diagonal line is drawn from one vertex of the base to the opposite vertex of the same base. Another dashed diagonal line is drawn from one vertex of the base to the opposite vertex of the other base.
The given cylindrical wind chime of dimensions 16 inches long will not fit inside the box in the shape of rectangular-prism having dimensions 10 inches x 8 inches x 6 inches.
1)shorter than
2) shorter than
3)will not fit.
What is a rectangular-prism?
Rectangular prism also called as cuboid, is a closed three dimensional solid having six faces. This prism has eight vertices or corners and 12 edges. The perimeter of this shape is the total length of 12 edges. The area of this shape is the sum of area of for or six surfaces depending on the given situation.
Given dimensions of cuboid/rectangular prism:
Length=10 inches, height= 6 inches & Width=8 inches
Dimensions of cylindrical wind chime: Length of cylinder = 16 inches
Case I: comparision of diagonal of base of the box & length of wind chime.
Diagonal of base can be found using pythagorean theorem:
Diagonal=[tex]\sqrt{L^{2} +W^{2} }[/tex]
=[tex]\sqrt{(10)^{2} +(8)^{2} }[/tex]
=[tex]\sqrt{100+64\\}[/tex]
=[tex]\sqrt{164}[/tex]
=12.8 inches ≠ 16 inches
Diagonal of rectangular prism base shorter than length of wind chime.
Case II:comparision of interior diagonal of the box & length of wind chime.
Interior diagonal of box=[tex]\sqrt{L^{2} +W^{2} +H^{2} }[/tex]
=[tex]\sqrt{(10)^{2} + (8)^{2} +(6)^{2} }[/tex]
=[tex]\sqrt{100+64+36}[/tex]
=[tex]\sqrt{200}[/tex]
=14.14 inches ≠ 16 inches
Interior Diagonal of rectangular prism is shorter than length of wind chime.
As the wind chime is greater than both the diagonals of cuboid, the chime will not fit into the prism.
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Refer to the complete question attached below.
Campfire Girls troop has 18 members. In how many different ways can the leader appoint 3 members to clean up camp? different ways
Answer:
To determine the number of different ways the leader can appoint 3 members to clean up camp from a troop of 18 members, we can use the combination formula: n C r = n! / r!(n - r)! where n is the total number of members in the troop (18), and r is the number of members to be chosen (3). Plugging in the values, we get: 18 C 3 = 18! / 3!(18 - 3)! = (18 x 17 x 16) / (3 x 2 x 1) = 816 Therefore, there are 816 different ways the leader can appoint 3 members to clean up camp from a troop of 18 members.
The figure below is a square. Find the length of side x in simplest radical
form with a rational denominator.
Answer:
x = 4
Step-by-step explanation:
Use Pythagoras theorem
Paul drove during a snowstorm for 40 miles. When it stopped snowing, he increased his
speed by 30 miles per hour and drove for an additional 132 miles. If Paul drove for a total
of 4 hours, which equation can be used to find his average rate of speed (x) in miles per
hour during the snowstorm?
what is 60% of $60 pls
find the equation of the tangents to the curve (9x^(2)+16y)/(2)=26 that are parallel to 4.5x-4y=0.5
The equation is 9x + 8y = 81.
Martha was collecting stickers. She got 38 stickers for her birthday, 75, stickers from her friend, and 18 stickers from her brother. She gave 25 stickers to her sister before putting them all in her sticker book. How many stickers did Martha put in her sticker book?
Answer: 106 stickers
Step-by-step explanation:
38 + 75 + 18 = 131 (total amount of stickers she got)
131 - 25 = 106
Find the angles that have the following tangents:
I don't know how I'm meant to figure out these questions.
The angles that have the given tangents, correct to 2 significant figures, are: 45°, 30° and 15°
How to find the angles?To find the angles that have the given tangents, we can use the inverse tangent function (tan⁻¹) or the arctan function on a calculator. Remember that the arctan function gives an angle in radians, so we need to convert the result to degrees.
[tex]\tan \theta = 1.000\ \theta = \tan^{-1}(1.000)\\\\ \theta \simeq 45^{\circ}[/tex]
tan θ = 0.577
θ = tan⁻¹(0.577)
θ ≈ 30°
tan θ = 0.268
θ = tan⁻¹(0.268)
θ ≈ 15°
tan θ = 4
θ = tan⁻¹(4)
θ ≈ 75.96°
θ ≈ 76° (rounded to 3 significant figures)
tan θ = 2.747\\θ = tan⁻¹(2.747)
θ ≈ 70.53°
θ ≈ 71° (rounded to 3 significant figures)
tan θ = 3.732
θ = tan⁻¹(3.732)
θ ≈ 75.89°
θ ≈ 76° (rounded to 3 significant figures)
Therefore, the angles that have the given tangents, correct to 2 significant figures, are:
45°
30°
15°
And the angles that have the given tangents, correct to 3 significant figures, are:
76°
71°
76°
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Given: Circle k(O) with OT ⊥ XY, and OU ⊥ WZ and X=Z. Prove: XY=WZ
The answer of the given question based on circle is we have shown that XY = WZ, since both line segments are equal to XM.
What is Pythagorean theorem?The Pythagorean theorem is fundamental concept in mathematics that relates to sides of right triangle. It states that square of the length of hypotenuse (the side opposite the right angle) is equal to sum of the squares of lengths of the other two sides (the legs).
To prove that XY = WZ, we will use the fact that the line segment connecting the center of a circle to the midpoint of a chord is perpendicular to the chord.
Let M be the midpoint of XY, and N be the midpoint of WZ. Since X = Z, we know that XM is parallel to ZN, and therefore MZNX is a parallelogram.
Let O be center of circle k(O). Then, by the perpendicularity property mentioned above, we know that OM is perpendicular to XY, and ON is perpendicular to WZ.
Since M and N are midpoints of XY and WZ respectively, we have:
XM = MY, and ZN = NW
Therefore, by the Pythagorean theorem, we have:
OX² = OM² + MX², and OZ² = ON² + NZ²
Adding these two equations gives:
OX² + OZ² = OM² + MX² + ON² + NZ²
Since MZNX is a parallelogram, we know that OM = ON, and MX = NZ. Substituting these values gives:
OX² + OZ² = 2OM² + 2MX²
But since OM is perpendicular to XY and ON is perpendicular to WZ, we have:
OM² + MX² = XM²/4, and ON² + NZ² = ZN²/4
Substituting these values gives:
OX² + OZ² = XM²/2 + ZN²/2
But since MZNX is a parallelogram, we know that XM = ZN. Therefore:
OX² + OZ² = XM²
Thus, we have shown that XY = WZ, since both line segments are equal to XM.
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What is the equation of the trend line in the scatter plot?
10
9
8
7
6
5
4
2
0
1 2 3 4 5 6 7 8 9 10
Use the two yellow points to write the equation in slope-intercept form. Write any coefficients
as integers, proper fractions, or improper fractions in simplest form.
Video
The equation in slope-intercept form y = x - 1
What is linear regression?In general, to find the equation of a trend line in a scatter plot, you can use linear regression. Linear regression is a statistical method that finds the line of best fit through a set of data points. The equation of the trend line can then be written in the slope-intercept form, y = mx + b, where "m" is the slope of the line and "b" is the y-intercept.
The yellοw pοints are lοcated at the pοints (1, 0) and (8, 7)
Fοr a linear equatiοn that pases trοugh the pοints (x1, y1) and (x2, y2), the slοpe is:
s = (y₂ - y₁)/(x₂ - x₁)
Then, fοr οur equatiοn, the slοpe is
s = (7-0)/(8-1) = 1
Then οur linear equatiοn is
y = 1 × x + b
where b is the y-axis intercept.
Tο find b, we can use that when x = 1, y = 0
0 = 1 × 1 + b
b = -1
Then οur equatiοn is y = x - 1
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Complete question:
What is the equation of the trend line in the scatter plot?
Use the two yellow points to write the equation in slope-intercept form. Write any coefficients as integers, proper fractions, or improper fractions in simplest form.
Jason lives on 750m plot .express the size of his plot as a fraction of the hectare if 1 hectare =10000m
By metric system it can be determined that Jasοn lives οn 0.075 hectare.
What is metric system?The metric system is a prοcess οf measuring things. The measurements are used wοrldwide in calculatiοns and research. Sοme examples οf measurements οf things using the metric system are given belοw:
Meter is the unit οf measuring distances and lengths.
Kilοgram οr Gram is used tο measure weight.
Jasοn lives οn 750m plοt .
The given prοblem can be sοlved by metric system.
By metric system
1 hectare =10000m
10000m = 1 hectare
1m= 1/10000 hectare
750m= 750/10000 hectare
Hence,
Jasοn lives οn 0.075 hectare.
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If I have an range for -00,00 is my equation linear or exponential or quadratic
The type of the function in the question is a linear function
Identifying the type of the functionThe range of -∞ to +∞ means that the equation is valid for all real numbers.
This range alone provides enough information to determine whether the equation is linear, exponential, or quadratic.
Because only a linear function can have this range
A linear equation is of the form y = mx + b, where m and b are constants and x is the independent variable.
And it always have a range of -∞ to +∞ , otherwise stated
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hats 2030=230.09+23x
The equation Hats 2030=230.09+23x represents a linear relationship between the number of years since 2010 (x) and the projected number of hats sold in the year 2030.
The slope of the equation, 23, indicates that for each additional year since 2010, the projected number of hats sold in 2030 increases by 23. The y-intercept of the equation, 230.09, represents the projected number of hats sold in 2030 if x is equal to zero (i.e., in the year 2010). This equation can be used to forecast future hat sales based on past trends and can help businesses and organizations plan for inventory and production needs. However, it is important to note that unforeseen events, changes in consumer behavior, and other factors can impact actual hat sales and should be considered when making business decisions.
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Solve the system for x and y
-8x-5y=-1
-8x-5y=-45
So the solutions for x and y are: (x, y) = (0, 1/5), (5, -39/5), (10, -79/5), (15, -119/5), ...
What are parallel lines?
Parallel lines are two lines in a plane that never intersect. This means that they maintain the same distance between each other at all points.
Subtracting the first equation from the second, we get:
-8x - 5y = -45 - (-1)
-8x - 5y = -44
Now we can see that the two equations are equivalent, which means that there are infinitely many solutions for x and y that satisfy both equations. Geometrically, the two equations represent two parallel lines in the x-y plane that never intersect, so there is no unique solution.
However, we can still find a solution for x and y in terms of one variable. Let's solve for y in terms of x:
-8x - 5y = -1
-5y = 8x - 1
y = (-8/5)x + 1/5
Now we can choose any value for x and plug it into this equation to get the corresponding value of y. For example, if we set x = 0, then y = 1/5. If we set x = 5, then y = -39/5.
So the solutions for x and y are:
(x, y) = (0, 1/5), (5, -39/5), (10, -79/5), (15, -119/5), ...
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13. In cell A17, use the SUMIF function and structured references to display the total membership in 2023 for groups with at least 40 members.
The SUMIF function in excel combines a condition and a sum of the values which meet the stated condition. The SUMIF statement required on Cell A17 would be =SUMIF(C1:C50, '>=40')
How to write functions in Excel?
In Microsoft Excel, formulas are usually used to generate results or values for a cell.
Now, there are different ways that a formula can be written but then the formula must start with an equal sign. The formula to be used for the cells given has the function "SUMIF" and this is used to provide a sum of numeric values based on a condition.
The SUMIF syntax goes thus ; SUMIF(row_range, condition)
Assume the total membership count is in cells A1 to A50.
Cells, where the values are greater than or equal to 40, should be summed.
Hence, the required formula would be =SUMIF(C1:C50, '>=40')
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The area of a round parachute can be represented by the expression
SEE QUESTION IN PICTURE
Step-by-step explanation:
pi ( x^2 - 36 )
pi ( x-6)(x+6) Done.
Consider a circle whose equation is x2 + y2 – 2x – 8 = 0. Which statements are true? Select three options. The radius of the circle is 3 units. The center of the circle lies on the x-axis. The center of the circle lies on the y-axis. The standard form of the equation is (x – 1)² + y² = 3. The radius of this circle is the same as the radius of the circle whose equation is x² + y² = 9.
The statements which are true are that the radius of the circle is 3 units, the center of the circle lies on the x-axis and the radius of this circle is the same as the radius of the circle whose equation is x² + y² = 9.
What is a circle?
A circle is formed by all points in a plane that are at a particular distance from the center. To put it another way, it is the curve that a moving point in a plane draws to maintain a constant distance from another point.
We are given a circle whose equation is [tex]x^{2}[/tex] + [tex]y^{2}[/tex] - 2x - 8 = 0.
We know that the general form of a circle is [tex]x^{2}[/tex] + [tex]y^{2}[/tex] + 2gx + 2fy + C = 0, where (-g, -f) is the center and √[tex]g^{2}[/tex] + [tex]f^{2}[/tex] - C is the radius.
So, in the equation, g is -1 and f is 0.
So, the center is (1, 0) which represents that the center lies on the x - axis.
Now,
⇒ Radius = √[tex]g^{2}[/tex] + [tex]f^{2}[/tex] - C
⇒ Radius = √1 + 0 - (-8)
⇒ Radius = √1 + 8
⇒ Radius = √9
⇒ Radius = 3 units
So, the radius of the circle is 3 units.
Now, radius of circle x² + y² = 9 is
[tex]r^{2}[/tex] = 9
r = 3
So, the radius of this circle is the same as the radius of the circle whose equation is x² + y² = 9.
Hence, the required solution has been obtained.
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The complete question has been attached below.
Write the polynomial the product and sum of whose zeroes are -9/2and -3/2 respectively
x²+(9/2)x-3/2
=2x²+9x-3
8₁
6) A triangular roof is built so that its height
is half its base, If the base of the roof is 32
feet long, what is the area of the roof? Show some work
If the height of the triangular roof is half its base, then the height of the roof is:
h = (1/2) * 32 = 16 feet
The area of a triangle is given by the formula:
A = (1/2) * base * height
Plugging in the values we have:
A = (1/2) * 32 * 16
A = 256 square feet
Therefore, the area of the triangular roof is 256 square feet.
Dilations are another way to transform quadratic functions. In addition dilation, the parent function is _____ by a given value and can be presented by ____ where a ≠ 0.
Answer: In dilation, the parent function is scaled or stretched by a given value, and can be presented by the equation y = a*x^2 where "a" is the dilation factor and a ≠ 0.
When a > 1, the dilation stretches the graph vertically, making the parabola narrower and taller. When a < 1, the dilation compresses the graph vertically, making the parabola wider and shorter. When a is negative, the dilation reflects the graph across the x-axis.
Step-by-step explanation:
Mr. DeWitt carved a wooden boat for his granddaughter. He began with a piece of wood that was 20.3 centimeters long. The boat is 16.7 centimeters long. How many centimeters did Mr. DeWitt carve off the length of the piece of wood when he made the boat?
In the word problem , the length of wood Mr. DeWitt carved off is 3.6cm.
What is word problem?
Word problems are often described verbally as instances where a problem exists and one or more questions are posed, the solutions to which can be found by applying mathematical operations to the numerical information provided in the problem statement. Determining whether two provided statements are equal with respect to a collection of rewritings is known as a word problem in computational mathematics.
According to the given ,
Length of wood = 20.3 cm
The length of boat = 16.7 cm.
Now Length of wood he carved off is,
=> Length of wood - length of boat
=> 20.3 - 16.7
=> 3.6 cm.
Hence the length of wood Mr. DeWitt carved off is 3.6cm.
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Please see attachment
The exponential function that relates the mass of the substance (y) to the number of years since 2004 (t) is: y = 450 * (0.923) ^ t
How to explain the functionIn this function, 0.923 is the decimal equivalent of 1 - 0.077, which represents the decay factor of 7.7%. The exponent t represents the number of years since 2004.
So, for example, if we want to know the mass of the substance in 2010 (6 years after 2004), we can plug t = 6 into the function:
y = 450 * (0.923) ^ 6 = 298.8 milligrams
Therefore, the mass of the substance in 2010 would be approximately 298.8 milligrams.
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Note that the exponential function that shows the relationship between the mass y of the substance and the number of years t since 2004 is:
y = 450 * 0.923^t
What is the explanation for the above response?The general formula for exponential decay is:
y = a * (1 - r)^t
where:
y = final amount
a = initial amount
r = decay rate
t = time elapsed
In this problem, the initial amount is 450 milligrams, and the decay rate is 7.7% per year, or 0.077 as a decimal. We can use the formula above to write an exponential function for the relationship between y and t:
y = 450 * (1 - 0.077)^t
Simplifying this expression, we get:
y = 450 * 0.923^t
Therefore, the exponential function that shows the relationship between the mass y of the substance and the number of years t since 2004 is:
y = 450 * 0.923^t
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Full Question:
Although part of your question is missing, you might be referring to this full question:
In 2004, a sample of a radioactive substance had a mass of 450 milligrams. Since then, the sample has decayed by 7.7% each year. Let t be the number of years since 2004. Let y be the mass of the substance in milligrams. Write an exponential function showing the relationship between y and t.
What is the equation X equals Y plus Z solved for Z
Answer:
To solve for Z in the equation X = Y + Z, we need to isolate Z on one side of the equation. We can do this by subtracting Y from both sides:
X - Y = Y + Z - Y
Simplifying the right-hand side, we get:
X - Y = Z
Therefore, the equation X = Y + Z solved for Z is:
Z = X - Y
Find the area of semicircle
Answer:
Area of semicircle = πr2/2. ..I know how to find the exponential function from two points, but I’m not able to actually identify two points on this graph. Could someone find two points, preferably whole number?
The required coordinate points are (3, 0) and (-4, 0)
Finding exponential function:To find the exponential function from two points, you need to know the coordinates of two points on the graph of the function.
If you have access to the graph, you can try to estimate the coordinates of two points by using the grid lines on the axes.
Choose two points that lie on the curve and try to read their coordinates as accurately as possible from the gridlines.
Here we have an exponent graph
In the given picture,
The curve cuts the y-axis at 3 and the x-axis at - 4
Hence,
The required coordinate points are (3, 0) and (-4, 0)
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a) The graph of y=g(x) is shown. Draw the graph of y=2g (x) +3
B) The graph of y=h(x) is shown. Drawn the graph of y= -h(x-2)
a) To graph y=2g(x)+3, we need to take the graph of y=g(x) and vertically stretch it by a factor of 2, then shift it upward by 3 units.
Start by identifying some key points on the graph of y=g(x). For example, if we take x=0, we see that y=g(0)=2, so the point (0,2) is on the graph. If we take x=1, we see that y=g(1)=4, so the point (1,4) is on the graph. If we take x=-1, we see that y=g(-1)=1, so the point (-1,1) is on the graph.
Now, to graph y=2g(x)+3, we take these key points and apply the transformation.
- Vertically stretch: Multiply the y-coordinates by 2. So (0,2) becomes (0,4), (1,4) becomes (1,8), and (-1,1) becomes (-1,2).
- Shift upward: Add 3 to each y-coordinate. So (0,4) becomes (0,7), (1,8) becomes (1,11), and (-1,2) becomes (-1,5).
These points give us enough information to sketch the graph of y=2g(x)+3. It will look like the original graph of y=g(x), but stretched vertically and shifted upward.
b) To graph y=-h(x-2), we need to take the graph of y=h(x) and reflect it across the y-axis, then shift it to the right by 2 units, and finally reflect it again across the x-axis.
Again, start by identifying some key points on the graph of y=h(x). For example, if we take x=0, we see that y=h(0)=1, so the point (0,1) is on the graph. If we take x=1, we see that y=h(1)=3, so the point (1,3) is on the graph. If we take x=-1, we see that y=h(-1)=2, so the point (-1,2) is on the graph.
Now, to graph y=-h(x-2), we take these key points and apply the transformation.
- Reflect across y-axis: Multiply the x-coordinates by -1. So (0,1) becomes (0,1), (1,3) becomes (-1,3), and (-1,2) becomes (1,2).
- Shift to the right: Add 2 to each x-coordinate. So (0,1) becomes (2,1), (-1,3) becomes (1,3), and (1,2) becomes (3,2).
- Reflect across x-axis: Multiply the y-coordinates by -1. So (2,1) becomes (2,-1), (1,3) becomes (1,-3), and (3,2) becomes (3,-2).
These points give us enough information to sketch the graph of y=-h(x-2). It will look like the original graph of y=h(x), but reflected across the y-axis, shifted to the right, and reflected across the x-axis.
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