2.2.18Find the vertex of the graph of the quadratic function. Determine whether thegraph opens upward or downward, find the y-intercept, and sketch the graph.f(x) = - x2 - 2x+3The vertex is(Simplify your answer. Type an ordered pair.)
Answers
The quadratic function is given by the following expression:
[tex]f(x)=-x^2-2x+3[/tex]The direction at which the graph opens is determined by the signal of the number multiplying x². If the number is positive then the graph opens upwards, if it is negative it opens downward. In this case it is negative so it opens donward.
The vertex of a quadratic expression can be found by the following expression:
[tex]x=\frac{-b}{2a}[/tex]Where a is the number multiplying "x²", while b is the number multiplying "x". Applying the data from the problem we have:
[tex]x=\frac{-(-2)}{2\cdot(-1)}=\frac{2}{-2}=-1[/tex]To find the value of "y" for the vertex we need to apply the coordinate for x on the expression. We have:
[tex]\begin{gathered} f(-1)=-(-1)^2-2\cdot(-1)+3 \\ f(-1)=-1+2+3=4 \end{gathered}[/tex]The coordinates of the vertex are (-1,4).
To sketch a graph we need to find the x-intercept and y-intercept of the function. These are given when f(x) = 0 and x=0 respectively. Let's find these points.
[tex]\begin{gathered} 0=-x^2-2x+3 \\ -x^2-2x+3=0 \\ x_{1,2}=\frac{-(-2)\pm\sqrt[]{(-2)^2-4(-1)(3)}}{2\cdot1} \\ x_1=-3 \\ x_2=1 \end{gathered}[/tex][tex]f(x)=-0^2-2\cdot0+3=3[/tex]The x intercept happens in two points -3 and 1, while the y intercept happens in the point 3. With this and the vertex we can sketch the function.
Related Questions
PLEASE ANSWER Given: a = 7 and b = 2 Then the m∠A=_?_ . ROund to the nearest degree. Enter a number answer only.
Answers
Answer:
A = 74 degrees
Step-by-step explanation:
a = b tan A
tan A = a/b
tan A = 7/2
A = arctan 7/2
A = 74 degrees
The sum of six consecutive integers is -9. What are the integers?
Answers
Therefore the six consecutive integers are -4,-3,-2,-1,0,1 .
What is consecutive integer ?
Integers that follow one another are referred to as consecutive integers. They proceed in a certain order or sequence. For instance, a sequence of natural numbers are all integers. In mathematics, the term "consecutive" refers to an uninterrupted chain of events or a continuous progression, such as when integers follow one another in a sequence where each succeeding number is one more than the one before it. The mean and median are both identical in a collection of successive integers (or in numbers). x + 1 and x + 2 are two consecutive integers if x is an integer.
Here sum of six consecutive integer is -9.
Let us take the 6 consecutive numbers are,
=> x,x+1, x+2, x+3,x+4, x+5
Now sum is -9. Then ,
=> x+x+1+x+2+x+3+x+4+x+5 =-9
=> 6x + 15 = -9
=> 6x = -9-15
=> 6x = -24
=> x = -24/6
=> x= -4
Then 6 integers are ,
=> -4,-3,-2,-1,0,1
Therefore the six consecutive integers are -4,-3,-2,-1,0,1 .
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Find an equation of a parabola that satisfies the given conditions.
Focus at (8,0), directrix x = -8
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The equation of a parabola for the focus at (8,0), directrix x = -8 is found as y² = 32x.
What is meant by the term parabola?A parabola is an open plane symmetrical curve created by the intersection of the a cone and a plane parallel towards its side. A projectile's path under the effect of gravity ideally continues to follow a curve of the this shape.The standard equation of parabola,
(y−n)² = 4p(x−m),
In which,
Vertex is (m,n)Axis of symmetry is y = mFocus is (p+m,n)Directrix is x =m−p.For the given value in question;
p+m = 8
n = 0
m−p = -8
m = 0
p = 8
Put the obtained values in general equation;
y² = 4×8(x+0)
y² = 32x
Thus, the equation of a parabola for the focus at (8,0), directrix x = -8 is found as y² = 32x.
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Figure 1 and figure 2 below are similar. which po8nt corresponds to point U.
Answers
SOLUTION
Step 1 :
In this question, we have that Figure 1 and Figure 2 are similar.
The point that corresponds to point U is Point E.
10. Write the slope-intercept form of the equation of the line through the given points. Write answer as y=mx+b. 1 po through: (0, 2) and (-5, -5) Your answer
Answers
First of all, remember what the equation of a line is:
y = mx+b
Where:
m is the slope, and
b is the y-intercept
First, let's find what m is, the slope of the line...
[tex]\begin{gathered} m=\frac{y2-y1}{x2-x1} \\ m=\frac{-5-2}{-5-0} \\ m=\frac{-7}{-5} \\ m=\frac{7}{5} \\ \end{gathered}[/tex]Now, what about b, the y-intercept?
[tex]\begin{gathered} b=y-mx \\ b=2-\frac{7}{5}(0) \\ b=2 \end{gathered}[/tex]The equation of the line that passes through the points
[tex]y=\frac{7}{5}x+2[/tex]can someone please help me find the valu of X?
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We are asked to find the value of x.
As you can see, the two sides are parallel and when these parallel sides intersect sides of overlapping triangles then the intercepted segments are proportional.
So, we can set up the following proportion.
[tex]\frac{15}{6}=\frac{(3x+10)-8}{8}[/tex]Let us solve the above equation for x.
[tex]\begin{gathered} \frac{15}{6}=\frac{3x+10-8}{8} \\ \frac{15}{6}=\frac{3x+2}{8} \\ 15\cdot8=6\cdot(3x+2) \\ 120=18x+12 \\ 120-12=18x \\ 108=18x \\ \frac{108}{18}=x \\ 6=x \\ x=6 \end{gathered}[/tex]Therefore, the value of x is 6
I need the answer to this use fractions and pi
Answers
We are to find the positive and negative angles that are coterminal with
[tex]\frac{2\pi}{3}[/tex]By definition
Coterminal Angles are angles that share the same initial side and terminal sides. Finding coterminal angles is as simple as adding or subtracting 360° or 2π to each angle, depending on whether the given angle is in degrees or radians
Hence,
The positive coterminal angle is
[tex]\frac{2\pi}{3}+2\pi[/tex]Simplifying this we get
[tex]\begin{gathered} \frac{2\pi}{3}+2\pi \\ =\frac{2\pi+6\pi}{3} \\ =\frac{8\pi}{3} \end{gathered}[/tex]Therefore, the positive coterminal angle is
[tex]\frac{8\pi}{3}[/tex]The negative coterminal angle is
[tex]\begin{gathered} \frac{2\pi}{3}-2\pi \\ =\frac{2\pi-6\pi}{3} \\ =-\frac{4\pi}{3} \end{gathered}[/tex]Therefore, the negative coterminal angle is
[tex]-\frac{4\pi}{3}[/tex]find the slope/rate of the line represented by each table
Answers
The correct answer is 1/2 or 0.5
Pick 2 points in the table; (x = -10, y = 1) and ( x = 0, y = 6)
The Rate of Change is given by;
[tex]\text{slope = }\frac{y_2-y_1}{x_2-x_1}=\frac{6-1}{0--10}=\frac{5}{10}=\frac{1}{2}[/tex]The rate of change is 1/2 or 0.5
Hence, the correct answer is 1/2 or 0.5
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Convert the following equation
into slope intercept form.
x-13y = 26
y = x -
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10
Enter
Answers
Answer:
y=x+2
Step-by-step explanation:
x-13y=26
Bring the x to the other side
-13y=x+26
To get y by itself divide both sides by -13
-13y = x-26
----------- -----------
-13 -13
y=x+2
Answer the following questions.(a) 12 is 15% of what?.(b) 45% of 60 is what number?.
Answers
ANSWER:
(a) 80
(b) 27
STEP-BY-STEP EXPLANATION:
(a)
We must calculate what number 15% is equal to 12, therefore, we do the following operation:
[tex]100\cdot\frac{12}{15}=80[/tex](b)
In this case we must calculate 45% of 60, therefore:
[tex]60\cdot\frac{45}{100}=27[/tex]let f(x)=2x+1. write a function g(x) whose grass is a rotation of f (x) about (0,1) by a factor of 3 , followed buy a vertical translation 6 units down
Answers
1) Considering f(x)= 2x+1, And g(x) be f(x) rotated about y=1, with a factor of 3 this
A vertical translation 6 units down= g(x) = 2x -5
A factor of 3: g(x)= 3(2x)-5
Part a: How many pieces are in the step functionpart b: how many intervals make up the step function? What are the interval valuespart c: why do we use open circles in some situations and closed in otherspart e are the pieces of this piecewise function linear or non linear?part f what is the range of this piecewise function?
Answers
a) The step function seen in the figure has 6 pieces, one for each step
b) There are 6 intervals, one for each piece. Their values are:
(0, 1]
(1, 2]
(2, 3]
(3, 4]
(4, 5]
(5, 6]
c) The open circles indicate that the endpoint is not included in the interval. The closed circles indicate the endpoint is included in the interval.
For example, in the second interval, 1 is not included (open circle) and 2 is included (closed circle).
d) This is a function because for eac value of x there iss one and only one of y. If the open circles were closed circles, then thi wouldnot be a function.
e) All the pieces are linear because their graph is a line (flat horizontal line)
f) The range of the function is the set of output values:
Range = {46, 48, 50, 32, 54, 56}
¿Qué es 1/3 x5 / 6? ¿Qué es 2/5 x 3/7?
Answers
La primera expresión es
[tex]\frac{1}{3}\times\frac{5}{6}[/tex]Para resolver esta multiplicación de fracciones, tenemos que multiplicar numerador con numerador y denominador con denominador.
[tex]\frac{1\times5}{3\times6}=\frac{5}{18}[/tex]Hence, the first product is 5/18.La segunda expresión es
[tex]\frac{2}{5}\times\frac{3}{7}[/tex]Repetimos el mismo proceso para multiplicar.
[tex]\frac{2\times3}{5\times7}=\frac{6}{35}[/tex]Hence, the second product is 6/35.4y-3x=-16 I need help with this problem and the problem in the picture please help me this work is already late I need to turn it in
Answers
Answer
x = 1.6
y = -2.8
Explanation
The question wants us to solve the simultaneous equation
4y - 3x = -16
8x - 4y = 24
Adding the two equations, we have,
4y - 3x + 8x - 4y = -16 + 24
5x = 8
Divide both sides by 5
(5x/5) = (8/5)
x = 1.6
If x = 1.6,
8 (1.6) - 4y = 24
12.8 - 4y = 24
-4y = 24 - 12.8
-4y = 11.2
y = -2.8
Hope this Helps!!!
(G.11a, 1 point) Points A, C, D, and E are on circle P. 136° 1340 Ε E If arc AD measures 136° and 2 ABD measures 134º, what is the measure of arc CE? o A 152 O B. 67 o C. 116 D. 132
Answers
D. 132º
1) Given that in this circle crossed by two secant lines we can state the following Theorem:
2) then we can write:
[tex]\begin{gathered} m\angle ABD=\frac{AD\text{ +CE}}{2} \\ 134=\frac{136+CE}{2} \\ 134\text{ }\times2=136+CE \\ 268=136+CE \\ CE\text{ =268-136} \\ CE=132 \end{gathered}[/tex]So the measure of the arc CE = 132º (D)
If the lateral areas of two similar prisms are in a ratio of 8 to 18, what is the ratio of the volumes? Enter answers in the same format and order as the original ratio. Round any decimals to the nearest 10th.
Answers
The ration of the lateral areas of the smaller prism to the larger prism is 8 to 18
The first step is t find the scale factor. Recall,
area = square of scale factor
Thus, scale factor = square root of area
Thus,
[tex]\begin{gathered} \text{scale factor = }\sqrt[]{\frac{8}{18}} \\ \text{Dividing the numerator and denominator by 2, we have} \\ \text{scale factor = }\sqrt[]{\frac{4}{9}} \\ scale\text{ factor = 2/3} \end{gathered}[/tex]Volume = cube of scale factor. Thus,
[tex]\begin{gathered} \text{volume = (}\frac{2}{3})^3 \\ \text{Volume = }\frac{8}{27} \\ \end{gathered}[/tex]the top of a rectangular box has an area of 42cm^2. The sides of the box have areas of 30cm^2 and 35cm^2. what are the dimensions of the box?
Answers
We have a rectangular box where we know the area of the faces and we have to find the width w, length l and height h.
The area of the top of the box is equal to the length times the width (l*w) and we also know that it is 42 cm², so we can write:
[tex]l\cdot w=42[/tex]With the same logic, we can write the equations for the other two areas:
[tex]\begin{gathered} l\cdot h=30 \\ w\cdot h=35 \end{gathered}[/tex]NOTE: the area we choose for l or w is indistinct,so we can relate it as we like.
Then, we can solve this system of equations substituting variables as:
[tex]\begin{gathered} l\cdot h=30\longrightarrow l=\frac{30}{h} \\ w\cdot h=35\longrightarrow w=\frac{35}{h} \\ l\cdot w=(\frac{30}{h})(\frac{35}{h})=\frac{1050}{h^2}=42 \\ h^2=\frac{1050}{42} \\ h^2=25 \\ h=\sqrt[]{25} \\ h=5 \end{gathered}[/tex]With the value of h, we can calculate l and w:
[tex]\begin{gathered} l=\frac{30}{h}=\frac{30}{5}=6 \\ w=\frac{35}{h}=\frac{35}{5}=7 \end{gathered}[/tex]Answer:
The dimensions of the box are: length = 6 cm, width = 7 cm and height = 5 cm.
Triangle A B C has vertices (1,4),(5,6) and (3,10) It is reflected across the y axis forming triangle A’B’C’. What are the vertices of the new triangle?
Answers
Given:
The coordinates of triangle ∆ABC are (1,4), (5,6) and (3,10).
The triangle is reflected across y axis forming ∆A'B'C'.
The objective is to find the vertices of the new triangle.
Explanation:
If a triangle with coordinate (a,b) is reflected across y axis, then the change in reflected coordinate will be (-a,b).
If a triangle with coordinate (a,b) is reflected across x axis, then the change in reflected coordinate will be (a,-b).
To find vertices:
Since, the given triangle is reflected across y axis, then the vertices of new triangle will be,
[tex]\begin{gathered} A^{\prime}=(-1,4) \\ B^{\prime}=(-5,6) \\ C^{\prime}=(-3,10) \end{gathered}[/tex]Hence, the vertices of the new triangle are (-1,4), (-5,6) and (-3,10).
The width of a rectangular slab of concrete is 16 m less than the length. The area is 80m^2Part 1 of 3(a) What are the dimensions of the rectangle?The length of the slab is?
Answers
width = w
length = l
w = l - 16
area = w*l
(l - 16)*l = 80
l^2 - 16l = 80
l^2 - 16l - 80 = 0
(l + 4)(l - 20) = 0
then length = 20
w = 20 - 16 = 4
the width is 4
Which equation represents the vertical line passing through (1,-9)?
A. x = -9
B.x = 1
C. y=-9
D. y = 1
Answers
Answer:
B is the correct equation.
Use the regression calculator to compare the teams’ number of runs with their number of wins.
A 2-column table with 9 rows. Column 1 is labeled R with entries 808, 768, 655, 684, 637, 619, 613, 609, 563. Column 2 is labeled W with entries 93, 94, 66, 81, 86, 75, 61, 69, 55.
What is the y-intercept of the trend line, to the nearest hundredth?
Answers
The y-intercept of the trend line is equal to -23.08.
How to determine the y-intercept of the trend line?In order to determine a linear equation of the trend line that models the data points contained in the table, we would have to use an Excel regression calculator (scatter plot).
In this scenario, the teams’ number of runs would be plotted on the x-axis of the scatter plot while the teams’ number of wins would be plotted on the y-axis of the scatter plot.
On the Excel worksheet, you should right click on any data point on the scatter plot, select format trend line, and then tick the box to display an equation for the trend line on the scatter plot.
From the scatter plot (see attachment) which models the relationship between data points in the table, a linear equation of the trend line is given by:
y = 0.15x - 23.08
In conclusion, a standard linear equation is given by:
y = mx + c
Where:
m represents the slope i.e 0.15.x and y are the points.c represents the y-intercept i.e -23.08.Read more on scatter plot here: brainly.com/question/28605735
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Solve the equation: 3(2y-5)=9 for y
Answers
Applying the distributive property in this case:
[tex]6y-15=9[/tex]Adding 15 at both sides of the equation:
[tex]6y-15+15=9+15\rightarrow6y+0=24[/tex]Then
[tex]6y=24[/tex]Dividing both sides by 6, we finally have:
[tex]\frac{6}{6}y=\frac{24}{6}\rightarrow y=4[/tex]Therefore, the value for y = 4.
use the given graph to find the mean, median and mode of the following distribution: the mean is _______the median is _______the mode(s) is/are: __________Note: when the data is presented in a frequency table, the formula to find the mean is:
Answers
Solution:
Given:
From the graph above, a frequency table can be made as shown below;
To calculate the mean;
[tex]\begin{gathered} \text{Mean}=\frac{\Sigma fx}{\Sigma f} \\ \text{Mean}=\frac{189}{20} \\ \text{Mean}=9.45 \end{gathered}[/tex]Therefore, the mean is 9.45
To calculate the median;
Median is the middle term when the data is arranged in rank order.
Since we have 20 terms, then the middle terms will be the 10th and 11th terms.
The median will be the mean of these two numbers.
[tex]\begin{gathered} 10th\text{ term=9} \\ 11th\text{ term=10} \\ \text{Median}=\frac{9+10}{2} \\ \text{Median}=\frac{19}{2} \\ \text{Median}=9.5 \end{gathered}[/tex]Therefore, the median is 9.5
To calculate the mode;
The mode is the data that appears most in the set. It is the data with the highest frequency.
From the graph
From the graph
8. Find the area of the shaded portion of the figure. 10.5cm
Answers
The area = area of the big rectangel - sum of the areas of the two circles
the diameter of each of the circles is 10.5cm
Hence, each circle has a radius of 10.5cm / 2 = 5.25cm
The length of the rectangle = the sum of the diameters of the circles
Therefore
The length of the rectangle = 10.5cm + 10.5cm = 21cm
The width of the rectangle = 10.5cm
Hence,
[tex]\begin{gathered} \text{area of shaded portion = 21}\times10.5\text{ - (}\pi\times5.25^2+\pi\times5.25^2) \\ =220.5-(55.125\pi)\approx47.32 \end{gathered}[/tex]Hence the area of the shaded portion is 47.32 square centimeters
State with the equation has one solution, new solution, or infinitely many solutions.
Answers
SOLUTION:
The equation is;
[tex]\begin{gathered} -2g+10=8g \\ 10=10g \\ g=1 \\ \end{gathered}[/tex]The equation has one solution
Which equation represents a line which is perpendicular to the line y - 6x = -3? Submit Answer Oy= 6x +3 Oy= 1/6x +4 O y= - 1/6x + 8 O y = -6x + 2
Answers
When we have two perpendicular lines their slopes are opposite and inverse.
In this case, we have the equation of the line:
y-6x= -3
Let's solve for y and arrange this equation to make it look like the general form of a linear equation: y=mx+b, where m is the slope of the line.
y-6x= -3
y-6x+6x= -3+6x
y= 6x-3
As we can see, the number that is multiplying the x variable in our equation is 6, the slope of this line is 6.
As mentioned, a perpendicular line to the line y= 6x-3 would have a slope opposite and inverse, then the slope of the line perpendicular to the first line (m2) would be:
[tex]m2=-\frac{1}{6}[/tex]from the options that we have, we can see that the only line that has a slope of -1/6 is the line y= -1/6+8, so that is the right option.
1. Find the slope 2.what is wrong with the following slopes ?
Answers
Given two points (x₁, y₁) and (x₂, y₂), the slope (m) is:
[tex]m=\frac{y_2-y_1}{x_2-x_1}[/tex]So, to solve this question. Follow the steps.
Step 01: Substitute the points in the equation and solve part 1.
(x₁, y₁) = (-3, -1)
(x₂, y₂) = (-2, 6)
[tex]\begin{gathered} m=\frac{6-(-1)}{-2-(-3)} \\ m=\frac{6+1}{-2+3} \\ m=\frac{7}{1} \\ m=7 \end{gathered}[/tex]The slope is 7.
Step 02: Find what is wrong.
The points are:
(x₁, y₁) = (3, 5)
(x₂, y₂) = (-2, 6)
So, substituting in the equation:
[tex]m=\frac{6-5}{-2-3}=\frac{1}{-5}=-\frac{1}{5}[/tex]What is wrong is that the numerator was substituted by (y₁ - y₂), while the denominator was substituted by (x₂ -x₁).
find the 8th term of geometric sequence where a1=5, r= -2
Answers
for Given:
[tex]\begin{gathered} a_1=5 \\ r=-2 \end{gathered}[/tex]You need to remember that "r" is the Common ratio between the terms of the Geometric Sequence and this is the first term:
[tex]a_1_{}_{}[/tex]The formula the nth term of a Geometric Sequence is:
[tex]a_n=a_1\cdot r^{(n-1)}[/tex]Where "n" is the number of the term, "r" is the Common Ratio, and the first term of the sequence is:
[tex]a_1[/tex]In this case, since you need to find the 8th term, you know that:
[tex]n=8[/tex]Then, you can substitute all the values into the formula:
[tex]a_8=(5)(-2)^{(8-1)}[/tex]Evaluating, you get:
[tex]\begin{gathered} a_8=(5)(-2)^{(7)} \\ a_8=(5)(-128) \\ a_8=-640 \end{gathered}[/tex]Hence, the answer is:
[tex]a_8=-640[/tex]
Give two examples that illustrate the difference between a compound interest problem involving future value and a compound interest problem involving presentvalue.Choose the correct answer below.A. In a compound interest problem involving present value the goal is to find how much money has to be invested initially in order to have a certain amount inthe future. In a problem involving future value the goal is to find how much money there will be after a certain amount of time has passed given an initialamount to invest.B. In a compound interest problem involving present value the goal is to find how much money there will be after a certain amount of time has passed given aneffective annual yield. In a problem involving future value the goal is to find how much money has to be invested initially in order to have a certain effectiveannual yield in the future.OC. In a compound interest problem involving present value the goal is to find how much money there will be after a certain amount of time has passed given aninitial amount to invest. In a problem involving future value the goal is to find how much money has to be invested initially in order to have a certain amountin the future.
Answers
From the list of statements, let's select the examples that illustrate the difference between a compound interest problem involving future value and a compound interest problem involving present value.
Apply the compound interest formula:
[tex]A=P(1+\frac{r}{n})^{nt}[/tex]Where:
A represents the future value which is the final amount after a given period of time.
P represents the present value which is the initial amount invested.
When you are required to find the present value, the goal is to find how much money has to be invested initially in order to have a certain amount in the future. Here, the future value is always given.
When you are required to find the future value, the goal is to find how much money there will be after a certain amount of time has passed given an initial amount to invest. Here, the present value is always given.
Therefore, the correct examples are:
In a compound interest problem involving present value the goal is to find how much money has to be invested initially in order to have a certain amount in
the future. In a problem involving future value the goal is to find how much money there will be after a certain amount of time has passed given an initial
amount to invest.
ANSWER: A.
In a compound interest problem involving present value the goal is to find how much money has to be invested initially in order to have a certain amount in
the future. In a problem involving future value the goal is to find how much money there will be after a certain amount of time has passed given an initial
amount to invest.
While your family is visiting Deep Creek Lake, you and your mother decide to go to boating. The rangers charge $6.50 per hour in addition to a $25 deposit to rent a canoe. If the total cost to rent the canoe from 12:30 pm to 3:30pm, write and solve a linear equation to find the total cost to rent the canoe.
Answers
Given: The cost of renting a canoe is $6.50 per hour in addition to a $25 deposit.
Required: If the family rented the canoe from 12:30 PM-3:30 PM, write and solve a linear equation to find the total cost to rent the canoe.
Explanation: Let x denote the number of hours the family rented the canoe. Then the linear equation representing the total cost of renting the canoe is given by-.
[tex]Cost,\text{ }C=6.50x+25[/tex]Now, since the family rented the canoe for 3 hours. Putting x=3 in the above equation gives,
[tex]\begin{gathered} Cost=6.50\times3+25 \\ Cost=\text{\$}44.50\text{ } \end{gathered}[/tex]Final Answer: The equation representing the total cost of renting the canoe is-
[tex]C=6.50x+25[/tex]And the total cost of renting the canoe for 3 hours is $44.50.