2.1 & 2.2 Assessment: Vertex & Standard Form of Quadratic Functions
What is the equation, written in vertex form, of a parabola with a vertex of (1, -9) that
passes through the point (2, -7)?
Work to Support Your Answer (Required for Credit!)
The equation of the parabola, in vertex form is
Answers
[tex]~~~~~~\textit{vertical parabola vertex form} \\\\ y=a(x- h)^2+ k\qquad \begin{cases} \stackrel{vertex}{(h,k)}\\\\ \stackrel{"a"~is~negative}{op ens~\cap}\qquad \stackrel{"a"~is~positive}{op ens~\cup} \end{cases} \\\\[-0.35em] ~\dotfill[/tex]
[tex]\begin{cases} h=1\\ k=-9 \end{cases}\implies y=a(x-1)^2 - 9\qquad \textit{we also know that} \begin{cases} x=2\\ y=-7 \end{cases} \\\\\\ -7=a(2-1)^2-9\implies 2=a(1)^2\implies 2=a \\\\[-0.35em] ~\dotfill\\\\ ~\hfill {\Large \begin{array}{llll} y=2(x-1)^2 -9 \end{array}} ~\hfill[/tex]
Related Questions
We want 3 pizzas, each topped with 0, 1, 2, 3, 4, 5, or 6 of 6 available whole-pie toppings. How many different combinations of 3 such pizzas are possible if different pizzas need not have different toppings?
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Which graph represents the polynomial function?
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The graph which represents the polynomial function f(x) = x^3 - 2x^2 - 5x + 6 is in option A.
First, let us understand the polynomial function:
A polynomial function is one that uses only non-negative integer powers or positive integer exponents of a variable in an equation such as the quadratic equation, cubic equation, and so on. For example, 9y + 6 is a polynomial with an exponent of 1.
We are given:
f(x) = x^3 - 2x^2 - 5x + 6
We know that, when the graph intersects the x-axis, it satisfies the equation that is f(x) = 0 at that point.
Let us check the points one by one;
Option A;
The graph intersects the x-axis at -2, 1, and 3 it should satisfy the equation f(x) = 0.
Let's put x = -2 in f(x), we will get;
f(-2) = -2^3 - 2(-2)^2 - 5(-2) + 6 = -8 - 8 + 10 + 6 = -16 + 16 = 0
f(-2) = 0.
Let's put x = 1 in f(x), we will get;
f(1) = 1^3 - 2(1)^2 - 5(1) + 6 = 1 - 2 - 5 + 6 = 7 - 7 = 0
f(1) = 0.
Let's put x = 3 in f(x), we will get;
f(3) = 3^3 - 2(3)^2 - 5(3) + 6 = 27 - 18 - 15 + 6 = 33 - 33 = 0
f(3) = 0.
So, the points -2, 1, and 3 satisfy the equation f(x).
Thus, the graph which represents the polynomial function f(x) = x^3 - 2x^2 - 5x + 6 is in option A.
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What is the slope of line KH if H(-4, 2) and K(2, -2)? Remember to reduce!
Answers
The slope of a line KH if H(-4, 2) and K(2, -2) is -2/3.
What is a slope?A mathematical word used to define a line's direction and steepness is its slope or gradient.
If a line is rising from left to right, it is growing. The slope is positive, or m>0
If a line decreases from left to right, it is said to be diminishing. The slope is negative, indicating that m < 0
A line has zero slopes if it is horizontal. This function is constant.
The slope of a line is y - y₁ = m(x - x₁)
Slope m = (y₂ - y₁)/(x₂ - x₁)
m = (- 2 - 2)/(2 + 4)
m = - 4/6
m = - 2/3
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five friends shared 8 1/3 pounds of apples equally. How many pund of apples do each of them get
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Answer:
A fruit seller buys 712 fruits of which 3/4 are apples. The apples that he bought 1/3 were found to be rotten if he sold all the good apples at Rs.5 1/4 each.
Step-by-step explanation:
Franks’s annual salary is $80000. About 25% of her salary is deducted for taxes.
How much tax is deducted?
Explain.
Answers
A rectangular tank with a square base, an open top, and a volume of 37044 ft^3 is to be constructed of sheet steel. Find the dimensions of the tank that has the minimum surface area.
Answers
The dimensions of this rectangular tank that has the minimum surface area are 42 feet and 21 feet.
How to calculate the dimensions of the tank that has the minimum surface area?First of all, we would determine the surface area of this rectangular tank with a square base by using this formula:
h = V/s²
Where:
h represents the height of a rectangular tank.s represents the side of the square base.V represents the volume of a rectangular tank.For the surface area of this rectangular tank with a square base, we have:
Surface area, A = base area + 4(lateral area)
Surface area, A = s² + 4(s × h)
Surface area, A = s² + 4(s × V/s²)
Surface area, A = s² + 4V/s
Next, we would take the first derivative of the surface area as follows:
A' = 2s - 4V/s²
Substituting the given parameters into the formula, we have;
A' = 2s - 4(37044)/s²
For the minimum surface area dimension, the first derivative of the surface area would be equated to zero:
2s - 148,176/s² = 0
148,176/s² = 2s
148,176 = 2s³
s³ = 74,088
s = ∛74,088
s = 42 feet.
For the height, we have:
Height, h = V/s²
Height, h = 37044/42²
Height, h = 37044/1764
Height, h = 21 feet.
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Two sides are labeled 5,244 feet. The other two sides are labeled 1,495 feet. What is the perimeter to the nearest hundred?
Answers
Answer:
Step-by-step explanation:
42feet
The formula for simple interest is I=Prt,I=Prt, where II is the interest earned, PP is the principal, rr is the interest rate and tt is the number of years. Solve the formula for PP in terms of r,r, II and t.t.
Answers
Answer:
[tex]P=\cfrac{I}{rt}[/tex]----------------------------
GivenI = PrtSolve for P[tex]I = Prt[/tex][tex]\cfrac{I}{rt} = \cfrac{Prt}{rt}[/tex][tex]\cfrac{I}{rt} =P[/tex][tex]P=\cfrac{I}{rt}[/tex]Graph the given equation by evaluating integer values of x from −2 to 2 and plotting the resulting points
y= _2x_1
Answers
The graph of the linear equation can be seen in the image at the end.
How to graph the equation?Here we have the linear equation:
y = -2x - 1
To graph this, we need to evaluate from x = -2 to x = 2, this means replacing the value of x by the correspondent number.
if x = -2, then:
y = -2*(-2) - 1 = 4 - 1 = 3
So we have the point (-2, 3)
if x = -1
y = -2*-1 - 1 = 2 - 1 = 1
We have the point (-1, 1)
If x = 0
y = -2*0 - 1 = -1
We have the point (0, -1)
If x = 1
y = -2*1 - 1 = -3
We have the point (1, -3)
If x = 2
y = -2*2 - 1 = -5
We have the point (2, -5)
Now we just need to graph these 5 points on a coordinate axis and then connect them with a line, the graph of the linear equation can be seen below.
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he residence of a city voted on whether to raise property taxes the ratio of yes votes to no votes was 4 to 7 if there was a 5918 total votes how many yes votes were there
Answers
There is a total of 2152 yes votes
How to determine the number of yes votes?From the question, the statement is given as
The ratio of yes votes to no votes was 4 to 7
Mathematically, this can be represented as
Ratio = 4 : 7
Alternatively, we have
Yes : No = 4 : 7
Express the ratio as fraction
Yes = Yes/Yes + No
So, we have
Yes = 4/4 + 7
Evaluate the sum
Yes = 4/11
From the question, we have
Total votes = 5918
So, we have
Yes = 4/11 * 5918
Evaluate
Yes = 2152
Hence, the number of yes votes is 2152
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-8 What is 9.15 x 10°° written in standard form?
Answers
The value of the number 9.15 x 10° written in standard form is equal to 9.15.
As given in the question,
Given number is equal to :
9.15 x 10°
Given number 9.15 x 10° is written in standard form.
The value of the given number 9.15 x 10° written in standard form is given by:
9.15 x 10°
Here 10° = 1
Using property of number: Any number raise to zero is always equal to 1.
9.15 x 1
Using Multiplicative identity , number multiply to 1 remain number itself.
= 9.15
Therefore, the value of the number 9.15 x 10° written in standard form is equal to 9.15.
The complete question is:
What is the value of 9.15 x 10° written in standard form?
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Pls help I’m failing
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Answer:
{-9,-8,7,8,10}
The graph of the absolute value parent function, f(x) = |x), is stretched
horizontally by a factor of 3 to create the graph of g(x). What function is g(x)?
OA. g(x)= x+3|
OB. g(x) = 31x1
C. g(x) = 13x1
OD. g(x)= |||
Answers
Option A is the correct answer g(x) = |x+3| by using the relation with the parent function.
Explain absolute value function.
A function that wraps an algebraic expression in absolute value symbols is known as an absolute value function. Remember that a number's distance from 0 on the number line represents its absolute value.
The definition of the absolute value parent function, f(x)=| x |, is as follows:
f(x) = [tex]\left \{ {{x ~, ~ x > 0} \atop {0 ~ , ~x = 0}} \atop {-x ~, ~x < 0} \right.[/tex]
Given parent function is f(x) = |x|
g(x) is obtained by stretching the graph of f(x) by a factor of 3.
Use the function g(x)=f(x−h) to translate the absolute value function f(x)=| x | horizontally.
The graph of f(x) is moved h units to the right to obtain g(x) for h>0
The graph of f(x) is translated h units to the left to obtain g(x) when h>0
Therefore, g(x) = |x+3|
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Write the slope-intercept form for the equation of a line with slope m = -9 and y-intercept (0, 5).
The equation of this line is
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The equation of the line is ,y= -9x+5.
What is an equation?A mathematical equation is a formula that uses the equals symbol (=) to connect two expressions and express their equality. A well-formed formula consisting of two expressions linked by an equals sign is considered to be an equation in English, but the word's cognates in other languages may have slightly different definitions. For instance, an équation is defined as having one or more variables in French.Finding the variables' values that cause the equality to be true is the first step in solving an equation with variables. The variables for which the equation must be solved are also known as the unknowns, and the unknowns' values that fulfill the equality are known as the equation's solutions. Equations can be categorized as either identities or conditional equations.Acc to our question-,
We should use the slope-intercept formula to solve this problem.
The point-slope formula states:= [ y = mx + c]
Where m is the slope and c is the y-intercept for the point (0,y1)
Substituting the information provided we get the equation: y= -9x+5.
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PLEASE HELP!!!! I WILL GIVE YOU BRAINLIST!! PLEASE ASAP :)
Which graph shows the image of ΔABC after a rotation about the origin?
Answers
Answer:
D.
Step-by-step explanation:
just click the rotate button until it matches one of the answers
PLEASE HELP ASAP!! Represent the following expressions as a power of the number a where a≠0.
(a^3)^-2
(a^-1 • a^-2)^-3
((a^2)^-2)^2
Answers
After using the property of power, the expressions as a power of the number a is [tex](a^3)^{-2} = a^{-6}[/tex] , [tex](a^{-1}\cdot a^{-2})^{-3} = a^{9}[/tex] and [tex]((a^2)^{-2})^2=a^{-8}[/tex] .
In the given question we have to solve the given expressions as a power of the number a where a≠0.
The first given expression is [tex](a^3)^{-2}[/tex].
Using the property of multiplication of power; [tex](a^m)^n=a^{mn}[/tex].
Simplifying the expression.
[tex](a^3)^{-2} = a^{3*(-2)}[/tex]
[tex](a^3)^{-2} = a^{-6}[/tex]
The second expression is [tex](a^{-1}\cdot a^{-2})^{-3}[/tex]
Using the property of multiplication of power; [tex](a^m)^n=a^{mn}[/tex].
[tex](a^{-1}\cdot a^{-2})^{-3} = a^{(-1)\times(-3)}\cdot a^{(-2)\times(-3)}[/tex]
[tex](a^{-1}\cdot a^{-2})^{-3} = a^{3}\cdot a^{6}[/tex]
Using the property of sum of power [tex]a^m\cdot a^n=a^{m+n}[/tex]
[tex](a^{-1}\cdot a^{-2})^{-3} = a^{(3+6)}[/tex]
[tex](a^{-1}\cdot a^{-2})^{-3} = a^{9}[/tex]
The third expression is [tex]((a^2)^{-2})^2[/tex].
Using the property of multiplication of power; [tex](a^m)^n=a^{mn}[/tex].
[tex]((a^2)^{-2})^2=((a^2)^{(-2)\times2})[/tex]
[tex]((a^2)^{-2})^2=(a^2)^{-4}[/tex]
Again using the property of multiplication of power; [tex](a^m)^n=a^{mn}[/tex].
[tex]((a^2)^{-2})^2=a^{2\times(-4)}[/tex]
[tex]((a^2)^{-2})^2=a^{-8}[/tex]
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Notebook paper is about 0.003 in. thick. If you cut a sheet of paper in half, stack one piece on top of the other, cut those in half, stack all the pieces together, repeating this process 20 times, how high will the stack of paper be?
Answers
Answer:
765
Step-by-step explanation:
A new gaming chair costs $309.99. You have already saved $131.49 and earn $25.50 each week babysitting. Write and solve an equation to determine how many weeks, w, you must babysit to earn enough money to buy the new gaming chair.
25.5w − 131.49 = 309.99; w = 17
25.5 + 131.49w = 309.99; w = 7
25.5w − 309.99 = 131.49; w = 17
25.5w + 131.49 = 309.99; w = 7
Answers
The linear equation we need to solve is:
$25.50*w = $309.99 - $131.49
And the solution is w = 7.
How to write and solve the equation?We know that the cost of the gaming chair is $309.99.
To buy this, you already have saved $131.49, and save another $25.50 per week, then the amount you have after w weeks is:
f(w) = $131.49 + $25.50*w
So the linear equation we need to solve is:
$131.49 + $25.50*w = $309.99
To solve this we need to isolate w.
$131.49 + $25.50*w = $309.99
$25.50*w = $309.99 - $131.49
w = ($309.99 - $131.49)/$25.50 = 7
So they can buy the chair after 7 weeks.
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Find the distance from the point (1,2) to the line-y= 1/2x-3. Round your answer to the nearest tenth.
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Answer:
the graphical explanation please
Deshaun drove 469 miles in 7 hours.
At the same rate, how long would it take him to drive 335 miles?
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We're assuming Deshaun is driving at a constant speed/at the same rate,
He drove 469 miles in 7 hours -> his speed = 469 : 7 = 67 (mph)
Therefore, it would take him 335 : 67 = 5 hours to drive 335 miles.
Please help me thank you :))))
Answers
Answer: y= (2/3)x - 2
Step-by-step explanation:
The biggest way to remember equations like these is to know rise over run which means however many times it goes up until it intersects with both x and Y will be on the top of the fraction and however many times it goes to the right until it intersects with both x and y will be on the bottom of the fraction. So since it hits (0, -2) then intersects at (3,0) it will be 2/3x and the Y is where X equals zero so it would be -2. Hope that helps!
Answer:
y = [tex]\frac{2}{3}x - 2[/tex]
Step-by-step explanation:
Step 1:
Choose two points on the graph, (x1, y1) and (x2, y2), to find the slope.
Find the ratio between change in y-values and change in x-values.
m = (y2 - y1) / (x2 - x1)
Points I picked:
(x1, y1) = (0, -2)
(x2, y2) = (3, 0)
Then,
m = (0 - (-2)) / (3 - 0)
m = 2/3
Step 2 :
Find the y-intercept from the graph. That is the point at which the line intersects the y-axis.
The y-intercept is -2.
b = -2
Step 3 :
Use the slope (m) and y-intercept (b) to write the equation in slope-intercept form:
y = mx + b
Substitute m = 2/3 and b = -2
y = 2/3 x - 2
algebra help pleaseee
Answers
Answer:
3x^2-2x+3+1/x+2
Step-by-step explanation:
WHat is 5/1 x 4/5 This is a hard one
Answers
4
Step-by-step explanation:
[tex]{ \tt{ \frac{5}{1} \times \frac{4}{5}}} [/tex]
[tex]{ \tt{ \frac{5 \times 4}{5}}} [/tex]
[tex]{ \tt{ = \frac{20}{5}}} [/tex]
[tex] = { \green{ \boxed{ \pink{ \sf{4}}}}}[/tex]
Which expression can be placed on the right side of the equals sign to form an equation?
2+3³ =
1. O 5³
2. 3 +4:3
3. 10-2-9
4. 52 +4
Answers
Expression which placed on the right side of equal sign is,
[tex]2+3^3 =2 +3\times 3\times 3[/tex]
[tex]2+3^3= 2+27[/tex]
[tex]2+3^3= 29[/tex]
What is expression?
Mathematical expressions consist of at least two numbers or variables, at least one arithmetic operation, and a statement. It's possible to multiply, divide, add, or subtract with this mathematical operation.
What do you mean by equation?
A mathematical statement known as an equation is made up of two expressions joined together by the equal sign. A formula would be 3x - 5 = 16, for instance. When this equation is solved, we discover that the value of the variable x is 7.
Given, a expression 2 + [tex]3^3[/tex] = ?
what can be placed right side of the expression?
so, first we calculate the [tex]3^3[/tex] which means 3 x 3 x 3 = 27.
then , add 2 and 27 and the answer is 29.
but the answer is not present into the option.
Final answer will be 29 which placed right side of the equation 2 + [tex]3^3[/tex] = 29.
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In a set of eight consecutive integers, the smallest integer is more than $\frac35$ of the largest integer. What is the smallest possible value of the sum of the eight integers?
Answers
Answer:
116.
Step-by-step explanation:
Your welcome <3.
Answer:
116
Step-by-step explanation:
A manufacturer produces 25-pound lifting weights. The lowest actual weight is 24 pounds, and the highest is 26 pounds. Each weight is equally likely so the distribution of weights is uniform. A sample of 100 weights is taken. Find the probability that the mean actual weight for the 100 weights is greater than 25.2
Answers
The probability that the mean actual weight for the 100 weights is greater than 25.2 is equal to 0.0136%.
How to determine the probability that the mean actual weight for the 100 weights?In order to determine the probability that the mean actual weight for the 100 weights is greater than 25.2, we would have to calculate the mean and standard deviation.
In a uniform distribution, the mean of a data set can be calculated by using this formula:
Mean, E(x) = [a + b]/2
Where:
a is the lowest boundary.b is highest boundary.Substituting the given parameters into the formula, we have;
Mean, E(x) = [24 + 26]/2
Mean, E(x) = 50/2
Mean, E(x) = 25.
For the standard deviation, we have:
Standard deviation, S(x) = √((b - a)²/12)
Standard deviation, S(x) = √((26 - 24)²/12)
Standard deviation, S(x) = √((2)²/12)
Standard deviation, S(x) = √(4/12)
Standard deviation, S(x) = 0.5774.
For the sample standard deviation, we have:
Standard deviation = S(x)/√n
Standard deviation = 0.5774/√100
Standard deviation = 0.5774/10
Standard deviation = 0.05774.
By applying the central limit theorem, we would determine the z-score as follows:
Z-score = (x - μ)/σ
Z-score = (25.2 - 25)/0.05774
Z-score = 0.2/0.05774
Z-score = 3.64
Therefore, the p-value is equal to 0.000136.
Probability (x > z) = 0.0136%.
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Represent the following expressions as a power of the number a where a ≠ 0.: (a^3)^-2
Answers
1/[tex]a^{6}[/tex] is the required number.
What is an expression?Numbers, symbols and operators (such as and times) grouped together that show the value of something.
Given an expression (a³)^-2
(a³)^-2 = [tex]a^{-6}[/tex] = 1/[tex]a^{6}[/tex]
Hence, 1/[tex]a^{6}[/tex] is the value of the expression (a³)^-2
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You deposit $500 into a savings account that earns interest annually. The function g(x) = 500(1.06)x can be used to find the amount of money in the savings account, g(x), after x years. What is the range of the function in the context of the problem?
ℝ
[500, ∞)
[0, 500]
[0, ∞)
Answers
Omar deposited d dollars into a savings account y years ago. ... following expressions could be used to determine the zeros of the function f (x) when f (x).
Step-by-step explanation:
Martin had 6 magazines. Then m more magazines came in the mail. Write an expression that shows how many magazines Martin has now.
Answers
Martin had 6 magazines. Then m more magazines came in the mail. Write an expression that shows how many magazines Martin has now.
Martin now has: 6 + m
A fisherman catchs 12 fish in 5 days. How many fish does he get in
5 + x days?
Answers
( 60+12x )/5 fish fisherman will catch in 5+x days .
What is unitary method ?The unit method is a method of solving a problem by first obtaining the value of a single unit, then multiplying the value of the single unit to obtain the necessary value.
Assume you go to the market to buy six apples. The shopkeeper told you that he sells 10 apples for 100 rupees. In this case, the apple is the unit and the apple price is the value. Knowing the units and values is important when solving problems using unitary methods.
For simplicity, always write what you calculate on the right and what you know on the left. In the problem above, we know the amount of apples, but we don`t know the value of the apples. Note that the concept of ratio and ratio is used for problems related to this method.
Calculation5 days ------------> 12 fish
1 day --------------> 12/5 fish
5 + x days --------------> 12/5 x (5+x)
= ( 60+12x )/5 fish
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