Two functions are shown in the table below. Function 1 2 3 4 5 6 f(x) = −x2 + 4x + 12 g(x) = −x + 6 Complete the table on your own paper, then select the value that is a solution to f(x) = g(x).
Answers
For [tex]\fbox{\begin \\\math{x}=6\\\end{minispace}}[/tex] the function [tex]f(x)=-x^{2} +4x+12[/tex] and [tex]g(x)=-x+6[/tex] has same value.
Step by step explanation:
The given functions are,
[tex]f(x)=-x^{2}+4x+12[/tex]
[tex]g(x)=-x+6[/tex]
Step 1:
Substitute [tex]x=1[/tex] in [tex]f(x)=-x^{2} +4x+12[/tex] to obtain the value of [tex]f(1)[/tex].
[tex]f(1)=-1^{2} +4(1)+12\\f(1)=-1+4+12\\f(1)=15[/tex]
Substitute [tex]x=1[/tex] in [tex]g(x)=-x+6[/tex] to obtain the value of [tex]g(1)[/tex] .
[tex]g(1)=-1+6\\g(1)=5[/tex]
Step 2:
Substitute [tex]x=2[/tex] in [tex]f(x)=-x^{2} +4x+12[/tex] to obtain the value of [tex]f(2)[/tex].
[tex]f(2)=-2^{2} +4(2)+12\\f(2)=-4+8+12\\f(2)=16[/tex]
Substitute [tex]x=2[/tex] in [tex]g(x)=-x+6[/tex] to obtain the value of [tex]g(2)[/tex] .
[tex]g(2)=-2+6\\g(2)=4[/tex]
Step 3:
Substitute [tex]x=3[/tex] in [tex]f(x)=-x^{2} +4x+12[/tex] to obtain the value of [tex]f(3)[/tex].
[tex]f(3)=-3^{2} +4(3)+12\\f(3)=-9+12+12\\f(3)=15[/tex]
Substitute [tex]x=3[/tex] in [tex]g(x)=-x+6[/tex] to obtain the value of [tex]g(3)[/tex] .
[tex]g(3)=-3+6\\g(3)=3[/tex]
Step 4:
Substitute [tex]x=4[/tex] in [tex]f(x)=-x^{2} +4x+12[/tex] to obtain the value of [tex]f(4)[/tex].
[tex]f(4)=-4^{2} +4(4)+12\\f(4)=-16+16+12\\f(4)=12[/tex]
Substitute [tex]x=4[/tex] in [tex]g(x)=-x+6[/tex] to obtain the value of [tex]g(4)[/tex] .
[tex]g(4)=-4+6\\g(4)=2[/tex]
Step 5:
Substitute [tex]x=5[/tex] in [tex]f(x)=-x^{2} +4x+12[/tex] to obtain the value of [tex]f(5)[/tex].
[tex]f(5)=-5^{2} +4(5)+12\\f(5)=-25+20+12\\f(5)=7[/tex]
Substitute [tex]x=5[/tex] in [tex]g(x)=-x+6[/tex] to obtain the value of [tex]g(5)[/tex] .
[tex]g(5)=-5+6\\g(5)=1[/tex]
Step 6:
Substitute [tex]x=6[/tex] in [tex]f(x)=-x^{2} +4x+12[/tex] to obtain the value of [tex]f(6)[/tex].
[tex]f(6)=-6^{2} +4(6)+12\\f(6)=-36+24+12\\f(6)=0[/tex]
Substitute [tex]x=6[/tex] in [tex]g(x)=-x+6[/tex] to obtain the value of [tex]g(6)[/tex] .
[tex]g(6)=-6+6\\g(6)=0[/tex]
Step 7:
As per the given condition [tex]f(x)=g(x)[/tex].
(a). Substitute [tex]f(x)=-x^{2} +4x+12[/tex] and [tex]g(x)=-x+6[/tex] in above equation.
[tex]-x^{2} +4x+12=-x+6[/tex]
(b). Multiply with [tex]-1[/tex] on both sides.
[tex]x^{2} -4x-12=x-6[/tex]
(c). Shift the term [tex]x-6[/tex] to left hand side.
[tex]x^{2} -4x-12-x+6=0\\x^{2} -5x-6=0[/tex]
(d). Split the middle term in such a way that its sum is 5 and multiplication is 6.
[tex]x^{2} -(6-1)x-6=0\\x^{2} -6x+x-6=0\\x(x-6)+1(x-6)=0\\(x+1)(x-6)=0\\x=-1 ,6[/tex]
It is observed from the above solution that for [tex]x=6[/tex] both the functions [tex]f(x)[/tex] and [tex]g(x)[/tex] has same value.
Direct method:
[tex]f(x)=g(x)\\\Leftrightarrow-x^{2} +4x+12=-x+6\\\Leftrightarrow-x^{2} +4x+12+x-6=0\\\Leftrightarrow-x^{2} +5x+6=0\\\Leftrightarrow-x^{2} +6x-x+6=0\\\Leftrightarrow x^{2} -6x+x-6=0\\\Leftrightarrow x(x-6)+1(x-6)=0\\\Leftrightarrow(x+1)(x-6)=0\\\Leftrightarrow x=6,-1[/tex]
The table for the function [tex]f(x)=-x^{2} +4x+12[/tex] and [tex]g(x)=-x+6[/tex] is attached below.
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Answer details:
Grade: Middle school.
Subjects: Mathematics.
Chapter: Function.
Keywords: Function, Middle term split method, Binomial,Quadratic, Polynomial, Factorized, Perfect square, Zeros, Zeros of a function, Expression, Equation, x, x^2, x^3, -x^2+4x+12, -x+6, roots of equation.
Answer:
The answer would be D.
Step-by-step explanation:
Related Questions
In triangle def, FE = 13 and angle f equals 34. Find DF to the nearest tenth.
Answers
Micah takes a job with a starting salary of $78,000 for the first year. He earns a 4% increase each year. How much does Micah make over seven years? $567,840 $616,067 $718,710 $718,499
Answers
Fv=pmt [((1+r)^(t)-1)÷r]
Fv how much he made ?
PMT yearly salary 78000
R rate of increases 0.04
T time 7 years
Fv=78,000×(((1+0.04)^(7)−1)÷(0.04))
Fv=616,066.96 round your answer to get 616067
Answer:
b
Step-by-step explanation:
05.02)
Two right triangles are graphed on a coordinate plane. One triangle has a vertical side of 2 and a horizontal side of 6. The other triangle has a vertical side of 10 and a horizontal side of 30.
Could the hypotenuses of these two triangles lie along the same line?
No, because they are not similar triangles
No, because one is larger than the other
Yes, because they are both right triangles
Yes, because they are similar triangles
Answers
Answer:
Yes, because they are similar triangles
Step-by-step explanation:
I got the answer right on the test
Convert 7.2 ⋅ 10–3 to standard form.
72,000
7,200
0.0072
0.00072
Answers
................................
0.0072 hope that helped
If BC = 4.5, CD = 7.7, and AD = 16.7, find AB to the nearest tenth.
A. 2.1
B. 28.6
C. 9.8
D. 2.2
Answers
[tex]9.8[/tex]
EXPLANATION
Triangle ABC is similar to triangle ADC,
therefore the corresponding sides sides are in the same proportion.
[tex] \frac{AB}{AD} = \frac{BC}{CD} [/tex]
We substitute the values to obtain,
[tex] \frac{AB}{16.7} = \frac{4.5}{7.7} [/tex]
We multiply through by
[tex]16.7[/tex]
to get,
[tex] AB= \frac{4.5}{7.7} \times 16.7[/tex]
This simplifies to give us,
[tex] AB= 9.75940259402....[/tex]
To the nearest tenth,
[tex] AB= 9.8[/tex]
The correct answer is C
how much would an 185 astronaut weigh on the moon?
Answers
Determine if the solution set for the system of equations shown is the empty set, contains one point or is infinite. 3x + 2y = 6 x - y = 2
Answers
Hope it helps.
Solve the inequality -2x is less than or equal to 3x + 1 is less than or equal to 10
Answers
[tex] \frac{-2x-1}{3} \leq x \leq 3[/tex]
this is what i understood from what you typed...
Write 121 raised to the one-half power equals 11 as a logarithmic function
Answers
so
[tex]121^\frac{1}{2}=11[/tex] translates to [tex]log_{121}(11)=\frac{1}{2}[/tex]
Evaluate 9 y 1 − y 0 9y1−y0 if y=−8 y=−8 .
Answers
9y - y^0...when y = -8
9(-8) - 1......anything to the ^0 = 1
-72 - 1 =
-73 <==
Answer:
Step-by-step explanation:
When factored, the trinomials x^2-2x-15 and x^2-9x+20 have one binomial factor in common. What is this factor?
1. x-7
2. x+7
3. x-5
4. x+5
Answers
(x-5)(x+3)
Proof:
-5+3=-2
-5*3=-15
x^2-9x+20
(x-4)(x-5)
Proof:
-4-5=-9
-4*-5=20
Final answer: (x-5)
What are the amplitude, period, and midline of f(x) = 3 cos(4x − π) + 2?
Answers
Answer:
The amplitude of the function is 3, period of the function is [tex]\frac{\pi}{2}[/tex] and midline of the function is 2.
Step-by-step explanation:
The given function is
[tex]f(x)=3\cos(4x-\pi)+2[/tex] .... (1)
The standard form of a Cosine function is
[tex]g(x)=a\cos(bx+c)+d[/tex] ... (2)
where, a is amplitude, period is [tex]\frac{2\pi}{b}[/tex] and d is midline.
From (1) and (2), we get
[tex]a=3,b=4,d=2[/tex]
The amplitude of the function is
[tex]a=3[/tex]
The period of the function is
[tex]\frac{2\pi}{b}=\frac{2\pi}{4}=\frac{\pi}{2}[/tex]
The midline of the function is
[tex]d=2[/tex]
Therefore the amplitude of the function is 3, period of the function is [tex]\frac{\pi}{2}[/tex] and midline of the function is 2.
If s is a sequence of consecutive multiples of 3, how many multiples of 9 are there in s? (1) there are 15 terms in s. (2) the greatest term of s is 126.
Answers
[tex]\it a_{15} =126=3\cdot42 \\\;\\a_1=3n=? \\\;\\ 3, 6, 9, 12, ... , 3n, ..., 3\cdot42 [/tex]
[tex]\it 42-(n-1) =15 \Rightarrow n-1=42-15=27 \Rightarrow n=28 \Rightarrow \\\;\\ \Rightarrow a_1=3\cdot28= 84[/tex]
Now, we have the sequence :
[tex]\it a_1= 3\cdot28 = 84 \notin M_9 \\\;\\ a_2= 3\cdot29 = 87 \notin M_9 \\\;\\ a_3= 3\cdot30 = 90 \in M_9 \\\;\\ \vdots[/tex]
[tex]\it a_{15} =3\cdot42=126 \in M_9 \\\;\\ 90\leq9\cdot k\leq126 |:9 \Rightarrow 10\leq k\leq14 \Rightarrow k \in \{10, 11, 12, 13, 14\}[/tex]
It implies 5 multiples of 9
At a glance : {90, 99, 108, 117, 126}
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Answers
In the equation 50t + 3w = 150, 3w represents the distance Jennifer travels by walking in 3 hours.
Explanation:In the equation 50t + 3w = 150, the term 3w represents the distance Jennifer travels by walking. Since w represents the number of hours she spends walking, 3w represents the distance she walks in 3 hours. This is because the rate of walking is constant, so if Jennifer walks for 1 hour and covers a certain distance, she would walk 3 times that distance in 3 hours.
can someone help me with this factoring problem? I forgot how to factor.
Factor completely −3x2 + 6x − 9.
−3(x2 − 2x + 3)
−1(x2 − 6x + 9)
−3x(x2 − 2x + 3)
−3(x2 + 2x − 3)
Answers
= -3(x^2 - 2x + 3)
answer
-3(x^2 - 2x + 3) first one
-3(x^2-2x+3)
To check, multiply what you took out by each term in the trinomial and make sure it's the same
-3*x^2=-3x^2
-3*-2x=6x
-3*3=-9
Write the equation of the line that passes through the points (-2,3) and (-2,5)
Answers
One week, Katie Hopper earned the following commissions: Monday, $124; Tuesday, $88; Wednesday, $62; Thursday, $137; Friday, $195. Find her total amount of commissions for the week
Answers
This is the answer⬆️
add them together:
124 + 88 + 62 +137 +195 = $606 total
HELP PLEASE THANKS:(
Answers
4+3y=13
3y=9
y=3
The answer is B
The students in the fourth grade sold 684 erasers for a fund-raiser. They sold 4 times as many erasers as the students in fifth grade. How many erasers did the students in 5 grade sell?
Answers
so
684 / 4 = 171
answer
students in 5 grade sold 171 erasers
The students in fifth grade sold 171 erasers
How Many Erasers Did the Fifth Grade Students Sell?
The question asks us to find out how many erasers the students in fifth grade sold based on the sales of fourth graders. We are told that the fourth grade students sold 684 erasers and that this number is 4 times the amount sold by the fifth graders. To find the number of erasers sold by the fifth graders, we need to perform a division operation.
Setting up the equation, let x be the number of erasers the fifth-grade students sold. Thus, 4x = 684. To solve for x, we divide both sides of the equation by 4. Therefore, x = 684 / 4 = 171.
So, the students in fifth grade sold 171 erasers for the fund-raiser.
a lab researcher wants to find out whether mice will run through a maze quicker during the day or at night, after training. He has 100 mice available. He randomly assigns 50 of them into ech group. He trains the first group to run the maze ar 9:00am and trains the sevond group to run the maze ar 9:00 pm. Each mouse is trained the same way, and the last 3 years run times are recored. Describe what is being measured in this experiment and what variable is being manipulated?
Answers
In this case, since the run times are the one being compared and recorded, therefore the data being measured in the experiment is the run times.
The difference of the two groups of mice is the time of the day that the mice run, day or night. Therefore the variable being manipulated is the time of the day.
Express 257.36 multiplied by 10 in scientific notation
Answers
__________________________________________________________
The height of the oversized container is 3 m it's diameter is 2 m what is its area?
Answers
i think
( 3.14×radius×radius) + (2×3.14×height)
thus,
(3.14×1×1) + (2×3.14×3)
=21.98 m
Jack is building a deck that is 8 ft by 27.63 ft. He has the base built; he just needs to cover the surface with wood. When he gets to the hardware store he finds sheets of wood that are four ft by eight ft. Use estimation to determine how many sheets of wood he should buy to cover the surface of the deck.
Answers
A ( deck ) = 8 * 27.63 = 221.04 ft²
The area of the sheet of wood:
A ( sheet ) = 4 * 8 = 32 ft²
221.04 : 32 = 6.9 ≈ 7 ( we need a whole number )
Answer: He should buy 7 sheets of wood to cover the surface of the deck.
The area (A) of a circle with a radius of r is given by the formula A=pier^2 and its diameter (d) is given by d=2r. Arrange the equations in the correct sequence to rewrite the formula for diameter in terms of the area of the circle.
Answers
Final answer:
The diameter d of a circle can be expressed in terms of the area A by first solving the area formula A = πr^2 for r, giving r = √(A/π), and then substituting r into the diameter formula d = 2r, resulting in d = 2√(A/π).
Explanation:
To rewrite the formula for the diameter of a circle in terms of the area, we start with the given formula for area A = πr^2 and the equation for diameter d = 2r. Our aim is to express the diameter d solely in terms of the area A. To achieve this, we rearrange the area formula to solve for the radius r:
A = πr^2r^2 = A/πr = √(A/π)Then, we substitute r into the diameter formula:
d = 2rd = 2√(A/π)Thus, the diameter d can be expressed in terms of the area A as d = 2√(A/π).
to the nearest whole degree, what angle measure has a sine of .9762
Answers
and if you just want the answer it is:
77.475 Degrees
and
it is 102.525 Degrees
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Answers
Sin corresponds to the y-value, so sin would be positive at Quadrants 1 and 2.
Tan, which is y/x (sin/cos), is positive at Quadrants 1 and 3.
So at quadrant 1 is where both sin and tan are positive.
In the diagram, which two points are shown to be collinear?
Answers
In the given diagram, the points that are shown to be collinear are H and E.
Collinearity in geometry refers to the condition where three or more points lie on the same straight line. In this case, points H and E are collinear as they lie on the same straight line, which is represented by a vertical double-headed arrow passing through them.
The diagram consists of two parallelograms, labeled 'I M' and 'E P', with three points inside each. In the parallelogram 'I M', the points are labeled as I, H, and J from top to bottom. Similarly, in the parallelogram 'E P', the points are labeled as F, E, and G from top to bottom. Among these points, only H and E share the same straight line, making them collinear. The other points do not share this line and hence are not collinear with H and E. This concept is fundamental in understanding geometric relationships and properties.
The length of a rectangle is 2 yd longer than its width. if the perimeter of the rectangle is 24 yd , find its area.
Answers
Length = x +2
Perimeter = 24
Area = Width • Length = ?
Perimeter = 24 = 2(x +2) + 2(x)
= 2x + 4 + 2x
= 4x + 4
= 20 = 4x
= 5 = x
Width = x = 5, therefore Length = (5) + 2 = 7
Area = Width • Length = 5 • 7 = 35
Area = 35 yards
What is the lateral surface area of the prism below?
1200 in2
2400 in2
1392 in2
2592 in2
Answers
so, the lateral, sides, area, is just the area of the sides, well, the sides are just those three rectangles
now, the rectangle is the middle is length = 25 as well as the others, but we dunno the width, say width = c
[tex]\bf c=\sqrt{16^2+12^2}\implies c=\sqrt{256+144}\implies c=\sqrt{400}\implies c=20\\\\ -------------------------------\\\\ \textit{lateral area}\implies (25\cdot 16)+(25\cdot 12)+(25\cdot 20) \\\\\\ 400+300+500[/tex]
Which system of inequalities is represented by the graph?
y ≤ –1/2x
y > –1
y ≥ –1/2x
y < –1
y < –1/2x
y ≥ –1
y > –1/2x
y ≤ –1
Answers
ur other line has a y int of (0,0), it has a slope of -1/2 ....it has a dashed line...it is shaded below the line.....so the equation for this line is : y < -1/2x
so ur answer is : y > = -1 and y < -1/2x
the answer is D)y ≥ 2x
y > 1
on ed
just took the test