Which represents the solution of the system of equations, y=x^2-x+1 and y=x? Determine the solution set by graphing
Answers
Answer:
(1,1)
Step-by-step explanation:
We have the following system:
[tex]y=x^2-x+1[/tex]
[tex]y=x[/tex]
We are going to plug the 2nd equation into the 1st eqaution giving us:
[tex]x=x^2-x+1[/tex]
Now time to solve [tex]x=x^2-x+1[/tex].
[tex]x=x^2-x+1[/tex]
Subtract x on both sides:
[tex]0=x^2-2x+1[/tex]
Now this actually not too bad. This a perfect square trinomial. That is it is of the form [tex]a^2x^2+2abx+b^2[/tex] which equals [tex](ax+b)^2[/tex].
So solving [tex]0=x^2-2x+1[/tex] is equivalent to solving [tex]0=(x-1)^2[/tex]
In order to solve [tex]0=(x-1)^2[/tex], we just need to know when x-1=0 which is at x=1. I got by adding 1 on both sides of x-1=0
Now remember y=x means we have the solution (1,1).
Okay, I'm going to leave all of this algebra here. I'm going to graph this without a graphing calculator.
[tex]y=x[/tex] is a line with slope 1 and y-intercept 0.
[tex]y=x^2-x+1[/tex] is a parabola. The vertex isn't obvious to me right now without the algebra but I do know the parabola is open up because the coefficient of x^2 being positive.
So let's find the vertex.
I'm going to start with the x-coordinate which is -b/(2a).
Compare [tex]x^2-x+1[/tex] to [tex]ax^2+bx+c[/tex] and you should see that [tex]a=1,b=-1,c=1[/tex].
So [tex]-\frac{b}{2a}=-\frac{-1}{2(1)}=\frac{1}{2}[/tex]
So the vertex occurs when the x-coordinate is 1/2.
To find the correspond y-value just use the equation that relations x and y.
That is use [tex]x^2-x+1[/tex].
Replace x with 1/2.
[tex](\frac{1}{2})^2-(\frac{1}{2})+1[/tex]
[tex]\frac{1}{4}-\frac{1}{2}+1[/tex]
Find a common denominator.
[tex]\frac{1}{4}-\frac{2}{4}+\frac{4}{4}[/tex]
[tex]\frac{1-2+4}{4}[/tex]
[tex]\frac{3}{4}[/tex]
So the vertex is at (1/2,3/4) and the parabola is open up.
When you plug in 0 into [tex]x^2-x+1[/tex] you get [tex]0^2-0+1=1[/tex] so the ordered pair (0,1) is on the parabola.
Using symmetry about the line x=1/2 we know that (1,1) is also on this graph.
Let's use this information to produce a graph now.
Now our answer appears to be (1,1) in the graph.
We can check this answer by plugging in (1,1) into our system and see if both equations check out:
[tex]y=x[/tex] gives us 1=1 which is true. So we are good there.
[tex]y=x^2-x+1[/tex] gives us [tex]1=1^2-1+1[/tex] which is true. So we are good there.
The work for showing [tex]1=1^2-1+1[/tex].
[tex]1^2-1+1[/tex]
[tex]1-1+1[/tex]
[tex]0+1[/tex]
1
So the point (1,1) checks out for both of our equations which means it is a common point amongst the equations given.
Related Questions
The graph shows the function f(x) = |x – h| + k. What is the value of k? k = –2.5 k = –1 k = 1 k = 2.5
Answers
Answer:
k= - 2.5
Step-by-step explanation:
If the graph is the attached picture then according to this:
The graph is in V- shape. So if the corners of the graph are not at (0,0) than there must be a constant.
k is the constant which means that it is the amount the function is vertically shifted. The graph is shifted 2.5 units down, therefore the value of k is -2.5.
Thus option 1 is correct....
Y’all what do i write
Answers
Answer:
y = - 0.5x + 4
Step-by-step explanation:
Calculate the slope m using the slope formula
m = (y₂ - y₁ ) / (x₂ - x₁ )
with (x₁, y₁ ) = (0, 4) and (x₂, y₂ ) = (4, 2) ← 2 points on the line
m = [tex]\frac{2-4}{4-0}[/tex] = [tex]\frac{-2}{4}[/tex] = - 0.5
Note the line crosses the y- axis at (0, 4) ⇒ c = 4
y = - 0.5x + 4 ← equation of boundary line
Answer:
y = -0.5 x + 4.
Step-by-step explanation:
Slope-intercept form of the equation of a line on a Cartesian Plane:
[tex]y = mx + b[/tex].
[tex]m[/tex] is the slope of the line.
If two different points [tex](x_1, y_1)[/tex] and [tex](x_2, y_2)[/tex] on the line are given, the slope of the line will equal
[tex]\displaystyle m = \frac{y_2 - y_1}{x_2 - x_1}[/tex].
For this line, the two points are:
[tex](4,2)[/tex], and [tex](0, 4)[/tex].As a result,
[tex]\displaystyle m = \frac{2 - 4}{4 - 0} = - \frac{1}{2}[/tex].
[tex]b[/tex] is the [tex]y[/tex]-coordinate (the second coordinate) of the point at the intersection of the line and the [tex]y[/tex]-axis (the vertical axis.) The [tex]x[/tex]-coordinate (the first coordinate) of that point shall equals [tex]0[/tex]. For this line, the coordinates of the intersection is [tex](0, 4)[/tex]. As a result, [tex]b = 4[/tex].
The slope-intercept form of the equation of this line will thus be
[tex]\displaystyle y = -\frac{1}{2}x + 4[/tex].
Let's investigate
You have an appointment at 2 o'clock at the dentist. Then you need to catch a
train one hour after your appointment. The walk to the train station takes
20 minutes. What time will you have to leave the dentist to catch your train?
Answers
Which fraction is not equivalent to 5/6?
A-15/18
B-20/30
C-10/12
D-30/36
Answers
Answer: B. 20/30
Step-by-step explanation: 20/30 simplified to 2/3. Multiply each number by 2 to get 4/6. 4/6 is not the same number as 5/6.
Answer:
B. 20/30
Step-by-step explanation:
I need dire help, I'm almost done with summer school!
A store owner tracks the number of online orders she receives each week since the start of a new advertising campaign and graphs the data.
Use the graph to select the true statement about the function that models the given situation.
A. The function is linear and is growing by equal differences over equal intervals.
B. the function is exponential and is growing by equal percentages over equal intervals .
C. the function is exponential and is growing by equal differences over equal intervals
D. the function is linear and is growing by equal percentages over equal intervals
Answers
Answer:
A. The function is linear and is growing by equal differences over equal intervals.
Answer:
A. The function is linear and is growing by equal differences over equal intervals.
Step-by-step explanation:
As you can see in the graph, the function is linear since if draw a line that conects the dots, the graph would turnout linear, and as you can see it is growing 100 orders every 2 weeks, so it is growing by equal differences, over equal intervals.
In each of the following pairs, determine the temperature change, and whether the temperature increased or decreased. a. 45° F to –10° F b. 35° C to 80° F c. –19° C to 49° C d. 137° C to 0° C e. 5° C to 19° F f. 68° F to 29° C
Answers
A: decreased
B: increased
C: increased
D: decreased
E: decreased
F: increased
Answer:
∵ Change = final value - initial value,
If change > 0, then the value is increased,
if change < 0, then the value is decreased,
a. 45° F to -10° F,
Change = -10° F - 45° F = -55° F < 0° F,
Thus, temperature is decreased,
b. 35° C to 80° F
∵ 35° C = 95° F ( by the formula (°C × 9/5) + 32 = °F )
Change = 80° F - 95° F = -5° F < 0° F,
Thus, temperature is decreased,
c. -19° C to 49° C,
Change = 49° C - (-19)° C = 68° C > 0° C,
Thus, temperature is increased,
d. 137° C to 0° C,
Change = 0° C - 137° C = -137° C < 0° C,
Thus, temperature is decreased,
e. 5° C to 19° F,
∵ 5°C = 41°F
Change = 19° F - 41° F = -22° F < 0° F,
Thus, temperature is decreased,
f. 68° F to 29° C,
∵ 29°C = 84.2°F
Change = 84.2°F - 68° F = 16.2° F > 0° F,
Thus, temperature is increased.
The segments shown below could form a triangle.
Answers
Answer:
True
Step-by-step explanation:
Answer:
The answer is false.
Step-by-step explanation:
Given line segments are :
AC = 9
CB = 8
AB = 17
As per the triangle inequality theorem the sum of any 2 sides should be greater than the third side. This should be true to all the three combinations.
[tex]AC+CB>AB[/tex] => [tex]9+8>17[/tex] => [tex]17>17[/tex] (not true)
[tex]AC+AB>CB[/tex] => [tex]9+17>8[/tex] => [tex]26>8[/tex] ( true)
[tex]AB+CB>AC[/tex] => [tex]17+8>9[/tex] => [tex]25>9[/tex] (true)
As we can see only two conditions are fulfilled not the third one, so the triangle cannot be formed.
The answer is false.
Clyde simplified ( j^-8)^-2
Answers
ANSWER
[tex]{j}^{16} [/tex]
EXPLANATION
The given expression is
[tex]( {j}^{ - 8} )^{ - 2} [/tex]
To simplify this expression, we use the law of exponents.
Recall that:
[tex] {(a}^{m} )^{n} = {a}^{mn} [/tex]
We need to multiply the exponents to get:
[tex]( {j}^{ - 8} )^{ - 2} = {j}^{ - 8 \times - 2} [/tex]
[tex]( {j}^{ - 8} )^{ - 2} = {j}^{16} [/tex]
Therefore after simplification, Clyde got:
[tex]{j}^{16} [/tex]
Answer:
Clyde should have applied the power of a power rule and multiplied the exponents instead of adding them. The simplified form should be j to the 16th power.
Step-by-step explanation:
The exponents should be multiplied.
The simplified form should be j^16
Ronalds furniture house is currently making 50 sofas in a day. How many sofas will be produced if prudction is to be increased by 150%
Answers
Answer:
125
Step-by-step explanation:
50 x 150% = 75
75 + 50 = 125
Answer:
125
Step-by-step explanation:
50 × 150% = 50 × 1.5 = 75
75 + 50 = 125
125 sofas is the answer
rewrite square root of -49 -4 in complex number notation please help
Answers
Note that if [tex]i^2=-1[/tex] than [tex]i=\sqrt{-1}[/tex],
Also note that [tex]\sqrt{ab}=\sqrt{a}\sqrt{b}[/tex]
So we have,
[tex]\sqrt{-49}=\sqrt{-1}\sqrt{49}=\boxed{7i}[/tex]
and,
[tex]\sqrt{-4}=\sqrt{-1}\sqrt{4}=\boxed{2i}[/tex]
Hope this helps.
r3t40
If f(x)=x^2+2x−5 and g(x)=2x+4, what is (f⋅g)(x)
Answers
Answer:
Step-by-step explanation:
f(x)=x²+2x−5 and g(x)=2x+4
(f⋅g)(x) = (x²+2x−5)(2x+4) = 2x^3 +4x² -10x +4x²+8x-20
(f⋅g)(x) = 2x^3+8x² -2x -20
For this case we have the following functions:
[tex]f (x) = x ^ 2 + 2x-5\\g (x) = 2x + 4[/tex]
We must find [tex](f * g) (x)[/tex]. By definition we have to:
[tex](f * g) (x) = f (x) * g (x)[/tex]
So:
[tex](f * g) (x) = (x ^ 2 + 2x-5) (2x + 4)[/tex]
Applying distributive property we have:
[tex](f * g) (x) = 2x ^ 3 + 4x ^ 2 + 4x ^ 2 + 8x-10x-20[/tex]
We add similar terms:
[tex](f * g) (x) = 2x ^ 3 + 8x ^ 2-2x-20[/tex]
Answer:
[tex](f * g) (x) = 2x ^ 3 + 8x ^ 2-2x-20[/tex]
p(x) is a polynomial with integer coefficients and p(-3) = 0. Which statements must be true? Choose all that apply. (MORE THAN ONE ANSWER)
x + 3 is a factor of the polynomial.
x - 3 is a factor of the polynomial.
-3 is the constant term of the polynomial.
p(x) can have at most 3 linear factors.
Answers
Answer:
x+3 is a factor of P
Step-by-step explanation:
By the factor theorem:
P(a)=0 <-> x-a is a factor of P.
You have P(-3)=0.
This implies x-(-3) is a factor of P.
Note: x-(-3)=x+3.
We know this from the factor theorem, which states that if f(a) = 0, then (x - a) is a factor.
Thus we know that (x - -3) is a factor, which simplifies to (x + 3).
All the other answer options do not have to be true, because (x - 3) doesn’t have to be factor, -3 doesn’t have to be the constant of the polynomial, and p(x) can have any number of linear factors, which means the only answer you should select as definitely true is the first option.
I hope this helps! Let me know if you have any questions :)
what is an irrational number that is between 5.2 and 5.5.
Answers
Answer:
The irrational number could be √30
Step-by-step explanation:
An irrational number is a number which can not be written as a ratio between two numbers and it can not be written as a simple fraction. In simple words we can not write it in p/q form.
Find an irrational number that is between 5.2 and 5.5.
Mostly irrational numbers are square root numbers.
So square the both numbers.
(5.2)^2 (5.5)^2.
27.04 , 30.25
It means that the number is between 27.04 and 30.25
We will pick number 30
Square root 30
√30 = 5.48
The irrational number could be √30
It is 5.48 to the hundredth....
F = {(-1, 2), (3, 2), (4, 2), (0, 2)} Is F a function and why/why not?
Yes, no two ordered pairs in this list have the same first coordinate.
No, each set of ordered pairs in this list has the same second coordinate.
Yes, there is more than one ordered pair in this list.
No, there is a limited number of ordered pairs in this list.
Answers
Answer:
Yes, no two ordered pairs in this list have the same first coordinate
Step-by-step explanation:
we know that
A function is: a relation from a set of inputs to a set of possible outputs where each input is related to exactly one output.
That means that there can be only one ordered pair with the same value of x.
therefore
In this problem F is a function, because all the ordered pairs of the list don't have the same first coordinate
factor the trinomial below. 4x^2+23-6. which two binomials are factors of the trinomial
Answers
Answer:
4x - 1 and x + 6 are the factors of given Trinomial
Step-by-step explanation:
The given Trinomial is:
[tex]4x^{2}+23x-6[/tex]
We can factorize this trinomial by mid-term splitting. The mid-term(23x) can be split up in two terms (24x and -x) such that their sum is 23x and their product(-24x²) is equal to the product of other two terms (4x² and -6)
So, the above expression can be written as:
[tex]4x^{2}+23x-6\\\\ =4x^{2}+24x-x-6\\\\\text{Taking out commons, we get}\\\\=4x(x+6)-1(x+6)\\\\ =(4x-1)(x+6)[/tex]
Therefore, 4x - 1 and x + 6 are the factors of given Trinomial
Answer: 4x-1 and x+6
Step-by-step explanation: a p e x
A train can hold 78 passengers. The train starts out empty and picks up one passenger at the first stop,2 passengers at the second stop,3 passengers at the third stop,and so forth. After how many stops will the train be full.
Answers
Answer:
After the 12th Stop
Step-by-step explanation:
2nd stop 1+2=3
3rd 3+3=6
4th 6+4=10
5th 10+5=15
6th 15+6=21
7th 21+7=28
8th 28+8=36
9th 36+9=45
10th 45+10=55
11th 55+11=66
12th 66+12=78
After the 12th stop.
The train will be full after 12 full stops. If we include the 13th stop, the train would exceed its capacity. Therefore, the train can handle 12 full stops if it picks one passenger at the first stop and increases the number of passengers picked by one at each stop.
Explanation:The problem you're describing is a summation problem in mathematics. You're incrementing the number of passengers at each stop starting at one and increasing by one each time. This can be summed up using the formula for the sum of the first n positive integers (including one), which is n*(n+1)/2. To find out after how many stops the train will be full, you'll need to solve this formula for n such that this sum is less than or equal to 78 (the capacity of the train). You can do this by substitution or by solving the quadratic equation, n² + n - 156 = 0. The result is roughly 12.421, which means that after 12 full stops, the train will have 78 passengers (since we can't have a fraction of a stop). After the 13th stop, the train will exceed its capacity.
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x³-64 in prime factored form
Answers
[tex]\bf \textit{difference and sum of cubes} \\\\ a^3+b^3 = (a+b)(a^2-ab+b^2) \qquad \qquad a^3-b^3 = (a-b)(a^2+ab+b^2) \\\\[-0.35em] \rule{34em}{0.25pt}\\\\ x^3-64\implies x^3-4^3\implies (x-4)(x^2+4x+4^2)\implies (x-4)(x^2+4x+16)[/tex]
If R(x) = 2x - 5 and g(x) = x2 - 4x - 8, find (f + g(%).
O A (f+9)(x) = x2 - 2x-3
O B. (f+g)(x) = x2 - 2x-13
O C. (*+ g)(x) = 3x2 - 4x- 13
O D. (f+ g)(x) = x2 + 2x-3
C
PREVIOUS
Answers
Answer:
OB. (f+g)(x) = x^2-2x-13
Step-by-step explanation:
Given
f(x) = 2x-5
and
g(x) = x^2-4x-8
We have to find (f+g)(x)
So,
(f+g)(x) = f(x) + g(x)
= 2x-5 + (x^2-4x-8)
= 2x-5+x^2-4x-8
= x^2+2x-4x-5-8
=x^2-2x-13
Therefore, Option B is correct ..
Evaluate the expression -66/(-3)+(-4)-(-5)
Answers
Answer:
-21
Step-by-step explanation:
For this case we must evaluate the following expression:
[tex]\frac {-66} {(- 3)} + (- 4) - (- 5) =[/tex]
We solve the division:
[tex]22 + (- 4) - (- 5) =[/tex]
We eliminate parentheses keeping in mind that:
[tex]+ * - = -\\- * - = +[/tex]
So:
[tex]22-4 + 5 =[/tex]
Different signs are added and the same sign is placed, while different signs are subtracted and the sign of the major is placed:
[tex]18 + 5 =\\23[/tex]
Answer:
23
The equation of a line is 3x + 8y - 16 = 0.
What is the slope and y-intercept?
slope =?
y intercept =?
Help confused !
Answers
3x+8y=16
8y=16-3x
Y= (16-3x)/8
Y= 2-3/8x
The y intercept is 2 and the slope is -3/8.
Hope this helps!
whole numbers to make 380 in four different ways.
Answers
Answer:
There are several ways to make 380 with whole numbers. For example:
100 + 80 = 380220 + 160 = 380190 + 190 = 3801 + 379 = 380What is the median of this data set?
22, 37, 49, 15, 82
O
<
A. 37
O B. 27
O C. 31
O D. 41
Answers
Answer:
A. 37
Step by step explanation:
The median is the middle value with high to low frequency and low to high frequency. How to determine the median is to sort the numbers from the smallest to the big number and the largest number to the smallest number, then count using two hands (one finger left hand and one finger right hand).
Method:
22, 37, 49, 15, 82 then sorted: 82, 49, 37, 22, 15 means the median is 37
-----------------------✨ Awesome ✨-----------------------The median of this data set is 49.
Option A is correct.
What is Median of the Data?The median is the value in the middle of a data set, which means that 50% of the data points have a value less than or equal to the median, and 50% of the data points have a value greater than or equal to the median. Given a small data set, count the number of data points (n) and arrange them in increasing order.
We have the data: 22, 37, 49, 15, 82
Now, arranging the data in Ascending order as
15, 22, 37, 49, 82
Now, the median is the middle term of the data.
Here there are 5 terms.
So, the median of data is 37.
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A football team started at the line of scrimmage. On the first play of the drive, the team lost 4 yards. On the second play of the drive, the team gained 45 yards. Taylor calculated |–4 – 45| = |–49| = 49, and said that the team gained a total of 49 yards on the two plays. Which best describes Taylor’s error?
A. Taylor did not find the correct distance between –4 and 45.
B. Taylor did not make an error. The team gained a total of 49 yards on the two plays.
C. Taylor correctly found the distance between –4 and 45, but this number represents the team’s total loss, not the team’s total gain.
D. Taylor correctly found the distance between –4 and 45, but this number represents the total distance the team moved in both directions, not the team’s total gain.
Answers
Answer:
D. Taylor correctly found the distance between –4 and 45, but this number represents the total distance the team moved in both directions, not the team’s total gain.
Step-by-step explanation:
The team only ended up gaining the 45 yards on the second play and had gained -4 on the first play. 45-4≠49, but the total distance between -4 and 45 is 49.
Answer:
D.
Step-by-step explanation:
find the absolute error of the measurement 10 meters. explain its meaning
Answers
Answer:
.5 m
Step-by-step explanation:
You can use maximum possible error to calculate absolute error
Maximum possible error of is half the unit of measure
This means it is ±0.5 meters
In this case, you find the place value the measurement 10 meters is rounded to .This measurement was measured to the nearest meter, thus the measuring unit is 1 meter.
Here you use the maximum possible error as the absolute error.
This means the measurement is 10 ± .5
The perpendicular bisector of a chord...
Answers
Choices A and C are both true statements.
But C is trivial. It's CALLED a perpendicular "bisector" of the chord, so OF COURSE it bisects the chord ... otherwise it would be silly to CALL it a "bisector".
The choice that's surprising but true is A .
The perpendicular bisector of a chord passes through the center of the circle.
What is a chord?A chord in plane geometry is a line segment that connects two points on a curve. A line segment whose endpoints are on a circle is frequently described using this phrase. A cycle chord of a graph cycle is an edge that is not in and whose endpoints are in, according to graph theory, which also uses the phrase.
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The description below represents Function A and the table represents Function B:
Function A
The function is 2 more than 5 times x.
Function B
x y
−1 2
0 5
1 8
Which statement is correct about the slope and intercept of the two functions?
A) Their slopes are equal but y-intercepts are not equal.
B) Their slopes are not equal but y-intercepts are equal.
C) Both slopes and y-intercepts are equal.
D) Neither slopes nor y-intercepts are equal.
Answers
Answer:
D is correct
Step-by-step explanation:
Step 1 : Find equation of function A in y = mx + c
2 more than 5 times x
f(x) = 2 + 5x or y = 5x + 2
Step 2 : Identify slope and y-intercept of function A
y = mx + c
y = 2 + 5x
Slope = m = 5
Y-intercept = c = 2
Step 3 : Find the equation of function B in y = mx + c
From my understanding, the number given in the question are coordinates
(-1, 2) , (0, 5) , (1, 8)
Slope = y2-y1/x2-x1
Lets take two coordinates : (-1, 2) , (0, 5)
Slope = 5-2/0-(-1)
Slope = 3/1 = 3
For y-intercept
Lets take (1, 8)
y = mx + c
8 = 3(1) + c
c = 5
The equation is f(x) = 3x + 5 or y = 3x + 5
Step 4 : Lets write both equations with their slope and y-intercept and check with the statements
y = 5x + 2
y = 3x + 5
A) Their slopes are equal but y-intercepts are not equal. Incorrect
B) Their slopes are not equal but y-intercepts are equal. Incorrect
C) Both slopes and y-intercepts are equal. Incorrect
D) Neither slopes nor y-intercepts are equal. Correct
!!
Find the missing value in the pair of similar polygons.
Answers
The scale factor is 2.25.
estimate 16.667 + 7.66 by first rounding each number to the nearest tenth
Answers
Answer:
24.4
Step-by-step explanation:
16.667 + 7.66 becomes 16.7 + 7.7, which, in turn, is 24.4
Answer:
24.4
Step-by-step explanation:
The nearest tenth is the first decimal place. We look at the second decimal place. If it is 5 or above we round up
16.667 to the nearest tenth = 16.7
7.66 to the nearest tenth = 7.7
16.7 + 7.7 = 24.4
The volume of a cube depends on the length of its sides. This can be written
in function notation as ws). What is the best interpretation of 3) = 27?
Answers
Answer:
Step-by-step explanation:
The given question is that the volume of a cube depends on the length of its sides.This can be written in function notation as v(s). What is the best interpretation of v(3)=27.
Solution:
According to the question the volume of a cube depends on the length of its sides. According to the statement we will apply the formula of volume of a cube.
V(s)=s³
In this question we have given s=3ft.
So, we will put the value of 's' in the formula.
V(s)=s³
V(3)=3³
Multiply 3 three times to get the answer.
V(3)=3*3*3
V(3)=27 ft³
This means that the cube has a volume of 27ft³ with the length of its sides 3ft....
Simplify (x-4)(x^2+5x+2).
Answers
Answer: x^3 + 1x^2 - 18x - 8
Step-by-step explanation: Using the box method, we get the values x^3, 5x^2, 4x^2, 20x, 2x, and 8. Then you add the like terms to get your final answer.
lol my diagram got glitched out when uploading, sorry if its hard to understand