The graph of the function f(x) = x^2 will be translated 3 units up and 1 unit left. What is the resulting function g(x)?

Answers

Answer 1
Answer:

Answer:  g(x) = (x + 1)² + 3

Step-by-step explanation:

The vertex form of a quadratic equation is: y = a(x - h)² + k    where

  • "a" is the vertical stretch
  • -a is a reflection over the x-axis
  • h is the horizontal shift (positive = right, negative = left)
  • k is the vertical shift (positive = up, negative = down)

f(x) = x²

Given: 3 units up   -->   k = 3

           1 unit left    -->   h = -1

                     g(x) = (x + 1)² + 3

Answer 2
Answer:

The translation of the graph of a function is one where the graph is moved to a different location on the plane that does not include a change in shape or rotation

The resulting function from the translation of the function f(x) = x², 3 units up and 1 unit left, g(x) = x² + 2·x + 4

The process by which the above value for g(x) is found is presented as follows:

The given function f(x) = x²

The vertical translation given to the function = 3 units up

The horizontal translation given to the function  = 1 unit left

The required parameter;

To find the resulting function g(x) that has results from the given translations

Solution:

A translation of a function y = f(x) vertically,k units upwards is the function y = f(x) + k

A translation of a function y = f(x) horizontally, k, units left, is the function y = f(x + k)

Therefore, we get

g(x) = f(x + 1) + 3 = (x + 1)² + 3 = x² + 2·x + 4

g(x) = x² + 2·x + 4

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Bruce earns 14.45/h and works 40 hours a week. What is his gross annual income

Answers

Answer:

$30056

Step-by-step explanation:

First, you have to do $14.45 times 40 to get how much he makes in a week, and you get $578. Then do 578 x 52 as there are 52 weeks in a year. Then you would get the answer $30056 as his yearly income.

Answer:

$30,056.00

Step-by-step explanation:

Took the quiz

A is 30% of b. b is 90% of c. what % of c is a

Answers

A = 30% of B = 0.3B
B = 90% of C = 0.9C

A = 0.3(0.9C) = 0.27C = 27% of C

Determine whether the lines L1 and L2 are parallel, skew, or intersecting. L1:x=9+6t,y=12-3t,z=3+9t L2:x=4+16s, y=12-8s, z=16+20s parallel skew intersecting If they intersect, find the point of intersection. (If an answer does not exist, enter DNE.)

Answers

Answer:

L1 and L2 are skew

Step-by-step explanation:

Since the equation of the line is

L1:x=9+6t,y=12-3t,z=3+9t

L2:x=4+16s, y=12-8s, z=16+20s

then if they intersect each other , they will have both in that point P=(xp , yp ,zp) then

1)9+6t = 4+16s

2) 12-3t =2-8s

3) 3+9t = 16+20s

adding 2*2) to 1)

9+6*t + 24-6t  = 4+16*s + 4-16*s

33 = 8

since this is not possible , the error comes from our assumption that the lines intersect each other

then they are skew or parallel. They are parallel if their corresponding vectors are parallel , that is

L1 (x,y,z) = (9,12,3) + (6,-3,9)*t

L1 (x,y,z) = (4,2,16) + (16,-8,20)*t

then if they are parallel

(16,-8,20)= k*(6,-3,9)

16=6*k

-8 = -3*k

20= 9*k

since there is no k that satisfy for x , y and z simultaneously then L1 and L2 are not parallel

therefore L1 and L2 are skew

Final answer:

The lines L1 and L2 are neither parallel nor intersecting. Upon comparing their direction vectors and attempting to find a common solution, it is determined that they are skew.

Explanation:

In order to determine whether two lines in three dimensions are parallel, skew, or intersecting, we compare their direction vectors. The given lines L1 and L2 are in the form of parametric equations. The direction vectors for the lines are d1 = <6, -3, 9> for L1 and d2 = <16, -8, 20> for L2. To determine if they are parallel, we check if there is a constant ratio between the corresponding terms. This isn't the case here, so the lines are not parallel.

For skew lines, they neither intersect or are parallel. Since we have already confirmed the lines are not parallel, to confirm if they are skew we must try to find a common solution (point of intersection). If we cannot, then they are skew. However, solving the equations does not give a common solution, so they do not intersect either. Hence, the lines are skew.

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Jared ate 1/4 of a loaf bread. He cut the rest of loaf into slices. How many slices of bread did he cut?

Answers

Answer:  He cut 6 slices of bread.

Step-by-step explanation:

Given : Jared ate (1)/(4) of a loaf of bread.

Then , the reaming portion of the bread will be 1-(1)/(4)=(4-1)/(3)=(3)/(4).

The size of each slice = (1)/(8) of a bread.

N ow , the number of slices he cut the remaining portion =(3)/(4)/(1)/(8)

=(3)/(4)*8=6

Hence, the number of slices of bread he cut = 6.

3 slices because he ate 1 slice ( 1/4) (2/4) (3/4) (4/4) so 3 slices

Find the exact values of the numbers c that satisfy the conclusion of the Mean Value Theorem for the interval [−2, 2]. (Enter your answers as a comma-separated list.)

Answers

Answer:

The answer is "\bold{c= \pm (2)/(√(3))}"

Step-by-step explanation:

If the function is:

\to f'(x) = 3x^2-2 \n\n\to f'(c) = 3c^2-2

points are:

\to  -2 \leq x \leq2

use the mean value theorem:

\to f'(c) = ( f(b)- f(a))/(b-a)

            = ( f(2)- f(-2))/(2-(-2))\n\n= (4-(-4) )/(4)\n\n= (8)/(4)\n\n= 2

\to 3c^2-2=2 \n\n\to 3c^2=4  \n\n\to c^2=(4)/(3) \n\nc= \pm (2)/(√(3))

Final answer:

The Mean Value Theorem states that for a continuous and differentiable function on a closed interval, there exists at least one 'c' within that interval where the average change rate equals the instantaneous rate at 'c'. In the given case of interval [-2,2], to find 'c', first calculate the average slope between the points (f(2)-f(-2))/4. Then equate this average slope to the derivative 'f'(c). The solution(s) to this equation are the c values for this problem.

Explanation:

The subject of this question pertains to the Mean Value Theorem in Calculus. According to this theorem, if a function f is continuous on a closed interval [a, b] and differentiable on the open interval (a, b), then there exists at least one number c in the open interval (a, b) such that the average rate of change over the interval equals the instantaneous rate of change at c.

In the given case, we're trying to find the 'c' value for the interval [-2,2]. First, we need to find the average slope between the two points. Assuming f is your function, that would be (f(2)-f(-2))/ (2 - -2). Subtract the function values of the two points and divide by the total interval length. Next, we need to see where this average slope equals the instantaneous slope 'f'(c), this entails solving the equation 'f'(c) = (f(2)-f(-2))/4. The solution to this equation will be the c values that satisfy the Mean value theorem within the provided interval.

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A rectangular garden measures 26ft by 37ft. Surrounding (and bordering) the garden is a path 2ft wide. Find the area of this path. Be sure to include the correct unit in your answer.

Answers

26*37=962sqft
Add 4 ft to both numbers (2ft per side)
30*41=1230sqft
Subtract out the garden area
1230-962=268sqft
The path is 268 sq ft