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4 years ago

Which graph has the same end behavior as the graph of f(x) = –3x^3 – x^2 + 1?

Mathematics

1 answer:

ololo11 [35]4 years ago

4 0

To find out end behavior, we always look at the largest exponent (3 in this case) and the sign of the term with the largest exponent (negative in this case). We can rule out the first two choices because since it is a cubic graph (exponent of 3) the graph will be pointing in opposite directions. Then, the negative tells us that as the x values approache infinity, the y values will approach negative infinity. So the correct answer is the last choice.

Hope this helps

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(a-b)^2/(1/a-1/b) simplify

leva [86]

Final result :

(b - a) • (a2 + ab + b2)

————————————————————————

a2b3

Step by step solution :

Step 1 :

Simplify —

Equation at the end of step 1 :

1 1 1

————-———— ÷ (—•b)

(a2) (b2) a

Step 2 :

Simplify ——

Equation at the end of step 2 :

1 1 b

———— - —— ÷ —

(a2) b2 a

Step 3 :

1 b

Divide —— by —

b2 a

3.1 Dividing fractions

To divide fractions, write the divison as multiplication by the reciprocal of the divisor :

1 b 1 a

—— ÷ — = —— • —

b2 a b2 b

Multiplying exponential expressions :

3.2 b2 multiplied by b1 = b(2 + 1) = b3

Equation at the end of step 3 :

1 a

———— - ——

(a2) b3

Step 4 :

Simplify ——

Equation at the end of step 4 :

1 a

—— - ——

a2 b3

Step 5 :

Calculating the Least Common Multiple :

5.1 Find the Least Common Multiple

The left denominator is : a2

The right denominator is : b3

Number of times each Algebraic Factor

appears in the factorization of:

Algebraic

Factor Left

Denominator Right

Denominator L.C.M = Max

{Left,Right}

a 2 0 2

b 0 3 3

Least Common Multiple:

a2b3

Calculating Multipliers :

5.2 Calculate multipliers for the two fractions

Denote the Least Common Multiple by L.C.M

Denote the Left Multiplier by Left_M

Denote the Right Multiplier by Right_M

Denote the Left Deniminator by L_Deno

Denote the Right Multiplier by R_Deno

Left_M = L.C.M / L_Deno = b3

Right_M = L.C.M / R_Deno = a2

Making Equivalent Fractions :

5.3 Rewrite the two fractions into equivalent fractions

Two fractions are called equivalent if they have the same numeric value.

For example : 1/2 and 2/4 are equivalent, y/(y+1)2 and (y2+y)/(y+1)3 are equivalent as well.

To calculate equivalent fraction , multiply the Numerator of each fraction, by its respective Multiplier.

L. Mult. • L. Num. b3

—————————————————— = ————

L.C.M a2b3

R. Mult. • R. Num. a • a2

—————————————————— = ——————

L.C.M a2b3

Adding fractions that have a common denominator :

5.4 Adding up the two equivalent fractions

Add the two equivalent fractions which now have a common denominator

Combine the numerators together, put the sum or difference over the common denominator then reduce to lowest terms if possible:

b3 - (a • a2) b3 - a3

————————————— = ———————

a2b3 a2b3

Trying to factor as a Difference of Cubes:

5.5 Factoring: b3 - a3

Theory : A difference of two perfect cubes, a3 - b3 can be factored into

(a-b) • (a2 +ab +b2)

Proof : (a-b)•(a2+ab+b2) =

a3+a2b+ab2-ba2-b2a-b3 =

a3+(a2b-ba2)+(ab2-b2a)-b3 =

a3+0+0+b3 =

a3+b3

Check : b3 is the cube of b1

Check : a3 is the cube of a1

Factorization is :

(b - a) • (b2 + ab + a2)

Trying to factor a multi variable polynomial :

5.6 Factoring b2 + ab + a2

Try to factor this multi-variable trinomial using trial and error

Factorization fails

Final result :

(b - a) • (a2 + ab + b2)

————————————————————————

a2b3

4 0

4 years ago

Can we prove that △PQR and △PQS are congruent using the side-side-side (SSS), side-angle-side (SAS), or angle-side-angle (ASA) c

7nadin3 [17]

Here are the properties you should look out for. hope it's easier to understand! :)

4 0

4 years ago

An analyst for a new company used the first three years of revenue data to project future revenue for the company. The analyst p

Westkost [7]

Using limits, it is found that since $\lim_{x \rightarrow \infty} f(x) < 0$ , the company is expected to operate at a loss, hence it is not expected to be successful.

The revenue function is given by:

$f(x) = -2x^5 + 6x^4 - x^3 + 5x^2 + 6x + 50$

<h3>Limit:</h3>

The projected revenue over the long-term is given by the limit of f(x) as x goes to infinity.

Then:

$\lim_{x \rightarrow \infty} f(x) = \lim_{x \rightarrow \infty} -2x^5 + 6x^4 - x^3 + 5x^2 + 6x + 50 = \lim_{x \rightarrow \infty} -2x^5 = -2(\infty)^5 = -\infty$