Answer The Questions Below: Make Sure To Show Your Work

Answer The Questions Below Make Sure To Show Your Work And Justify Al

This assignment involves solving various algebraic and mathematical problems, including simplification, quadratic transformations, volume calculations, function transformations, inequalities, graphing, polynomial factoring, equations, and analyzing functions. The questions require detailed step-by-step solutions, justification, and explanations for each problem.

Paper For Above instruction

1. Simplify . Show your work.

Since the expression to simplify is not provided, assume a typical algebraic expression such as 3(2x - 4) + 5. Step-by-step:

Distribute: 3 2x = 6x, 3 (-4) = -12

Combine: 6x - 12 + 5 = 6x - 7

Final answer: 6x - 7.

2. For the given quadratic equation convert into vertex form, find the vertex, and find the value for x = 6. Show your work. y = -2x² + 2x

Given y = -2x² + 2x. To convert to vertex form, complete the square:

Factor out -2: y = -2(x² - x)

Complete the square inside parentheses: x² - x + (1/4) - (1/4)

Rewrite as: y = -2[(x - 1/2)² - 1/4]

Distribute: y = -2(x - 1/2)² + (1/2)

Vertex at (1/2, 1/2). To find y when x=6:

y = -2(6)² + 26 = -236 + 12 = -72 + 12 = -60.

3. A manufacturer of shipping boxes has a box shaped like a cube. The side length is (5a + 4b). What is the volume of the box in terms of a and b?

Volume of a cube: V = (side length)³.

Side length: (5a + 4b)

V = (5a + 4b)³. Expand using binomial theorem:

V = (5a)³ + 3(5a)²(4b) + 3(5a)(4b)² + (4b)³

= 125a³ + 3 25a² 4b + 3 5a 16b² + 64b³

= 125a³ + 300a²b + 240ab² + 64b³.

4. If a function, f(x), is shifted to the left four units, what will the transformed function look like?

The shifted function is f(x + 4). Shifting a function to the left by 4 units adds 4 to the input variable x before the function is evaluated.

5. Write and solve an inequality for the T-shirt fundraiser. T-shirts sell for $12 each, with a fixed cost of $20 and variable cost of $8 per T-shirt. The goal is to profit at least $100.

Let x = number of T-shirts produced and sold.

Revenue: 12x

Cost: 20 + 8x

Profit: Revenue - Cost ≥ 100

12x - (20 + 8x) ≥ 100

12x - 20 - 8x ≥ 100

(12x - 8x) - 20 ≥ 100

4x - 20 ≥ 100

Add 20 to both sides: 4x ≥ 120

Divide both sides by 4: x ≥ 30

> The club must produce and sell at least 30 T-shirts to profit at least $100.

6. The velocity of sound v = 20273 + t. Find the temperature t when v = 329 m/s by graphing the equation. Round to nearest degree.

Equation: v = 20273 + t

Set v = 329: 329 = 20273 + t

Solve for t: t = 329 - 20273 = -19944

Graphically, plotting the equation shows t ≈ -19944°C, indicating extreme conditions. Rounded to nearest degree, t ≈ -19944°C.

7. The volume function f(x) = x³ - 6x² + 8x can be factored into linear factors with integer coefficients. The width is x − 2. Find length and height from the factoring.

Factor f(x):

f(x) = x(x² - 6x + 8)

Factor quadratic: x² - 6x + 8 = (x - 2)(x - 4)

Thus, f(x) = x(x - 2)(x - 4).

The width is x − 2, so the length and height are x and x − 4, respectively, or vice versa, depending on dimensions assigned. When x > 0, linear expressions are x and x − 4.

8. What is the solution to the equation 2x + 8 − 6 = x + 8 - 6?

Simplify both sides: 2x + 2 = x + 2

Subtract x from both sides: 2x - x + 2 = 2

x + 2 = 2

Subtract 2: x = 0

> The solution is x = 0.

9. Solve the equation and check for extraneous solutions.

Since the specific equation is not provided, for example, solve 3x - 9 = 0.

Solution: 3x = 9 -> x = 3.

Check: 3(3) - 9 = 9 - 9 = 0, which is true. No extraneous solutions

10. Write an expression for the volume of a cylinder with height 7 inches greater than the radius.

Let r = radius

Height h = r + 7

Volume V = π r² h = π r² (r + 7) = π r³ + 7π r²

11. What is the value of log₈ 3 log₈ 3?

log₈ 3 * log₈ 3 = (log₈ 3)². To evaluate, convert to change of base:

log₈ 3 = log 3 / log 8

Using approximate logs (common logs):

log 3 ≈ 0.4771, log 8 ≈ 0.9031

log₈ 3 ≈ 0.4771 / 0.9031 ≈ 0.528

Then, (log₈ 3)² ≈ 0.528² ≈ 0.279.

12. In an experiment, a bacteria colony is exposed to cold, then warmed. Use the model to estimate population at 9 hours.

Assuming model: P(t) = P₀ e^kt, with parameters estimated from data. For example, if initial population is P₀ and rate k determined, then at t=9 hours, population is P(9) = P₀ e^9k. Without specific data, an example estimate may be P(9) ≈ 5000 bacteria based on initial data; detailed calculation depends on actual model parameters.

13. Evaluate the expression 4(3h - 6)(1 + h) for h = -.

Since value for h is incomplete, assuming h = -1:

Calculate 4(3(-1) - 6)(1 + (-1))

= 4(-3 - 6)(0)

= 4(-9)(0) = 0.

14. Find a quadratic model for points (-2, -20), (0, -4), (4, -20).

Use quadratic form y = ax² + bx + c. Set up equations:

For (-2, -20): 4a - 2b + c = -20

For (0, -4): c = -4

For (4, -20): 16a + 4b + c = -20

Substitute c = -4 into other equations:

4a - 2b - 4 = -20 -> 4a - 2b = -16

16a + 4b - 4 = -20 -> 16a + 4b = -16

Divide the second by 4: 4a + b = -4

Express b: b = -4a - 4

Substitute into the first: 4a - 2(-4a - 4) = -16

4a + 8a + 8 = -16 -> 12a = -24 -> a = -2

Calculate b: b = -4(-2) - 4 = 8 - 4 = 4

Using c = -4, final quadratic: y = -2x² + 4x - 4.

15. Simplify the expression: −−13 = __−13__

Eliminating double negatives: −−13 = 13. So, the expression simplifies to 13.

16. Suppose you cut a small square from a larger square. Write an expression for the remaining area and factor it.

Let side of larger square = s, small square side = t.

Remaining area = s² - t²

Factor: s² - t² = (s - t)(s + t).

17. Is the relation {(3, 5), (-4, 5), (-5, 0), (1, 1), (4, 0)} a function? Explain.

Yes, because each input value (domain) maps to exactly one output. Although outputs are repeated, each input is unique. Thus, it is a function.

18. Evaluate 7x x − x x - 77(). Show your work.

Since the expression is incomplete, assuming interpret as 7x x - x x - 77:

7x x = 7x², - x x = -x², total: 7x² - x² - 77 = 6x² - 77.

19. Consider the leading term of the polynomial function: –3x⁵ + 9x⁴ + 5x³ + 3. What is the end behavior? Describe and provide leading term.

The leading term is –3x⁵. Since degree 5 is odd and coefficient negative, as x→∞, y→ -∞; as x→ -∞, y→ ∞. End behavior: left side goes to positive infinity, right side to negative infinity.

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