I need some help please, I have an assignment that evaluates my proficiency in analyzing linear and quadratic functions, including their domains and ranges, behavior, maxima, and minima. It also gauges the ability to interpret function graphs, as well as comprehension of whether functions are increasing or decreasing.

There is 2 tasks in this assignment, the first task involves questions about quadratic functions. These functions are essential for representing the movements of objects in space, such as throwing a ball up or down, projectile motion, and curvilinear movements. Based on the readings provided, answer the questions related to quadratic functions. The second task involves interpreting linear functions to connect different locations through the roads. Task 1.

Imagine a scenario involving a bungee jumper leaping from a bridge, with the jumper's height above the river surface modeled by the equation h(t) = -0.5t² + v₀t + h₀ , where h is measured in meters, t is in seconds, v₀ represents the jumper's initial velocity in meters per second, and h₀ is the initial height above the river. Given v₀ = 0 m/sec and h₀ = 210 meters.

Reference: Stitz, C., & Zeager, J. (2013). College algebra.Stitz Zeager Open Source Mathematics. https://stitz-zeager.com/szca07042013.pdf

*Reading section 2.1, pages 151-172

(i) Based on this scenario, answer the following questions that are related to the mathematical understanding of the concept:

(a) What is the domain and range of h(t)? What is the physical significance of domain and range in this scenario?

(b) What is the vertex of the given height function, h(t) = -0.5t² + v₀t + h₀? What does the vertex represent in this scenario?

(c) At what time does the bungee jumper reach maximum height and what is the maximum height? Explain using the formula and scenario.

(d) At what time does the bungee jumper reach the height of 11m?

(e) What is the height after 20 seconds of the jump, and what does this situation represent?

(f) When will the bungee jumper touch the river?

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(ii) Based on that scenario, answer the following questions that related to the graphical understanding of the concept:

(a) Draw the graph of the given height function, h(t) = -0.5t² + v₀t + h₀ (use https://www.geogebra.org/calculator) in graphing.

(b) By observing the graph, determine the time intervals in which the height is increasing or decreasing.

(c) Find the axis of symmetry on the graph and explain the above scenario using the axis of symmetry.

(d) What are the t and h intercepts on the axes here? What do they represent in this scenario?

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Task 2.

Imagine you are a city planner working on improving transportation routes in a bustling metropolis. Your task is solving several geometric and graphical challenges related to a proposed road project connecting linearly to different locations. Solving these challenges contributes to an efficient urban transportation network, benefiting residents and visitors alike.

(i) Optimal Route Planning: Determine the equation of the road that seamlessly connects critical locations, Points A(5,7) and B(6,5).

(ii) Traffic Flow Analysis: Calculate the road's slope between A and B for efficient traffic design.

(iii) Enhanced Traffic Safety: Determine the changes in road elevation from A to B, focusing on safety and convenience.

(iv) Alternate Routes Provision: Kindly make a parallel and perpendicular routes to offer commuters diverse travel options.

(v) Visual Infrastructure Mapping: Can you make a graphical map of the road alignment for use in planning and stakeholder presentations.

(vi) Access Points Identification: Locate intercepts on the x and y-axes, serving as vital road access and landmarks.

(vii) In this situation how many parallel and perpendicular to the proposed road are possible?

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Task 3.

Globalpetrolprices.com is a website that compares utility bills for households and businesses. It has been observed that Denmark, German and Italy have the highest household electricity costs. Suppose Italy's electricity pricing is as follows: Each household bears a minimum of 50$ fixed charge and an additional 0.78$ for each unit of electricity consumed by the household.

Reference: Yoshiwara, K. (2020). Modeling, functions and graphs. American Institute of Mathematics. https://yoshiwarabooks.org/mfg/MFG.html

*Reading section1.5 Linear Functions

Stitz, C., & Zeager, J. (2013). College algebra.Stitz Zeager Open Source Mathematics. https://stitz-zeager.com/szca07042013.pdf

*Reading section 2.1 Linear Functions pages 151-162

Based on the above scenario, please answer the following instructions:

(i) Formulate a linear function for the electricity pricing based on the consumption. Please clearly define each variable.

(ii) How does the average rate of change in electricity price with consumption impact a consumer's monthly bill? Answer by calculating the average rate of change.

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References:

https://stitz-zeager.com/szca07042013.pdf

https://yoshiwarabooks.org/mfg/MFG.html

https://yoshiwarabooks.org/mfg/linear-functions.html

https://my.uopeople.edu/pluginfile.php/1850606/mod_assign/intro/Stitz%20Zeager.pdf

https://www.geogebra.org/calculator