functions roller coaster project\nobjective: to create a continuous roller coaster using functions.\nyour…
functions roller coaster project\nobjective: to create a continuous roller coaster using functions.\nyour job is to create a roller coaster. your roller coaster must be made by graphing functions on a coordinate plane. here are the requirements you need to follow.\n1) you must use at least 5 different functions from the following list; linear, quadratic, cubic, absolute value, square root, cube root, exponential, log, greatest integer function, power and rational functions. the graphs of the functions must touch.\n2) you must have at least 10 equations to form your logo/roller coaster.\n3) you must have a list of equations you used to create your picture and turned in either together or separately from the picture. any equations you only used parts of must include the limits using inequalities. example: y = x; -3 < x < 1.\n4) you cannot have any lines or mark (other than the coordinate plane) that are not written as equations.\n5) you also must include a written description of your roller coaster such as its domain, range, x - intercept, y - intercept, maximum, minimum, positive, negative, increasing, decreasing, and the end behavior.\nyou will use www.desmos.com which is an online program to help you figure out how to do your picture. it will let you move your functions around using transformation until you have your roller coaster just the way you want it. when you are done, you can print the picture. be sure to write your functions on your picture.
Answer
Explanation:
Step1: Select 5 different functions
Let's choose linear ($y = x$), quadratic ($y=x^{2}$), cubic ($y = x^{3}$), absolute - value ($y=\vert x\vert$), and square - root ($y = \sqrt{x}$) functions.
Step2: Create 10 equations
We can modify the basic functions with transformations. For example, for the linear function: $y=x + 2$, $y=2x-1$; for the quadratic function: $y=(x - 1)^{2}$, $y = 2x^{2}+3$; for the cubic function: $y=(x + 2)^{3}$, $y=-x^{3}+1$; for the absolute - value function: $y=\vert x-3\vert$, $y = 2\vert x\vert$; for the square - root function: $y=\sqrt{x + 1}$, $y=3\sqrt{x}-2$.
Step3: Determine domain and range for each function
For $y = x$, domain is $(-\infty,\infty)$ and range is $(-\infty,\infty)$. For $y=x^{2}$, domain is $(-\infty,\infty)$ and range is $[0,\infty)$. For $y = x^{3}$, domain is $(-\infty,\infty)$ and range is $(-\infty,\infty)$. For $y=\vert x\vert$, domain is $(-\infty,\infty)$ and range is $[0,\infty)$. For $y=\sqrt{x}$, domain is $[0,\infty)$ and range is $[0,\infty)$. For transformed functions, adjust domain and range according to the transformation. For example, for $y=(x - 1)^{2}$, domain is $(-\infty,\infty)$ and range is $[0,\infty)$.
Step4: Find intercepts, maxima, minima, etc.
For $y = x$, $x$-intercept and $y$-intercept are both 0, it is increasing on $(-\infty,\infty)$. For $y=x^{2}$, $x$-intercept is 0, $y$-intercept is 0, minimum is at $(0,0)$, it is decreasing on $(-\infty,0)$ and increasing on $(0,\infty)$. For $y = x^{3}$, $x$-intercept and $y$-intercept are both 0, it is increasing on $(-\infty,\infty)$. For $y=\vert x\vert$, $x$-intercept and $y$-intercept are both 0, minimum is at $(0,0)$, it is decreasing on $(-\infty,0)$ and increasing on $(0,\infty)$. For $y=\sqrt{x}$, $x$-intercept and $y$-intercept are both 0, it is increasing on $[0,\infty)$.
Step5: Write the description
Describe the overall shape of the roller - coaster formed by these functions, mention how the functions connect, and summarize the key characteristics (domain, range, intercepts, maxima, minima, increasing and decreasing intervals, end - behavior) of the combined functions.
Answer:
The roller - coaster is formed by a combination of linear ($y = x,y=x + 2,y=2x-1$), quadratic ($y=x^{2},y=(x - 1)^{2},y = 2x^{2}+3$), cubic ($y = x^{3},y=(x + 2)^{3},y=-x^{3}+1$), absolute - value ($y=\vert x\vert,y=\vert x-3\vert,y = 2\vert x\vert$), and square - root ($y=\sqrt{x},y=\sqrt{x + 1},y=3\sqrt{x}-2$) functions. The domain and range, intercepts, maxima, minima, increasing and decreasing intervals, and end - behavior are determined as described above. A written description of the roller - coaster is provided summarizing these characteristics.