# The Bone Bender

## Our Master Piece

This beast is made up of 46 equations. X axis is time in seconds and y axis is height above ground in ft. Minimum height is 10 ft and maximum height is 300 ft. Roller coaster runs for 100 seconds.

## Rough Draft

Consists of: 3 linear functions, 7 quadratic functions, 9 cubic functions, 4 quartic functions, 3 rational functions, 9 sinusoidal functions, 6 exponential functions, 2 logarithmic functions, 1 square root function, and 2 circle equations.

## How we created our roller coaster

Our final roller coaster was a product of a rough draft, mathematical interpretation, and application. We made a rough draft of how we wanted the general shape of the coaster to look. We then analyzed the shape of each portion of the coaster and determined which type of function would best represent that portion of the coaster (i.e straight line- linear function, wave-sinusoidal function). Lastly, we applied our knowledge of the various transformations of the various functions (stretches, reflections, and translations) and restrictions (domain, range, limit x or y values to a certain range) to make the specific function fit the desired shape depicted in the rough draft. So we started off by applying the translations (left, right, up, down) on the function to get it in the general desired location. We then applied reflections to get the function in the desired direction (x-axis- to control up or down or y axis- to control right or left) (i.e if we wanted the parabola to open down, made the “a” value negative). We then applied stretches and/or compressions (horizontal and/or vertical) to the function to get a desired shape corresponding to our rough draft so each function aligned smoothly with its adjacent intersecting functions. Finally, we added restrictions to limit the functions to the desired time range in between the intersections of the adjacent functions (x values greater than the intersection to the left and x-values less than the intersection to the right).

## Difficulties we encountered

The only difficulty we encountered the during creation of the coaster was making the roller coaster smooth at the points where the functions intersected. This was done in order to make it look as if it were one continuous equation. So basically, we needed to make the instantaneous rates of change relatively equal of the two functions at the points of intersections. However, to do this, we had to play around with the magnitude of the stretch/compression factor. This is something we found challenging as the stretch or compression affected the entire function (so both its points of intersections). Hence, we had to choose a magnitude that made both intersections look smooth and that took a lot of trial and error. In a few situations where no compression/stretch factor seemed to work, in order to make the intersection points of two functions smooth, we inserted a small quadratic function in-between and made the instantaneous rate of change relatively equal at the intersection points (demonstrated by equation 4 and 5). All in all, we managed to overcome this difficulty and make the coaster flow relatively smoothly.

## Descriptions and Equations of Functions

The roller coaster is made of up numerous functions. A function is a relation for which every x value, there is only one unique y value. The numerous functions have varying rates of change to provide a joyful roller coaster ride. A tangent line represents the instantaneous vertical speed of the roller coaster and since the slope of the tangent would constantly change throughout the coaster, the riders experience acceleration inducing that thrill felling. We used all the functions learned in MHF4UO as well as previously learned functions (square root) and equations (circle).

## Linear Function:

Are those whose graph is a straight line (constant rate of change). Is a degree one polynomial (odd degree), hence end behaviours from either Q3-Q1 or Q2-Q4. A linear function is a function of the form f(x) = mx + b, where a and b are real numbers. Here, "a" is the slope of the line, and "b" is the y-axis intercept.

Equation 1: y=51x+10

• The function has a slope of 51 (m value).
• The y-intercept is (0, 10) (b value).
• Domain: x ∈ [0, 5.28246095]
• Range: y ∈ [10, 279.3]

Equation 17: y=250

• The function has a slope of zero, therefore is only a horizontal line, which would pass through all points (x, 250) if it weren’t restricted.
• Domain: x ∈ [31.939, 33]
• Range: y ∈ [250]

Equation 46: y=12

• Function has a slope of zero, thus making it a straight horizontal line, with a y-intercept of 12 if its domain was not restricted.
• Domain: x ∈ [99.676, 100]
• Range: y ∈ [12]

Is a second-degree polynomial function, (even degree), hence the end behaviours are from Q2-Q1 or Q3-Q4. Every quadratic function has a “U-shaped” graph called a parabola. A parabola either opens up or down depending on the leading coefficient. If a>0, the parabola will “open up.” If a<0 the parabola will “open down.” To make graphing easier, we used quadratic functions in the form:f(x)=a[k(x-d)]^2+c where (d,c) is the vertex. Rate of change increases as you move out from vertex.

Equation 2: y=-40(x-6)^2+300

• Reflection in the x-axis (- a value), also means the parabola opens downwards, making the vertex a maximum point.
• The function has been vertically stretched by a factor of 40 (a value).
• There is a horizontal translation to the right by 6 seconds (d value).
• A vertical translation upwards by 300 feet (c value).
• Domain: x ∈ [5.2824609, 6.9999878748]
• Range: y ∈ [261.6, 300]

Equation 4: y=20(x-11.8)^2+24

• Positive quadratic function and opens up, thus has a minimum point.
• The function has been vertically stretched by a factor of 20 (a value).
• Horizontally translated to the right by 11.8 seconds (d value).
• The function has been vertically translated upwards by 24 feet (c value).
• Domain: x ∈ [11.318, 12]
• Range: y ∈ [24, 28.6]

Equation 5: y=10.5(x-13.5)^2+12

• This is a positive quadratic function and parabola opens up, thus has a minimum point.
• The function has been vertically stretched by a factor of 10.5 (a value).
• Horizontally translated to the right by 13.5 seconds (d value).
• The function has been vertically translated upwards by 12 feet (c value).
• Domain: x ∈ [12.376, 14.093]
• Range: y ∈ [12, 25.3]

Equation 12: y=-20(x-20.55)^2+270

• Reflected in the x-axis, and so the resulting parabola opens downwards (- a value) and has a maximum point, its vertex.
• The function has been vertically stretched by a factor of 20 (a value).
• Horizontally translated to the right by 20.55 seconds (d value).
• The function has been vertically translated upwards by 270 feet (c value).
• Domain: x ∈ [20.049, 22.141]
• Range:y ∈ [219.4, 270]

Equation 13: y=22(x-25)^2+159

• This is a positive quadratic function, opens up, and has a minimum point.
• The function has been vertically stretched by a factor of 22 (a value).
• Horizontally translated to the right by 25 seconds (d value).
• The function has been vertically translated upwards by 159 feet (c value).
• Domain: x ∈ [24.7179, 25.216]
• Range: y ∈ [159, 160.7]

Equation 16: y=-20(x-31.939)^2+250

• Reflected in the x-axis, so parabola opens downwards (- a value) and has a maximum point, its vertex.
• The function has been vertically stretched by a factor of 20 (a value).
• Horizontally translated to the right by 31.939 seconds (d value).
• The function has been vertically translated upwards by 250 feet (c value).
• Domain: x ∈ [30.9999, 31.939]
• Range: y ∈ [232.36, 250]

Equation 35: y=15.7(x-71)^2+90

• Parabola opens up; vertex is the minimum point of the function.
• The function has been vertically stretched by a factor of 15.7 (a value).
• Horizontally translated to the right by 71 seconds (d value).
• The function has been vertically translated upwards by 90 feet (c value).
• Domain: x ∈ [68.996, 73.0031]
• Range: y ∈ [90, 158.2]

## Cubic Function:

Is a third degree polynomial function (odd degree), hence the end behaviours are from Q3-Q1 or Q2-Q4. The function was graphed in the form f(x)=a[k(x-d)]^3+c. Inflects around the origin as a number between -1 and 1 when cubed gets even smaller

Equation 22: y=(3(x-42.6))^3 +135

• This is a positive cubic function. A positive cubic functions runs from quadrant 3 to quadrant 1.
• The function has been horizontally compressed by a factor of 1/3 (k value).
• The function has been horizontally translated to the right by 42.6 seconds (d value).
• The function has been vertically translated upwards by 135 feet (c value).
• Domain: x ∈ [41.319, 43.999]
• Range: y ∈ [78.38, 207.51]

Equation 24: y=(-3(x-47.28))^3 +135

• This is a cubic function that has been reflected in the y-axis (- k value) so function goes from quadrant 2 to quadrant 4.
• The function has been horizontally compressed by a factor of 1/3 (k value).
• The function has been horizontally translated to the right by 47.8 seconds (d value).
• The function has been vertically translated upwards by 135 feet (c value).
• Domain: x ∈ [45.873, 48.562]
• Range: y ∈ [78.38, 210.18]

Equation 28: y=-(1.2(x-53.8))^3 +100

• Reflected in the x-axis (- a value). This means it runs from quadrant 2 to quadrant 4.
• The function has been horizontally compressed by a factor of 5/6 (k value).
• The function has been horizontally translated to the right by 53.8 seconds (d value).
• The function has been vertically translated upwards by 100 feet (c value).
• Domain: x ∈ [51.911, 55.947]
• Range: y ∈ [83.07, 111.12]

Equation 31: y=(3(x-60.7))^3 +72

• The function has been horizontally compressed by a factor of 1/3 (k value).
• The function has been horizontally translated to the right by 60.7 seconds (d value).
• The function has been vertically translated upwards by 72 feet (c value).
• Domain: x ∈ [59.904, 61.9297]
• Range: y ∈ [58.69, 121.03]

Equation 32: y=(3(x-62.8))^3 +140

• The function has been horizontally compressed by a factor of 1/3 (k value).
• The function has been horizontally translated to the right by 62.8 seconds (d value).
• The function has been vertically translated upwards by 140 feet (c value).
• Domain: x ∈ [61.9296, 64.0297]
• Range: y ∈ [122.22, 189.03]

Equation 33: y=(3(x-64.9))^3 +208

• The function has been horizontally compressed by a factor of 1/3 (k value).
• The function has been horizontally translated to the right by 64.9 seconds (d value).
• The function has been vertically translated upwards by 208 feet (c value).
• Domain: x ∈ [64.0295, 65.9995]
• Range: y ∈ [190.22, 242.97]

Equation 37: y=(-3(x-77.1))^3 +208

• Reflected in the y-axis (-k value), hence function goes from quadrant 2 to quadrant 4.
• The function has been horizontally compressed by a factor of 1/3 (k value).
• The function has been horizontally translated to the right by 77.1 seconds (d value).
• The function has been vertically translated upwards by 208 feet (c value).
• Domain: x ∈ [76.000503, 78.33]
• Range: y ∈ [157.76, 242.97]

Equation 38: y=(-3(x-79.2))^3 +140

• Reflected in the y-axis (-k value), hence goes from quadrant 2 to quadrant 4.
• The function has been horizontally compressed by a factor of 1/3 (k value).
• The function has been horizontally translated to the right by 79.2 seconds (d value).
• The function has been vertically translated upwards by 140 feet (c value).
• Domain: x ∈ [78.329, 80.43]
• Range: y ∈ [89.76, 157.8]

Equation 39: y= (-3 (x-81.3))^3 +72

• Reflected in the y-axis (-k value), so function goes from quadrant 2 to quadrant 4.
• The function has been horizontally compressed by a factor of 1/3 (k value).
• The function has been horizontally translated to the right by 81.3 seconds (d value).
• The function has been vertically translated upwards by 72 feet (c value).
• Domain: x ∈ [80.4295, 82.125]
• Range: y ∈ [57.11, 89.76]

## Quartic Function:

Is a fourth degree polynomial function (even degree) f(x)=a[k(x-d)]^4+c, hence the end behaviours are from Q2-Q1 or Q3-Q4. Function was graphed in the form Looks just like a quadratic except that it flattens more around the origin as a number between -1 and 1 when raised to the exponent 4 gets smaller than being squared.

Equation 21: y=0.48(2(x-40))^4+55

• Like a “U” shaped parabola. It has a minimum point since it opens up (+ leading coefficient). The function has been vertically compressed by a factor of 0.48. (a value).
• The function has been horizontally compressed by a factor of ½ (k value).
• The function has been horizontally translated to the right by 40 seconds (d value).
• The function has been vertically translated upwards by 55 feet (c value).
• Domain: x ∈ [38.966, 41.3195]
• Range: y ∈ [55, 77.62]

Equation 25: y=1.927(x-51)^4+11

• The function has been vertically stretched by a factor of 1.927 (a value).
• The function has been horizontally translated to the right by 51 seconds (d value).
• The function has been vertically translated upwards by 11 feet (c value).
• Domain: x ∈ [48.56, 53.6881]
• Range: y ∈ [11, 110.41]

Equation 34: y=-6.1(x-67)^4 +250

• Reflected in the x-axis, so the parabola opens down, making the vertex a maximum point and runs from quadrant 3 to quadrant 4 (- leading coefficient).
• The function has been vertically stretched by a factor of 6.1 (a value).
• The function has been horizontally translated to the right by 67 seconds (d value).
• The function has been vertically translated upwards by 250 feet (c value).
• Domain: x ∈ [65.9994,68.997]
• Range: y ∈ [154.34, 250]

Equation 36: y=-6.1(x-75 )^4 +250

• Reflected in the x-axis, so the function opens down, making it’s vertex a maximum point and goes from quadrant 3 to quadrant 4 (- leading coefficient)
• The function has been vertically stretched by a factor of 6.1 (a value).
• The function has been horizontally translated to the right by 75 seconds (d value).
• The function has been vertically translated upwards by 250 feet (c value).
• Domain: x ∈ [73.003, 76.000504]
• Range: y ∈ [154.34, 250]

## Exponential Function:

A base value is raised to the exponent of the independent variable. The function increases or decreases by the factor of the b value. The rate of change of the function increases or decreases from left to right, depending on the b value. We graphed the function in the form f(x)=ab^[k(x-d)]+c

Equation 3: y=0.6^(x-17.88567)

• Function is decaying as the b value is greater than 0 but less than 1.
• The function has a horizontal asymptote at y=0 (c value).
• The function has been horizontally translated to the right by 17.88567 seconds (d value).
• Domain: x ∈ [6.9998, 11.319]
• Range: y ∈ [28.6, 260]

Equation 6: y=3^(x-10)+15.8

• Function is growing as b value is greater than 1.
• The function has been horizontally translated to the right by 10 seconds (d value).
• The function has been shifted upwards by 15.8 feet. This also makes y=15.8 the equation of the horizontal asymptote (c value).
• Domain: x ∈ [11.999, 14.0255]
• Range: y ∈ [24.8 98.6]

Equation 7: y=3^-(x-15)+7.4

• Reflection in the y-axis, so although the b value is greater than 1, the function decays from left to right (- k value).
• The function has been horizontally translated to the right by 15 seconds (d value).
• The function has been vertically shifted upwards by 7.4 feet. This also makes y=7.4 the equation of the horizontal asymptote (c value).
• Domain: x ∈ [10.9213,12.38]
• Range: y ∈ [25.25, 95.71]

Equation 11: y=0.485^(x-28)+150

• This is an exponential function that decreases from left to right because the b value is less than 1 but greater than 0.
• The function has been horizontally translated to the right by 28 seconds (d value).
• The function has been vertically shifted upwards by 150 feet. This also makes y=150 the equation of the horizontal asymptote (c value).
• Domain: x ∈ [22.14, 24.719]
• Range: y ∈ [160.81, 219.4]

Equation 40: y=0.4^(x-86.2) +15

• This is an exponential function that decreases from left to right because the b value is less than 1 but greater than 0.
• The function has been horizontally translated to the right by 86.2 seconds (d value).
• The function has been vertically shifted upwards by 15 feet. This also makes y=15 the equation of the horizontal asymptote (c value).
• Domain: x ∈ [82.1248, 84.902]
• Range: y ∈ [18.29, 56.65]

Equation 41: y=2.5^(x-84) +16

• This is a growing exponential function that is increases from left to right as the b value is greater than 1.
• The function has been horizontally translated to the right by 84 seconds (d value).
• The function has been vertically shifted upwards by 16 feet. This also makes y=16 the equation of the horizontal asymptote (c value).
• Domain: x ∈ [84.9019, 89.5]
• Range: y ∈ [18.29, 170.41]

## Logarithmic Function:

Logarithmic functions are the inverse of exponential functions. This means instead of horizontal asymptotes, they have vertical asymptotes. For logarithmic functions, there are also no restrictions on range values. Logarithmic functions have the form: y=alog[k(x-d]+c. All the logarithmic functions we used were base 10 since that is the most commonly used base for simplicity.

Equation 10: y=50log(x-19.05)+265

• This is a positive logarithmic function that is increasing from left to right.
• The function has been vertically stretched by a factor of 50 (a value).
• The function has been horizontally translated to the right by 19.05 seconds. This also makes x=19.05 the equation of the vertical asymptote (d value).
• The function has been vertically shifted upwards by 265 feet (c value).
• Domain: x ∈ [19.134, 20.051]
• Range: y ∈ [211.21, 265]

Equation 20: y=-250log(x-34.75) +220

• Reflected in the x-axis so function decreases from left to right.
• The function has been vertically stretched by a factor of 250 (a value).
• The function has been horizontally translated to the right by 34.75 seconds. This also makes x=34.75 the equation of the vertical asymptote (d value).
• The function has been vertically shifted upwards by 220 feet (c value).
• Domain: x ∈ [35.385, 38.9666]
• Range: y ∈ [64.4, 268.5]

## Rational Function:

Is a ratio of two polynomial functions (numerator and denominator are both polynomials). The roots of the denominator are the vertical asymptotes and the c value (vertical shift) is the horizontal asymptote. We graphed our rational functions in the form f(x)= (a/x-b) + c

Equation 9: y=-500/x-21 -56.7

• This is a rational function with no x-intercepts as there is no variable in the numerator. Vertical asymptote at x=21.The function has been reflected in the x-axis (- a value).
• The function has been vertically stretched by a factor of 500 (a value).
• The function has been horizontally translated to the right by 21 seconds (d value).
• The function has been vertically translated down by 56.7 feet. Horizontal asymptote y=-56.7 (c value).
• Domain: x ∈ [14.09,19.1344]
• Range: y ∈ [15.8, 210.7]

Equation 15: y=1000/(x-37.83)^2 +18

• This is a rational function with no x-intercepts as there is no variable in the numerator. Also has a vertical asymptote at x=37.83 as that is the root of the denominator.
• The function has been vertically stretched by a factor of 1000 (a value).
• The function has been horizontally translated to the right by 37.83 seconds (d value).
• The function has been vertically translated upwards by 18 feet. Horizontal asymptote y=18 (c value).
• Domain: x ∈ [29.49, 31]
• Range: y ∈ [161.9, 232.9]

Equation 45: y=10/x-94.677 +10

• This is a rational function with no x-intercepts as there is no variable in the numerator. Vertical asymptote at x=94.677.
• The function has been vertically stretched by a factor of 10 (a value).
• The function has been horizontally translated to the right by 94.677 seconds (d value).
• The function has been vertically translated upwards by 10 feet. Horizontal asymptote y=10 (c value).
• Domain: x ∈ [94.739424, 99.676]
• Range: y ∈ [12, 168.73]

## Sinusoidal Function:

A function the repeats itself after a certain interval known as the period. We used the cosine and sine functions in our roller coaster as they have continuous domain (no vertical asymptotes like the tan functions) and realistic roller coaster shape unlike the tan function. We graphed the functions in the from f(x)= a sin(k(x-d))+c or f(x)= a cos(k(x-d))+c

Equation 8: y=75cos(x-12.496)+96

• This function has an amplitude (also vertical stretch) of 75 (a value).
• The period of the function is 2π (since k value =1)
• The function has a phase shift of 12.496 seconds to the right (d value).
• The function’s equation of axis (vertical translation up 96 feet) is at y=96 (c value).
• The maximum of the function is 171 and the minimum is 21 feet.
• Domain: x ∈ [10.9213, 14.0255]
• Range: y ∈ [96.4, 171]

Equation 14: y=-10 cos (3(x-25))+168

• This function has an amplitude (also vertical stretch) of 10 (a value).
• Reflection across x axis, therefore the cosine function starts at a minimum.
• The period of the function is 2 π /3, which is found by 2π/k.
• The function has a phase shift of 25 seconds to the right (d value).
• The function’s equation of axis (vertical translation 168 feet up) is at y=168 (c value).
• The maximum of the function is 178 and the minimum is 158 feet.
• Domain: x ∈ [25.2159, 29.5]
• Range: y ∈ [158, 178]

Equation 18: y=15cos(3(x-33))+235

• This function has an amplitude (also vertical stretch) of 15 (a value).
• The period of the function is 2 π /3, which is found by 2π/k.
• The function has a phase shift of 33 seconds to the right (d value).
• The function’s equation of axis (vertical translation up 235 feet) is at y=235 (c value).
• The maximum of the function is 250 and the minimum is 220 feet.
• Domain: x ∈ [33, 34.047]
• Range: y ∈ [220, 250]

Equation 19: y=30cos(3(x-33))+250

• This function has an amplitude (also vertical stretch) of 30 (a value).
• The period of the function is 2 π /3, which is found by 2π/k.
• The function has a phase shift of 33 seconds to the right (d value).
• The function’s equation of axis (vertical translation 250 feet up) is at y=250 (c value).
• The maximum of the function is 280 and the minimum is 220 feet.
• Domain: x ∈ [34.047, 35.3851]
• Range: y ∈ [220, 280]

Equation 23: y=20sin(5(x-44)) +209

• This function has an amplitude (also vertical stretch) of 20 (a value).
• The period of the function is 2 π /5, which is found by 2π/k.
• The function has a phase shift of 44 seconds to the right (d value).
• The function’s equation of axis (vertical translation 209 feet up) is at y=209 (c value).
• The maximum of the function is 229 and the minimum is 189 feet.
• Domain: x ∈ [43.998, 45.874]
• Range: y ∈ [189, 229]

Equation 30: y=50cos(1.5(x-57)) +75.82

• This function has an amplitude (also vertical stretch) of 50 (a value).
• The period of the function is 4 π /3, which is found by 2π/k.
• The function has a phase shift of 57 seconds to the right (d value).
• The function’s equation of axis (vertical translation 75.82 feet up) is at y=75.82 (c value).
• The maximum of the function is 125.82 and the minimum is 25.82 feet.
• Domain: x ∈ [57, 59.905]
• Range: y ∈ [25, 125.8]

Equation 42: y=20sin(3(x-89.5)) +170.4

• This function has an amplitude (also vertical stretch) of 20 (a value).
• The period of the function is 2 π /3, which is found by 2π/k.
• The function has a phase shift of 89.5 seconds to the right (d value).
• The function’s equation of axis (vertical translation 170.4 feet up) is at y=170.4 (c value).
• The maximum of the function is 190.4 and the minimum is 150.4 feet.
• Domain: x ∈ [89.499, 91.5945]
• Range: y ∈ [150.4, 190.4]

Equation 43: y=40sin(3(x-89.5)) +170.4

• This function has an amplitude (also vertical stretch) of 40 (a value).
• The period of the function is 2 π /3, which is found by 2π/k.
• The function has a phase shift of 89.5 seconds to the right (d value).
• The function’s equation of axis (vertical translation 170.4 feet up) is at y=170.4 (c value).
• The maximum of the function is 210.4 and the minimum is 130.4 feet.
• Domain: x ∈ [91.5943, 92.642]
• Range: y ∈ [170.4, 210.4]

Equation 44: y=20sin (3 (x-89.5)) +170.4

• This function has an amplitude (also vertical stretch) of 20 (a value).
• The period of the function is 2 π /3, which is found by 2π/k.
• The function has a phase shift of 89.5 seconds to the right (d value).
• The function’s equation of axis (vertical translation 170.4 feet up) is at y=170.4 (c value).
• The maximum of the function is 190.4 and the minimum is 150.4 feet.
• Domain: x ∈ [92.641, 94.7395]
• Range: y ∈ [150.4, 190.4]

## Square Root Function:

Is like half a parabola. It is only defined for x values greater or equal to 0 as you cannot square root a negative number. We graphed this function in the form f(x)=a(k(x-d))^1/2+c.

Equation 29: y=25(x-54.05787) +82.91

• This is a positive square root function and therefore is increasing from left to right.
• The function has been vertically stretched by a factor of 25 (a value).
• The function has been horizontally translated to the right by 54.05787 seconds (d value).
• The function has been vertically translated upwards by 82.91 feet (c value).
• Domain: x ∈ [x≤57.0001]
• Range: y ∈ [83.44, 125.82]

## Circle Equation:

Are not functions as one x value has more than 1 y value (fails vertical line test). We graphed the circle equation in the from [a(y-k)]^2+(x-h)^2+=r^2 for the loop portion of the graph. More of an ellipse due to the vertical stretch.

Equation 26: 1=(0.025(y-130)) ^2+(x-52.8) ^2

• A circle with a radius of 1.
• Vertically stretched by a factor of 40 (a value).
• Vertically translated 130 feet up (k value).
• Horizontally translated 52.8 seconds right (h value).
• Domain: x ∈ [51.8, 53.8]
• Range: y ∈ [111.22,170]

Equation 27: 1=(0.025(y-70)) ^2+ (x-55) ^2

• A circle with a radius of 1.
• Vertically stretched by a factor of 40 (a value).
• Vertically translated 70 feet up (k value).
• Horizontally translated 55 seconds right (h value).
• Domain: x ∈ [54,56]
• Range: y ∈ [30, 83.47]

## Height of 250 feet

Times our roller coaster reaches a height of 250 feet can be graphically determined by the x coordinate of the point of intersection of the linear line y=250 and the various functions of the roller coaster. Therefore, we let each of the functions that intersect with this line equal to 250 and solved for x to determine the time at which the function reached a height of 250 feet.

## Height of 12 feet

We let each of the functions that intersected the line y=12 equal to 12, and then solved for the x value; which corresponded to the x coordinate of the point of intersection (time when coaster is at a height of 12 feet).

## Average rate of change

AROC is equivalent to the slope of the secant line that passes through the point (10s, y ft) and (15s, y ft) for 10 to 15 seconds and (50s, y ft) and (60s, y ft) for 50 to 60 seconds. The slope of the secant line represents the average vertical speed of the roller coaster between the time intervals as it is equivalent to Δh/Δt.

## 10 to 15 seconds

The negative AROC indicates that on average, from 10 to 15 seconds, you're travelling downwards, and the low magnitude (5.90531ft/s) indicates that your vertical speed is relatively low (6.479778557 km/h).

## 50 to 60 seconds

The postive AROC indicates that on average, from 50 to 60 seconds, you're travelling upwards, and the low magnitude (4.98120 ft/s) indicates that your vertical speed is relatively low (5.46577114 km/h).

## Instantaneous Rate of Change

IROC is equivalent to the slope of the tangent line that intersects the coaster at (35s, y ft). However, to determine this slope, we need the tangent to intersect the coaster at 2 points really close to each other. Therefore, the IROC at 35 seconds is approximately equal to the slope of the tangent that intersects the points (35s, y ft) and (35.0001s, y ft) just like a secant line but really zoomed in. The slope of the tangent line represents the approximate instantaneous vertical speed of the roller coaster as it is equivalent to Δh/Δt.

## At 35 Seconds

The high magnitude positive instantaneous rate of change indicates the roller coasters height is rapidly increasing at 35 seconds.

## Loops? 0.o

Why does the roller coaster have loops?
We are aware of the fact that the x-axis represents time and that the roller coaster should not go back in time. Although, we added loops to make the roller coaster more visually appealing. After all the assignment is called "design a roller coaster" (i.e not vertical position time graph). We made sure that loops would not interfere with our AROC or IROC calculations for the times 10-15 seconds, 40-50 seconds, and 35 seconds as there would be 2 heights corresponding to one time. We this by making sure our loops did not intersect the lines x=10, x=15, x=40, x=50, x=35 because then the roller coaster would be at two heights at the same time, creating two possible AROCS and IROCS. :)