For this task you will be analyzing and sketching a Rational Function which will be created using the unique digits of your student number. Student number is 716818. Follow this set of instructions to create your Rational Function:
i. Take the first three digits of your student number and use them (in order) as the coefficients of a quadratic expression of the form 𝒂𝒙𝟐 + 𝒃𝒙 + 𝒄. For example, if your student number is 543210, the quadratic expression will look like 5𝑥2 + 4𝑥 + 3. You may decide to make any of the coefficients negative if you like. For example, the above expression could be changed to −5𝑥2 − 4𝑥 + 3. The resulting expression will be the numerator of your Rational Function.
ii. Take the last three digits of your student number and use them (in order) as the coefficients of a quadratic expression of the form 𝒅𝒙𝟐 + 𝒆𝒙 + 𝒇. For example, if your student number is 543210, then this quadratic expression will look like 2𝑥2 + 1𝑥 + 0, or 2𝑥2 + 𝑥. You may decide to make any of the coefficients negative if you like. The resulting expression will be the denominator of the Rational Function.
iii. Take the numerator you created in (i) and the denominator you created in (ii) and put them together to make your Rational Function, which you will call 𝑓(𝑥). For the above example, the resulting function
would be 𝑓(𝑥) = −5𝑥2−4𝑥+3. If your student number has some 0 digits in it, you may have missing terms. 2𝑥2+𝑥
This is not a problem, and you should be able to proceed through the remaining tasks normally. You should now have a unique Rational Function.
Use this Rational Function and complete a full analysis according to the listed instructions:
Write the equation of the Rational Function you created according to the instructions listed above.
State the Domain of your function.
Find and state the y-intercept of your function (if it has one). If there is none, explain why not.
Determine any discontinuities of your function and classify them (hole discontinuity, infinite discontinuity, jump discontinuity, etc.). If you have a hole discontinuity at 𝑥 = 𝑎 then evaluate lim 𝑓(𝑥) to determine
𝑥→𝑎
the co-ordinates. If your function has no discontinuities, make sure to fully demonstrate why not.
Find and state the x-intercepts of your function. If it has no x-intercepts, make sure to fully demonstrate why not.
Use lim 𝑓(𝑥) and lim 𝑓(𝑥) to determine whether there is a horizontal asymptote. If there is a
𝑥→+∞ 𝑥→−∞
horizontal asymptote, then determine whether your function approaches from above or below as 𝑥 → +∞ and as 𝑥 → −∞. If your function has an Oblique Asymptote instead, then determine the equation of the Oblique Asymptote.
Determine 𝑓′(𝑥) for your function and simplify the equation as much as possible. Make sure your derivative is expressed with positive exponents.
Determine all critical numbers for your function. If it has no critical numbers, make sure to fully demonstrate why not. If your equation cannot be solved algebraically, use technology to solve and make sure to indicate why it was necessary to do so.
Determine the intervals of increase and decrease for your function and state the co-ordinates of any local maxima or minima.
10. Determine 𝑓′′(𝑥) for your function and simplify the equation as much as possible. Make sure your derivative is expressed with positive exponents.
11. Determine all x-values where 𝑓′′(𝑥) = 0 or where 𝑓′′(𝑥) is undefined. If your equation cannot be solved algebraically, use technology to solve and make sure to indicate why it was necessary to do so.
12. Determine the intervals of concavity for your function and state the co-ordinates of any points of inflection.
At this point you should have all the information you need to sketch a graph of your function. Make your sketch and include as much detail as possible.
Important Check Points
Required
Met Approaching
Not Met
Correctly determined the equation of the function to be sketched
(Q1)
Determined the Domain of the function
(Q2)
Determined and classified any discontinuities on the function (or justified why there are none)
(Q4)
Determined the x-intercepts of the function (or justified why there are none)
(Q5)
Determined the horizontal (or oblique) asymptote and how
the function approaches the horizontal asymptote.
(Q6)
Determined the first derivative of the function
(Q7)
Determined the critical numbers of the function (or justified why there are none)
(Q8)
Determined the intervals of increase and decrease (Q9)
Determined the co-ordinates of any local maxima and minima (or justified why there are none)
(Q9)
Determined the second derivative of the function (Q10)
Determined where 𝑓′′(𝑥) = 0 and where 𝑓′′(𝑥) is undefined (Q11)
Determined the intervals of concavity (Q12)
Determined the co-ordinates of the points of inflection (or justified why there are none)
(Q12)
Sketch of graph is neat, properly labelled, and all necessary points are shown
3. Determined the y-intercept of the function (or justified why there is none)
(Q3)
