Determine when a horizontal level has a negative input

The curve CC has equation given by y=x−a+a2a(x−a)y=\frac{x-a+a^2}{a(x-a)}, where aa is a negative constant.

(i) Write down the equations of the asymptotes of CC. marks

(ii) Sketch the graph of CC, labelling clearly the asymptotes and axial intercepts. marks

(iii) Deduce the range of values of kk for which the equation x−a+a2a(x−a)=k\frac{x-a+a^2}{a(x-a)}=k has one negative real root. marks

(iv) State the range of values of xx for which the curve is concave downwards. mark

[2012/NJC/Promo/11(a)]

Total marks: 7

Solution

(i)

Asymptotes

The curve CC has equation y=x−a+a2a(x−a)y=\dfrac{x-a+a^2}{a(x-a)}, where aa is a negative constant. Write down the equations of the asymptotes of CC.

Hint

  1. Divide the numerator by the denominator: write y=constant+constantx−ay=\text{constant}+\frac{\text{constant}}{x-a}.

  2. The constant is the horizontal asymptote; the zero of the denominator gives the vertical one.

Answer · (i)

x=a, y=1/ax=a,\ y=1/a.

(ii)

Curve sketching

Sketch the graph of CC, labelling clearly the asymptotes and axial intercepts.

Explore this idea

Hint

  1. Intercepts: put y=0y=0 and x=0x=0.

  2. Is the curve increasing or decreasing? Look at the sign of dydx=−a(x−a)2\frac{dy}{dx}=-\frac{a}{(x-a)^2}.

Q11 source rational sketch
Q11 source rational sketchScroll across to see the full diagram.

Answer · (ii)

Intercepts (a−a2,0),(0,1/a−1)(a-a^2,0),(0,1/a-1); both branches increase.

a<0a<0; diagram is a representative negative-a sketch.

Explore the idea

A hyperbola with increasing branches

Drawn with a = −1: y = −1 − 1/(x + 1).

Switch on a feature to add it to the sketch and see how it is found. A full-mark sketch shows all of them.

  • Hyperbola C
  • X = a
  • Y = 1/a
  • Intercepts
Ideas to try
  1. Which branch is above the horizontal asymptote?
  2. Why are both branches rising?
(iii)

Roots from a graph

Deduce the range of values of kk for which the equation x−a+a2a(x−a)=k\dfrac{x-a+a^2}{a(x-a)}=k has one negative real root.

Explore this idea

Hint

  1. A root is an intersection of y=ky=k with CC. For a negative root, the intersection must have x<0x<0.

  2. Which parts of the curve have x<0x<0? Both the left branch and the part of the right branch left of the yy-axis.

  3. Read off which heights those parts reach.

Answer · (iii)

k<1/a−1k<1/a-1 or k>1/ak>1/a.

Explore the idea

Count only the roots on the left of the y-axis

Only intersections with x < 0 are counted. With a = −1, the cut-offs are k = −1 and k = −2.

Height of the horizontal linek = -1.5
Try an example
Test boundary heights

Buttons use full-precision boundary values; numbers shown here are rounded.

Number of distinct negative roots
0

No negative root. At k=−2k=-2 the only root is x=0x=0; at k=−1k=-1 the horizontal asymptote is not attained.

There are no isolated intersections at this value. A touching point counts once; hollow endpoints are excluded.

  • Hyperbola, drawn with a = −1
  • Y = k
  • Intersections with x < 0, the negative roots
  • Intersection selected in the coordinate table
Key idea

The root count changes at k=1a−1k=\frac1a-1 (the yy-intercept) and k=1ak=\frac1a (the horizontal asymptote). These are the two heights that bound the gap.

Ideas to try
  1. Find the two heights where the count changes. Which features of the curve do they match?
  2. Put k=−2k=-2: the root is exactly x=0x=0. Is it counted?
(iv)

Concavity

State the range of values of xx for which the curve is concave downwards.

Hint

  1. Concave downwards means the gradient is decreasing: d2ydx2<0\frac{d^2y}{dx^2}<0.

Answer · (iv)

x>ax>a.