A discriminant reveals a forbidden output interval

The curve CC has equation y=2x2+43(x+λ+1)y=\frac{2x^2+4}{3(x+\lambda+1)}. It is given that CC has a vertical asymptote x=0x=0.

(i) Determine the value of λ\lambda. mark

(ii) Find the equation of the other asymptote of CC. marks

(iii) Prove, using an algebraic method, that CC cannot lie between −4a3<y<4a3-\frac{4\sqrt a}{3}<y<\frac{4\sqrt a}{3}, a<0a<0, where aa is to be determined. marks

(iv) When λ=0\lambda=0, the graph of y=2x2+43(x+λ+1)+ky=\frac{2x^2+4}{3(x+\lambda+1)}+k cuts the xx-axis at 2 distinct points where kk is a positive integer.

(a) Sketch the graph of y=2x2+43(x+λ+1)+ky=\frac{2x^2+4}{3(x+\lambda+1)}+k when λ=0\lambda=0, indicating clearly all asymptotes and the coordinates of the axial intercepts. marks

(b) Deduce the least possible integer value of kk. mark

[2009/NYJC/Prelim/I/7]

Total marks: 11

Solution

If this part uses an earlier result, revisit (ii) Oblique asymptotes or open all working together.

(iii)

Range by discriminant

Prove, using an algebraic method, that CC cannot lie between −4a3<y<4a3-\dfrac{4\sqrt a}{3}<y<\dfrac{4\sqrt a}{3}, where aa is to be determined.

Explore this idea

Hint

  1. Let yy be a value the curve takes. Then 2x2+43x=y\frac{2x^2+4}{3x}=y must have a real solution xx.

  2. Rearrange into a quadratic in xx. A real solution exists only if the discriminant is ≥0\ge0.

  3. The heights with discriminant <0<0 are the ones the curve never reaches.

Answer · (iii)

a=2a=2; excluded outputs −42/3<y<42/3-4\sqrt2/3<y<4\sqrt2/3.

Printed a<0 is inconsistent; the real square-root answer requires a>0.

Explore the idea

Open interactive exploration

Which heights does the curve never reach?

A horizontal line y = k meets the curve exactly when k is a value the curve takes.

Height of the horizontal linek = 0
Try an example
Test boundary heights

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

Number of distinct intersections
0

No intersection: ∣k∣<42/3|k|<4\sqrt2/3.

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

  • Y = (2x² + 4)/(3x)
  • Horizontal line y = k
  • Intersections: the values of x with y = k
  • Intersection selected in the coordinate table
Key idea

A height kk is attained exactly when 2x2−3kx+4=02x^2-3kx+4=0 has a real root. The discriminant 9k2−329k^2-32 is negative for ∣k∣<423|k|<\frac{4\sqrt2}{3}, the gap you can see between the two branches.

Ideas to try
  1. Find the two heights where the line just touches the curve.
  2. What is the smallest value on the right branch?