Graph of y = x⁴/(x + 1) Compared with y = x²

Grades 10-11

Question: y=x^4/(x+1) y=x^2

To find the intersection points of the two curves

y = \frac{x^4}{x+1}

and

y = x^2

, we must solve for the values of

x

where the functions are equal.

We set the two expressions for

y

equal to each other:

\begin{align*} \frac{x^4}{x+1} &= x^2 \\ x^4 &= x^2(x + 1) \quad (\text{provided } x \neq -1) \\ x^4 &= x^3 + x^2 \\ x^4 - x^3 - x^2 &= 0 \end{align*}

Factoring out the common

x^2

term:

\begin{align*} x^2(x^2 - x - 1) = 0 \end{align*}

This gives us two cases:

Case 1: $x^2 = 0 \implies x = 0$ . Substituting back into $y = x^2$ , we get $y = 0$ . This gives the intersection point $(0, 0)$ .
Case 2: $x^2 - x - 1 = 0$ . Using the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ : $\begin{align*} x &= \frac{1 \pm \sqrt{1 - 4(1)(-1)}}{2} \\ x &= \frac{1 \pm \sqrt{5}}{2} \end{align*}$ These are the values of the Golden Ratio ( $\phi \approx 1.618$ ) and its negative reciprocal ( $-1/\phi \approx -0.618$ ).

Calculating the corresponding

y

-values using

y = x^2

For $x = \frac{1 + \sqrt{5}}{2} \approx 1.618$ , $y = \left(\frac{1 + \sqrt{5}}{2}\right)^2 = \frac{1 + 2\sqrt{5} + 5}{4} = \frac{3 + \sqrt{5}}{2} \approx 2.618$ .
For $x = \frac{1 - \sqrt{5}}{2} \approx -0.618$ , $y = \left(\frac{1 - \sqrt{5}}{2}\right)^2 = \frac{1 - 2\sqrt{5} + 5}{4} = \frac{3 - \sqrt{5}}{2} \approx 0.382$ .

In the diagram:

The Blue Curve represents the rational function $y = \frac{x^4}{x+1}$ . It has a vertical asymptote at $x = -1$ (indicated by the dashed line), where the denominator becomes zero.
The Red Curve represents the standard parabola $y = x^2$ .
The three points of intersection calculated above are marked: the origin and the two points related to the Golden Ratio. Note that for large positive values of $x$ , the blue curve grows much faster (behaving like $x^3$ ), while for values very close to the asymptote, it shoots toward infinity.

Observe that between

x = -0.618

and

x = 0

, the blue curve lies below the red parabola. After

x = 1.618

, the rational function grows more steeply than the quadratic, which is expected since the degree of the numerator is higher.