Determining the Equation of a Graph

Video lesson · from 34:102 worked examples

One lesson video covers all of Polynomials & Quadratics, so it opens at 34:10 for this topic — not from the beginning.

Given the roots and at least one other point on the graph, we can establish the graph's equation.

If the roots are $x = a, x = b$ and $x = c$ , the equation is of the form $y = k(x-a)(x-b)(x-c)$ . We can find $k$ using the other given point.

If a repeated root exists, then a stationary point lies on the x-axis.

Recall that a repeated root exists when two roots, and hence two factors, are equal.

If a graph has a root $x=a$ and a repeated root at $x=b$ (where it touches the x-axis), the equation is of the form $y = k(x-a)(x-b)^2$ .

You must find $k$ . Many candidates find the factors and then stop, with no strategy at all for finding k. Substitute another point from the graph — usually the $y$ -intercept — and solve.
Signs of the roots: sign errors here are extremely common. A root at $x = -3$ gives the factor $(x+3)$ .
A touch means a repeated factor: where the curve touches the axis and turns back, that factor is squared. Treating it as a single root gives a curve of the wrong degree.
Count the degree: a cubic needs three factors in total, counting the repeat twice.

Example 1

Find the equation of the cubic shown in the diagram:

The roots are $x = -6, x = -3$ and $x = 1$ .

So the equation is of the form:

y = k(x+6)(x+3)(x-1)

We are given the y-intercept $(0, -36)$ . Substitute $x=0$ and $y=-36$ :

-36 = k(0+6)(0+3)(0-1)

-36 = k(6)(3)(-1)

-36 = -18k

k = 2

Therefore, the valid equation is:

y = 2(x+6)(x+3)(x-1)

Example 2

Repeated Roots. Find the equation of the cubic shown in the diagram:

There is a root at $x = -2$ and a repeated root (touches the x-axis) at $x = 3$ .

The equation is of the form:

y = k(x+2)(x-3)^2

We use the point $(0, 9)$ to find $k$ . Substitute $x=0, y=9$ :

9 = k(0+2)(0-3)^2

9 = k(2)(-3)^2

9 = k(2)(9)

9 = 18k

k = \frac{1}{2}

Therefore, the Valid equation is:

y = \frac{1}{2}(x+2)(x-3)^2