Find quadratic equations whose graphs are similar to the shape of the given graph. Then multiply these quadratic equations. You may need to multiply the resulting equation by a constant so that its y-intercept is the same as the given graph.
y=1/8(x^2-4)^2
Practice makes perfect
We are given the following graph.
We can highlight the following features about the graph.
Zeros
y-intercept
Near to x=-2
Near to x=2
-2 and 2
2
Resembles a parabola
Resembles a parabola
Since the graph resembles a parabola with vertex ( -2, 0) near x=-2, let's begin by finding its quadratic equation.
y=(x-h)^2+k ⇒ & y = (x-( -2))^2 + 0
& y = (x+2)^2
Near x=2, the graph resembles a parabola with vertex ( 2, 0). Let's find its equation.
y=(x-h)^2+k ⇒ & y = (x- 2)^2 + 0
& y = (x-2)^2
Next, we multiply these two equations to get a quartic equation.
y = (x+2)^2(x-2)^2
Let's simplify the latter equation.
The graph of the function y=(x^2-4)^2 looks as follows.
We can see that it is similar (in shape) to the given graph, but the y-intercept is different. Multiply the equation above by a constant k so that the y-intercept is 2.
y = k(x^2-4)^2
Let's substitute x= 0 and y= 2 and solve the resulting equation for k.
Consequently, the equation we are interested in is the one shown below.
y = 1/8(x^2-4)^2
We can check that this is correct by comparing its graph with the given one.
