Summary
Highlights
The video introduces graphing polynomials using end behavior, zeros, and multiplicities. The first example, y = 2(x-2)(x+3)(x-4), is already in factored form. The first step is to find the zeros by setting each factor to zero, resulting in x=2, x=-3, and x=4. These are the x-intercepts where the y-coordinate is zero. The leading coefficient is positive (2), indicating the graph goes up to the right. The degree is odd (x*x*x = x^3), meaning the end behavior to the left is opposite, so it goes down to the left. Since all zeros have a multiplicity of 1, the graph will pass straight through each x-intercept like a line. The video also shows how to find the y-intercept by setting x=0 to get a more precise sketch.
The second example is y = -x^3 + 3x^2. First, factor out -x^2, resulting in -x^2(x-3). Setting the factors to zero gives x=0 (multiplicity of 2) and x=3 (multiplicity of 1). The leading coefficient is negative (-1), so the graph goes down to the right. The highest power (degree) is 3, which is odd, so the left end behavior is opposite, meaning it goes up to the left. At x=0, with a multiplicity of 2, the graph will have a parabolic bounce shape. At x=3, with a multiplicity of 1, the graph will pass straight through.
The third example is y = (x+1)^1 (x-3)^2 (x+2)^3. The zeros are x=-1 (multiplicity 1), x=3 (multiplicity 2), and x=-2 (multiplicity 3). The leading coefficient is positive. To find the degree, add the exponents of the factors: 1+2+3 = 6. Since the degree is 6 (even), both ends will have the same behavior, so it goes up to both the right and the left. At x=-2, with a multiplicity of 3, the graph will have a cubic shape. At x=-1, with a multiplicity of 1, the graph passes straight through. At x=3, with a multiplicity of 2, the graph bounces like a parabola. The video notes that for more precision, one could plot additional points between x-intercepts.