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Transcript of Graphing polynomials
We know the zeros are 4, with a multiplicity of 3, -1 with a multiplicity of 2, and 1. At (4,0) which has a multiplicity of 3, the graph will flatten. At (-1,0) the graph will bounce because it has a multiplicity of 2, and at (1,0) the graph will go through because it has a multiplicity of 1.
To find the y-intercept, you can just multiply each number together, so (1/9)(-4)^3(1)^2(-1) = about 7 Examples:
f(x)= 2x^3 g(x)=-2x ^3 If this is negative, then the graph is flipped over the x-axis If the degree is even, than the graph has the same Examples:
f(x)= 2x^2 g(x)=-2x^2 Again, if this is negative than it is flipped over the X-axis Every graph of a polynomial is continuous, meaning it is an unbroken curve without jumps, gaps, or holes, and no sharp corners If it has a leading coefficient that is greater than one, than the graph gets skinner, and if it is less than one, than it gets fatter. The y-intercept of the graph is the constant The x-intercepts of the graph are the real zeros Other stuff: Always has one and only one y-intercept (otherwise it would not be a function) example:
f(x)= (x-3)^3 (x-1) (x+1)^2 The zeros are: 3, 1 and -1
The multiplicities of each are: 3,1 and 2 Rules of multiplicity:
Multiplicity of 1, goes through the x-axis
Multiplicity that is even, bounces (tangent) at the x-axis
Multiplicity that is odd, it flattens
the higher the odd multiplicity, the more it flattens Local extrema is the local maximum or minimum, or where the graph has a peak or a valley
To find number of local extrema, take the degree n and it has at most n-1 local extrema. Points of inflection is when the concavity of a graph changes.
To find the number of points of inflection, take degree n, with n 2, has at most n-2 points of inflection
or a graph of an odd degree, with n> 2 has at least 1 point of inflection More stuff: f(x)=1/9(x-4)^3(x+1)^2(x-1) create a function for the following graph Because the infinities are different we know that the degree must be an odd number. We also know that the degree could be at least a 5, as there are 4 local extrema.
The zeros here are shown to be -2 with multiplicity 1, 0, with multiplicity 2, 4 with multiplicity 1 and 6 with multiplicity 1. Add the multiplicities together to get the degree of 5.
Right now, we know the equation is something like:
using a as the unknown coefficient.
In order to find the leading coefficient, we need to plug in another point for f(x). Here the given point is (1,1). We also know the leading coefficient must be positive because the graph starts in quadrant III and goes to I, the leading coefficient must be positive.
I got a to be 1/45, so the final equation is: # 2 Bibliography: https://www.desmos.com/calculator
The text book (1,1) f(x)=1/45(x+2)(x-4)(x-6)(x+0)^2