For each polynomial function, identify its graph from choices A–F...

Question

For each polynomial function, identify its graph from choices A–F. ƒ(x)=(x-2)^2(x-5)^2

Accepted Answer

Hello everybody. I hope you're doing all right. Today, today we're going to be looking at this question that states consider the given polynomial function and plot its graph where our function is F of X is equal to open parentheses, X minus 17, closed parentheses, squared, multiplying and open parentheses, X minus 29 closed parentheses squared. Now, when looking at a function like this, I see that we have two parentheses multiplying each other with X values in set of each parentheses. So I think that this function is in a factored form. So I think about solving for the different zeros of our graph using this information. And to do that, all we have to do is obtain the, whatever is inside of each parentheses and set that value equal to zero. So in one parentheses, we had X minus 17. So we'll set that equal to zero and we'll have the other parentheses which was X minus 29 was not equal to zero. Now, for X minus 17, we just move the 17 over from the left hand side to the right hand side by adding and we'll have X is equal to 17 and for X minus 29 we'll do the same thing. We'll move the 29 over from the left hand side to the right hand side by adding and we have X is equal to 29 or points at 17 comma zero and 29 comma zero. So those will be our zeros for our graph or where we intersect the X axis. Now, for the Y intercept, we just set our X equals to zero. So we'll have F 00 is equal to open parentheses, zero minus 17, closed parentheses, squared, multiplying an open parentheses, zero minus 29 close parentheses squared. Now, once you complete that math and you distribute everything, the value that you'll get is 243, 49 or A Y intercept at zero comma 243,049. Now, I can look at the multiplicity of my zeros. So X equals and X equals have multiplicity of one which is an odd number or actually they have a multiplicity of two because each of them is raised to the second power. So they have a multiplicity of two which is an even number. Therefore, they are tangent, our graph is tangent to those points and they will change direction, the graph will change direction. So once we get to drawing the graph, you'll see this illustrated a bit better. But basically those points will serve as tangents where our graph will touch those points and change direction. And now finally, I want to see what the both ends of my graph will look like. So the left hand side of my graph and the right hand side and to do that, I just look at my function. So I see that we have two parentheses squared, multiplying each other. So what I'll have 1/4 power because when you multiply two exponents, you add them. So I know I'll have 1/4 power that will be the greatest power in my function at the end, once everything is distributed and I know that it will apply to the X. So once I multiply the XS or one X will be squared, the other X will be squared as well. And once you multiply those, it'll have X to the fourth. So end of graph or ends of graph is determined by X to the fourth power. Therefore, ends will be up and up, both ends will be facing upwards. So now with all of this information, my zeros, my Y intercept my, the shape of the ends of the graph. And my tangents, I can draw a general idea of what my graph will look like. So I'll draw in a Y axis and an X axis and now plug in some points. So I know I had a point at 17 at X equals 17 and another point at X equals 29. Now the Y axis was very large 2, 243, those were on A Y intercept. So from X equals 17 or the point 17 comma zero will go upwards. Like we said, with our end of graph of X to the fourth power, we know that N is gonna go upwards and intersect the graph intersect the graph at 243,049 A Y of 243,049. And we know the other end of the graph from X equals 29 onwards will also be facing upwards and go infinitely up. And now we look at our multiplicity and the tangent. So we say that each X, it, each X works as a tangent. So at X equal 17, it, we don't cross the X axis. We change direction and stay in the first quadrant of this coordinate system. And same thing with X equals 29 we change direction instead of crossing the X axis and go back into our first quadrant. And we know at some point, the graph in between seven, the X equal equal 17 and X was 29. At some point, we will have to change direction again in the middle so that we can intersect both points. So once we do that our graphs should look like a rela relatively in the shape of A W where from the Y intercept to the X equals 17 point, we move downwards. Once we touch X equal 17, we change direction and move up. Then at some point we turn back downwards and touch the X axis at X equals 29 that also function in the tenant. So we once again at that point, change direction upwards and continue infinitely up. So that is a general idea of what the graph for this function should look like. So I hope that was helpful. And until next time.

For each polynomial function, identify its graph from choices A–F. ƒ(x)=(x-2)^2(x-5)^2

Key Concepts

Polynomial Functions

Factoring and Roots

Graph Behavior and Multiplicity

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