For each polynomial function, identify its graph from choices ...

Question

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

$ƒ (x) = (x - 2) (x - 5)$

Six graphs labeled A to F show different polynomial curves with x-intercepts at 2 and 5 for the function f(x) = (x-2)(x-5).

Accepted Answer

Hello everybody. Everything. All right. Today, today we're going to be looking at this math question that states consider the given polynomial function and plot its graph where the function is F of X is equal to open parentheses. X minus 17, closed parentheses, multiplying and open parentheses. X minus 29 closed parentheses. Now, when looking at this question, the first thing I can think of is finding out zeros for our graph. So I see that they gave us two parentheses with an X inside of them. It's kind of like a factored form. So I can go ahead and solve four zeros and to do this, we just have to grab whatever is inside of that parentheses and set each of them equal to zero. So we'll have an X minus 17, equal to zero and we'll also have an X minus 29 equals zero. So if we have X minus 17 equals zero, we know that X will be positive 17. We moved at 17 from the left hand side to the right hand side by adding it. And we do the same thing with the X minus 29. So we move the 29 from the left hand side to the right hand side by adding it, which means we'll have X is equal to four points at 17 comma zero and 29 comma zero. Those will be R zeros or where we intersect the X axis. No, I'm going to approach solving for a Y intercept. So to do that, I just plug in a zero for our axes. So we'll have F of zero is equal to zero minus 17 in the first parentheses, multiplying zero minus 29 inside of the second parentheses, once we distribute that we'll have that will be equal to 493 or eight points at zero, comma 400 93. So that'll be your Y intercept. Now, I can think of the multiplicity of my zeros and how that's going to affect our graph. So X equals and X equals have a multiplicity of one which is an odd number, which means that or therefore we will cross the X axis at those points. So that is what a multiplicity of one or a multiplicity of an odd number means it means you will be crossing the X axis. And finally, I'm going to, I want to find out the end points of my graph. So the left side of my graph on the right side of my graph, I wanna know what those would look like. And like I mentioned before our function is factored. So if we distribute everything, we will have a term where we multiply the X of the first parentheses and the X of the second parentheses leading to an X squared. So the end of our graph will be determined by X A positive X squared, which is a problem which means that our graph will be facing up and up on both ends because it's a parameter and a parameter, the left hand side is up facing up and the right hand side is also facing up. So now I have the zeroes of my graph, the Y intercept the multiplicity which will tell me that I'm crossing the X axis and I know what each end of my graph will look like. So with that information, I can draw a general idea of what my graph is. So we draw a Y intercept and then or a Y axis and an X axis and we had zeros at X equals and another zero at X equals 29. And like we said, the ends of our graph will be up up. So from the 17, the 170.17 comma zero, we will be going upwards and touch the Y axis at around 493 which will be way up in our graph, which is our Y intercept. So R Y 710 comma 4 93. And then from 29 the X at 29 or the zero at 29 we also extend upwards and now we're missing a space between the x of 17 or the point at 17 comma zero and the point at 29 comma zero. So we're missing this what happens to graph between those two points. And that's where we turn to our multi turn to our multiplicity. We know that we're going to be crossing the X axis. So that's exactly what we do. And then at some point, our graph is gonna have to turn like a parabola and go upwards again to intersect at the 29 comma zero and go upwards again, cross the X axis once more and go infinitely upwards. So in the end, we'll have a parabola facing upwards with points at X at 29 comma zero 17 comma zero A Y intercept of zero comma 493. And we cross the X axis in between 17 comma zero and 29 comma zero. So I hope that was helpful. And until next time.

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

Key Concepts

Polynomial Functions

Roots or Zeros of a Polynomial

Graphing Polynomial Functions

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