Lecture: Polynomial & Rational Functions
Lecture script · about 15–20 minutes spoken · Week 10
Opening
Picture a small print shop that makes custom tote bags. Setting up the press costs 200 dollars, and each bag then needs 5 dollars of material. Make 10 bags and each one costs 25 dollars on average. Make 200 and the average drops to 6 dollars. But it never drops below 5, because every bag still needs its material. A shop owner who knows that floor can plan a price. A nursing student watching a medicine level climb and then settle is reading the same kind of graph.
Here's where we are in this course. In Week 3 you learned that a rational function's domain leaves out any input that makes the denominator zero. In Week 6 you factored polynomials. In Week 7 you solved x squared minus 6x plus 5 equals 0 and got 1 and 5. In Week 9 you graphed that same parabola and saw both of its ends rise. In this lecture you learn to read a graph's shape from its formula, for polynomials of any degree and for one polynomial divided by another. Next week, in Week 11, you simplify and solve those fractions.
I'll take eight ideas in order: the leading term, the four end-behavior cases, zeros, multiplicity, sketching, vertical asymptotes and holes, horizontal asymptotes, and what a horizontal asymptote means in a setting. Our running example is the week's parabola, f of x equals x squared minus 6x plus 5.
1. Read the far ends from the leading term
Think about a road trip seen from a plane. Up close, the road bends around every hill. From high above, you only see which way it heads. A graph works the same way. Far from the middle, the small bends stop mattering, and only the overall direction is left.
So we need words for that direction. A polynomial function is a function whose rule is a polynomial, such as x squared minus 6x plus 5; it accepts every real number as an input. The leading term is the term with the highest power of the variable; far from the origin it outweighs all the other terms. And end behavior is the direction a graph heads at its far left and far right, described as rising or falling on each end. For a polynomial, the leading term alone decides it.
Let's read the week's parabola. The highest power is x squared, so the leading term is x squared. Its exponent, 2, is the degree. No number is written in front, so the leading coefficient is 1. The degree is even, so both ends point the same way. The coefficient is positive, so both ends rise. The table agrees. At x equals 0 the value is 5, and at x equals 6 it's 5 again, both higher than the minus 4 at the bottom.
Pause and try this. In minus 2x to the fourth plus 3x squared minus 1, which term is the leading term?
Find the highest power before you answer.
Here's the answer. The leading term is minus 2x to the fourth, because 4 is the highest power, even though the 3x squared term looks friendlier.
That friendly-looking term is a trap. It's positive, and it tempts you to say both ends rise. But it doesn't decide anything. The leading term does.
2. Sort end behavior into four cases
Think of a person's two arms. With an even degree, both arms point the same way: both up, like cheering, or both down, like a shrug. With an odd degree, the arms point opposite ways, one down on the left and one up on the right, or the reverse.
The sign of the leading coefficient picks the direction. Positive and even: both ends rise. Negative and even: both ends fall. Positive and odd: the graph falls on the left and rises on the right. Negative and odd: it rises on the left and falls on the right. That's the whole rule, four cases.
Take minus 2x to the fourth plus 3x squared minus 1. The degree, 4, is even. The coefficient, minus 2, is negative. So both ends fall. Test it far out at x equals 10. You get minus 20000, plus 300, minus 1, which is minus 19701. That's far below the axis, as promised.
Now take x cubed minus 5x plus 2. The degree, 3, is odd, and the coefficient, 1, is positive. So the graph falls on the left and rises on the right. Here's why the other terms don't matter far out. At x equals 10, x cubed is 1000, while minus 5x plus 2 is only minus 48. The whole thing is 952. The 1000 wins easily.
Pause and try this. Does x cubed minus 5x plus 2 head up or down at its far left?
Think odd degree, positive coefficient.
Here's the answer. It heads down, because an odd degree sends the ends opposite ways and a positive coefficient puts the rising end on the right; at x equals minus 10 the value is minus 948.
3. Find the zeros from factored form
Picture a hiking trail that dips below sea level and climbs back up. The spots where it crosses sea level are the interesting ones. On a graph, sea level is the horizontal axis, and those crossing spots are where the output is zero.
Here's the term. A zero of a function is an input that makes the output of a function equal 0; each real zero shows up on the graph as an x-intercept.
You already know how to find zeros, because you did it in Week 7. Factor the week's parabola. You need two numbers with product 5 and sum minus 6, which are minus 1 and minus 5. So x squared minus 6x plus 5 equals x minus 1 times x minus 5. Set each factor to zero. x minus 1 equals zero gives x equals 1. x minus 5 equals zero gives x equals 5. Check 1: 1 minus 6 plus 5 is zero. Check 5: 25 minus 30 plus 5 is zero. The graph meets the horizontal axis at 1 and at 5.
Pause and try this. The factor x plus 4 appears in a polynomial. Is the zero 4 or minus 4?
Set the factor equal to zero.
Here's the answer. The zero is minus 4, because x plus 4 equals zero only when x is minus 4.
That sign flip catches a lot of people. The number written inside the factor has the opposite sign of the zero.
4. Use multiplicity: cross or touch
Picture a skateboarder on a ramp that dips down to ground level. One path rolls straight through the low spot and keeps going. Another touches the ground and swoops back up the same side. Zeros come in those two kinds.
The term that sorts them is multiplicity, the number of times a factor appears in a factored polynomial, which is the exponent on that factor. At a zero of odd multiplicity, the graph crosses the axis, and the output changes sign. At a zero of even multiplicity, the graph touches the axis and turns back, and the output keeps its sign.
In the week's parabola, each factor appears once. One is odd, so the graph crosses at 1 and at 5. The table shows it: 5 at x equals 0, then minus 3 at x equals 2. The sign changes.
Now take x minus 1, squared, times x plus 4. The zero 1 has multiplicity 2, and the zero minus 4 has multiplicity 1. Test both sides of 1. At x equals 0 the output is 1 times 4, which is 4. At x equals 2 it's 1 times 6, which is 6. Both positive, so the graph touches at 1 and turns back. Test both sides of minus 4. At minus 5 the output is 36 times minus 1, which is minus 36. At minus 2 it's 9 times 2, which is 18. The sign changes, so the graph crosses at minus 4.
Pause and try this. Does the graph of x minus 1, squared, times x plus 4 cross the axis at x equals 1, or touch it?
Check the exponent on that factor.
Here's the answer. It touches and turns back, because the exponent 2 is even, and the outputs 4 and 6 on either side are both positive.
5. Sketch a polynomial from its features
Think about giving a friend directions: where to start, where to turn, and where to end up. A sketch needs the same three things. The ends tell you where to start and finish. The zeros tell you where to meet the axis. The y-intercept gives you one more landmark.
Let's sketch x minus 1, squared, times x plus 4. Multiply the leading parts of the factors, x squared times x, and you get x cubed. So the degree is 3 and the leading coefficient is 1, which means the graph falls on the left and rises on the right. It crosses at minus 4 and touches at 1. Put in zero for x and you get 1 times 4, which is 4, so the y-intercept is the point zero, 4. Now trace it in your head. The graph comes up from the lower left, crosses at minus 4, passes through the point minus 2, 18, comes down through zero, 4, touches the axis at 1, and rises off to the upper right.
One more, with three multiplicities at once: x squared, times x minus 3, times x plus 2, cubed. The zero at x equals 0 has multiplicity 2, so the graph touches there. The zero at 3 has multiplicity 1, so it crosses. The zero at minus 2 has multiplicity 3, so it crosses, flattening out as it goes through. For the degree, add the exponents: 2 plus 1 plus 3 is 6. Even degree, positive coefficient, so both ends rise.
Pause and try this. What is the degree of x squared, times x minus 3, times x plus 2, cubed?
Don't stop at the biggest exponent you see.
Here's the answer. The degree is 6, because multiplying the factors out adds their exponents, 2 plus 1 plus 3.
6. Find a rational function's domain: vertical asymptotes versus holes
Picture a bridge with one plank missing. You can't stand on that spot, but the bridge around it is fine. Now picture a cliff edge, where the ground drops away for good. A rational function, one polynomial divided by another, can have both kinds of trouble, and factoring tells you which is which.
Here are the two terms. A vertical asymptote is a vertical line that a graph approaches but never reaches, created by a zero of the denominator that's still there after common factors are cancelled. A hole is a single missing point on a graph, drawn as an open circle, at an input where a factor in the numerator and denominator cancelled out.
Take R of x equals x plus 1, over x squared minus 4. Factor the bottom: x minus 2 times x plus 2. It's zero at 2 and at minus 2, so the domain is every real number except minus 2 and 2. The top, x plus 1, shares no factor with the bottom. Nothing cancels, so both are vertical asymptotes, x equals minus 2 and x equals 2.
Now take x squared minus x, over x minus 1. Factor the top: x times x minus 1. The factor x minus 1 cancels, leaving plain x, for every x except 1. So x equals 1 is a hole, not an asymptote. Its height comes from the simplified form, which gives 1. The graph is the line y equals x with an open circle at the point 1, 1.
Pause and try this. In x squared minus x, over x minus 1, is x equals 1 a vertical asymptote or a hole?
Ask whether the factor cancels.
Here's the answer. It's a hole at the point 1, 1, because the factor x minus 1 cancels; only a denominator zero that survives cancelling makes an asymptote.
7. Find the horizontal asymptote by comparing degrees
Think about free throws over a long season. Early on, one miss swings the percentage a lot. Late in the season, one more shot barely moves it. The percentage settles toward a level, and that level is the idea of this section.
A horizontal asymptote is a horizontal line that a graph approaches at its far left and far right. For a rational function, you find it by comparing the degree of the top with the degree of the bottom, and there are three cases.
If the top's degree is smaller, the asymptote is y equals 0. R of x has degree 1 over degree 2, so its horizontal asymptote is y equals 0. If the degrees are equal, the asymptote is the ratio of the leading coefficients. 3x plus 1, over x minus 5, has degree 1 over degree 1, so it's y equals 3. Far out, at x equals 105, the value is 3.16, and at 205 it's 3.08, creeping toward 3. If the top's degree is larger, there's no horizontal asymptote. x squared plus 1, over x minus 1, keeps growing: at x equals 11 it's already 12.2.
That last case has a classic slip. Both leading coefficients are 1, so it's tempting to say y equals 1. But the ratio rule only applies when the degrees match.
Pause and try this. What is the horizontal asymptote of 2x over x squared plus 1?
Compare the two degrees first.
Here's the answer. It's y equals 0, because the top has degree 1 and the bottom has degree 2, so the top's degree is smaller.
8. Interpret a horizontal asymptote and sketch a rational function
Back to the print shop. Making x bags costs 5x plus 200 dollars. The average cost is the total cost of making a number of items divided by that number of items, so here it's 5x plus 200, over x. At 10 bags that's 25 dollars each. At 50 bags it's 9. At 100 it's 7. At 200 it's 6. The degrees are equal, so the horizontal asymptote is the ratio 5 over 1, which is 5. The average falls toward 5 dollars a bag and never reaches it, because the 200-dollar setup is always spread over the bags, however many there are.
A simplified medicine model works the same way. Its concentration, t hours after a steady drip starts, is 6t over t plus 1, in milligrams per liter. After 1 hour it's 3, after 2 hours it's 4, after 5 hours it's 5, and after 11 hours it's 5.5. The degrees are equal, so it levels off toward 6.
Now let's sketch R of x, x plus 1 over x squared minus 4, from its features. Vertical asymptotes at minus 2 and 2. Horizontal asymptote y equals 0. The top is zero at minus 1, so it crosses the axis at the point minus 1, zero. At zero the value is 1 over minus 4, which is minus one fourth. At 3 it's 4 fifths, positive, so to the right of 2 the graph sits above the axis and sinks toward zero. Type it into Desmos and you'll find three separate pieces, divided by the vertical lines.
Pause and try this. In the print shop, can the average cost per bag ever equal exactly 5 dollars?
Think about where the setup cost goes.
Here's the answer. No, because the 200 dollars of setup is always shared over the bags, so the average stays a bit above 5.
Closing
Let's pull it together in order. The leading term alone decides a polynomial's ends. Even degree sends both ends the same way, odd degree sends them opposite ways, and the coefficient's sign picks which. Zeros come from setting each factor to zero. Odd multiplicity crosses the axis, and even multiplicity touches and turns. A sketch combines the ends, the zeros, and the y-intercept. For a rational function, factor and cancel first: a cancelled factor leaves a hole, and a surviving denominator zero leaves a vertical asymptote. Compare degrees for the horizontal asymptote: smaller gives zero, equal gives the ratio, larger gives none. In a setting, that asymptote is the level a quantity approaches but never reaches.
Next in this module come the slide deck, the Readings and Resources page, the lecture tutorial, the practice set, the quiz, the discussion, and the assignment. Next week, in Rational Expressions and Equations, the factor-and-cancel rule returns, this time to simplify and solve.
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