How to Graph Cubics Quartics Quintics And Beyond

Sample Test Question

Let's look at a sample test question. Which graph represents the equation y = x^3 - 10x^2 - 11x + 180? Looking at these four multiple choice options, let's use what we know about graphing polynomials to select the correct answer. To begin, let's graph what we know about this equation on a blank Cartesian plane.

We know that our graph will go down to the left and up to the right because the leading coefficient, 1, is positive and its exponential power of 3 is odd. Answer choice B can be eliminated because it goes up to the left and down to the right.

We can also determine that there will be two bumps on the graph because the degree power of the equation is 3. To calculate the number of bumps, remember we subtract the degree power of the equation minus 1. So, 3 - 1 = 2 bumps. So the maximum number of bumps we will find in this equation will be two. So right away we can cancel out any graph that has more than two bumps. We can cancel out graph A because it has more than two bumps.

We are now left with two answer choices: C and D. The next way that we can determine which graph matches the equation is to find the y-intercept. To find the y-intercept, we will need to evaluate the equation for x = 0.

Starting with our equation, y = x^3 - 10x^2 - 11x + 180, we will plug in a zero for every x value we see. Now our equation looks like y = (0)^3 - 10(0)^2 - 11(0) + 180. To solve this equation, we will first do our exponents. Zero raised to any exponent is zero. Our equation now looks like y = 0 - 10(0) - 11(0) + 180. Next, we need to do all of the multiplication. Again, any number times zero equals zero. Our equation simplifies to y = 180. So our y-intercept for this equation is (0, 180).

It looks like the graphs of answer choices C and D both intersect the y-axis at (0, 180). The y-intercept does not help us in this case pick our correct answer. Let's graph some other points that we can find on a blank Cartesian plane to see which graph best matches the correct answer.

If our equations were easy to factor, we could find the x-intercepts. Unfortunately, we aren't that lucky with this equation. So we are going to plot some points to see how it looks in the middle, or between the ends of the graph. Easy points that I would try with virtually any equations would be when x = -1 and 1. We already know that when x = 0 that was our y-intercept.

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First let's solve for x = -1. To do so, we will evaluate our equation by plugging in -1 for all of the x values. Our equation was y = x^3 - 10x^2 - 11x + 180. When we plug in our -1 values for x, it becomes y = (-1)^3 - 10(-1)^2 - 11(-1) + 180. To work this equation, we will need to compute all of the exponents, making our equation y = -1 - 10(1) - 11(-1) + 180. Next, we simply need to multiply our equation, so now we get y = -1 - 10 + 11 + 180. After adding, our solution will be y = 180. So this ordered pair is (-1, 180). We can see that this point on the graph would be beside our y-intercept.

Now we can see that we have two pairs of coordinates in our table: (-1,180) and (0,180). Next, let's evaluate the equation for x = 1. When we let x = 1, we plug in 1 in for all x values. Our equation now looks like y = (1)^3 - 10(1)^2 - 11(1) + 180. We need to solve this equation the same way - by first completing the exponents and then multiplying.

After completing the exponents, our equation looks like y = 1 - 10(1) - 11(1) + 180. Then after multiplying we end up with y = 1 - 10 - 11 + 180. After we add, the solution to this equation will be y = 160. So this ordered pair is (1, 160). By graphing that point, we can see that this point will be to the right of and below our y-intercept.

We now have three pairs of coordinates: (-1, 180), (0, 180) and (1, 160). By knowing these three ordered pairs, we can now check our answer choices to see if we can eliminate either of our remaining choices. At this point, we can see that graph C is not our graph, so the graph that matches the equation would be answer choice D.