graphs
Introduction
This session is mainly about graphs and how their shapes relate to their equations. We'll look at some standard curves: parabolas, cubics and hyperbolas, and investigate polynomial graphs and how their shape can be established by looking at a factorisation. We'll also look at transformations of curves and relate these to the first two sessions. Each section below should take around 40-60 minutes to work through.
1. Investigating graphs of polynomial functions
This section is mainly investigative, and you will need to use Desmos or Geogebra. You should make notes on the key features of the graphs you plot.
Work through these questions; most of them can be checked directly with graphing software. Here's a discussion of the first problem on the second page.
2. Asymptotes
Work through this sheet. Again, you will need to use some graphing software.
4. Focus on cubics
In the first section above you looked at some cubic graphs. What do we notice about their shape? One of the key things is that they always have rotational symmetry of order two. In a depressed cubic (no x-squared term), the centre of this rotational symmetry is forced to be on the y-axis. Have a think about why that is.
You also looked at how the equation of a depressed cubic affects whether the graph has turning points or just an inflection point.
The discriminant
of a quadratic allows us to establish how many roots (i.e. x-axis crossing points) it has. Can we develop a similar idea for cubics?
Watch the video below, which does this in the case of a depressed cubic whose y-intercept is positive. Use the idea discussed to have a go at the questions here. Solutions here.
An optional challenge: thinking back to Tartaglia's method for solving a depressed cubic, can you show that when a depressed cubic has three solutions, the method always leads to trying to take the square root of a negative number? If you're taking further maths next year you will learn a method for progressing in these cases.
5. Optional section: cello tangents to quartics
In the picture above, the bow is a tangent to the cello at two different points, hence we'll call such a tangent a cello tangent.
Have a look at Mr Fellerman's animation here, which constructs this tangent for a quartic curve. An interesting question should pose itself: how do we construct this tangent, and can we always do so? Work through this sheet. A solution to the last question is here.
6. Bringing ideas together: graphs of rational functions
The last two sessions have included some techniques for looking at the shapes of graphs. In this section, we're going to think about sketching curves. In order to do this, and to practise some of the things we've learned, we're going to do so without graphing software, except as a final check. A rational function is one that can be written as a fraction, where both the numerator and denominator are polynomials.
When we're given the instruction sketch, it's useful to run through the following check list:
Where does the graph cross the axes? For the y-axis, substitute x = 0; for the x-axis, solve the equation y = 0.
Are there any turning points? We can often find these by considering the discriminant of a quadratic.
Are there any asymptotes? Vertical asymptotes can usually be seen easily from looking at the denominator of the function; for other asymptotes it's useful to check what happens when you substitute a large positive or large negative value into the expression. At other times, the technique of splitting the numerator can give us the asymptote.
In our final sketches, there's no need to plot points accurately. What we're interested in is the overall shape and general position of key features. Watch the videos below and then have a go at these questions. Check your work on desmos.
Example 1, example 2, example 3.