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Graph Sketching Made Easy

Expert reviewed • 21 July 2024 • 6 minute read

HSC Maths Advanced Syllabus

  • use graphical methods with supporting algebraic working to solve a variety of practical problems involving any of the functions within the scope of this syllabus, in both real-life and abstract contexts
    • select and use an appropriate method to graph a given function, including finding intercepts, considering the sign of 𝑓(𝑥) and using symmetry

Note:

Video coming soon!

How to Sketch a Curve

The ability to sketch a curve is a vital skill, however, some curves can seem challenging and almost impossible to sketch. Fortunately, by breaking down the information given to you, curve-sketching becomes easy.

Following the ordered steps below, is a great way to construct any graph.

  • Simplification and factorisation: It is important to first simplify any equation you have been asked to graph. This ensures that the following steps are easier for you to complete.
  • Find intercepts: The next step is to find your xx and yy intercepts.
  • Find signs: once your intercepts have been found, it is important to determine the the nature of the curve between points/intercepts. By doing so, you will start to understand the general shape of the graph. Remember, the signs can be found by using a table of signs.
  • Determine horizontal and vertical asymptotes: Find the discontinuities and plot them on a cartesian plane using a dotted line
  • Sketch the graph: Finally, by following your table of signs, using your xx and yy intercepts, and following the asymptotes, plot the graph. (ensure all important aspects of the graph are labelled)
  • Apply any additional graphical transformations (Eg. Any additional dilations)

These instructions only allow for constructing graphs we have encountered so far in the syllabus. The next chapter involves new concepts and graphs. This sketch-menu will be expanded upon then.

Practice Question 1

Sketch the graph f(x)=x2−1x−2f(x)=\frac{x^2-1}{x-2}

Following the steps listed above, we must first simplify the equation. This equation however, has already been simplified as much as possible.

Now to find the intercepts:

0=x2−1x−20=x2−1x2=1x=±10=\frac{x^2-1}{x-2}\\0=x^2-1\\x^2=1\\x=\pm 1

The zeros for this equation are x=±1x=\pm1

y=02−10−2=−1−2=12y=\frac{0^2-1}{0-2}\\=\frac{-1}{-2}\\=\frac{1}{2}

There is only one yy intercept for this equation at 12\frac{1}{2}

Now to create a table of signs:

x-3-1013
Sign---++

Finding the asymptotes:

0=x−2x=20=x-2\\x=2

There is a vertical asymptote at x=2x=2

As the numerators degree is greater than the denominators, there is no horizontal asymptote for this equation.

Now we can graph:

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Solving Equations and Inequalities

Return to Module 2: Graphs and Equations