2.3 Graphs of functions and sketchingIB Maths: Applications and Interpretation HL: Revision notes

The graph of a function $f$ is the set of all points $(x, f(x))$, drawn as the curve $y=f(x)$. Each $x$ in the domain gives one point, so a vertical line meets the graph at most once. To use a graph you need to read it correctly: the horizontal axis shows the input (e.g. time $t$) and the vertical axis shows the output (e.g. temperature $T$). Axes should be labelled with the quantity and its units.

These two command terms are different.

• Draw: an accurate, labelled graph or diagram, using a pencil, with a ruler for straight lines, correctly scaled and with points plotted and joined by a straight line or smooth curve.
• Sketch: a graph or diagram that gives the general idea of the shape or relationship, and includes the relevant features. It need not be to scale. A sketch must still show all axes and key features labelled: intercepts, maximum or minimum points (vertex), asymptotes and end points, with coordinates where they are known.

Enter the function into your GDC as $y_1$ and choose a window that shows all the key features: use the domain to set $x$ values, and check the range of $y$ values. You can also graph sums and differences of functions. If revenue is $R(x)=30x$ and cost is $C(x)=2x^2+50$, the profit is the difference $P(x)=R(x)-C(x)=-2x^2+30x-50$. Graphing $y=P(x)$ shows where the profit is positive and where it is greatest. Use the GDC tools for zeros, maximum or minimum, and intersections, and give answers to 3 significant figures unless told otherwise.

When you copy a graph from the GDC to your answer sheet:

1. Draw and label both axes, with their quantities.
2. Copy the overall shape accurately (increasing or decreasing, curved or straight, concave up or down).
3. Mark and label the key features with their coordinates, e.g. the $y$-intercept, the vertex, the zeros, any asymptote (as a dashed line with its equation), and the end points of the domain.
4. Keep the domain: do not extend the curve beyond the given values. Example: $f(x)=x^2-4x-5$ has zeros $x=-1$ and $x=5$, $y$-intercept $(0,-5)$ and vertex $(2,-9)$.

Sometimes there is no equation to graph, only a description. Decide on the shape the context suggests, then label the given values. Example: a drink is poured at 90 °C and cools towards a room temperature of 20 °C. The sketch is a decreasing curve, starting at $(0, 90)$ and flattening towards a horizontal asymptote $T=20$. For $T(t)=20+70\times0.9^t$, the point at $t=30$ is $(30, 23.0)$. Example: for $P(x)=-2x^2+30x-50$, the sketch is an upside-down U-shape (a maximum at $(7.5, 62.5)$) crossing the $x$-axis at $1.91$ and $13.1$.