Graphing functions and curves

See also: graphing calculator and usages on TI-84 Plus

A graph of a function is a visualization of input-output pairs as points on a coordinate plane. This page shows how to graph a function by hand and on a graphing calculator, and lists common features of graphs.

Contents

• Graphing from a table of values
• Graphing questions
• Zoom
• Interpretation of graphs
• Zeros
• Extrema and concavity
• Asymptotes
• HL: Non-function curves

Graphing from a table of values

See also: limits from a table of values

A table of values can be generated from select input values ($x$) and their corresponding output values ($y$). Plotting such $(x, y)$ points will show the general shape of the function. More available points typically means a more accurate graph.

This is exactly what graphing calculators do when they graph functions. On the TI-84 Plus for example, the number of points are limited by the Xres; changing it from 1 to 3 would result in a 3x graphing speed, but at the cost of only using a third as many points.

Graphing questions

In IB, questions that ask for a sketch of a function can span from 1 to over 4 marks. The amount of detail depends on the number of marks. 1 mark questions require only a generally correct shape, indicating presence of zeros, $y$-intercept, asymptotes, extrema, and points of inflexion. Additional marks will require specific points or values for such features. Questions will often explicitly state what to include. Typically, any relevant numerical values found in previous parts are to be included in the sketch.

Use a pencil for graphs. Erase as necessary. Do not trace or go over your graph; use one stroke for each part of the graph, or a single stroke if the function is continuous. Make sure graphs of functions pass the vertical line test.

Zoom

General procedure to setting the zoom on a calculator is to choose a desired $x$-interval, then let ZoomFit or ZOOM AUTO to include the min and max $y$-values in the interval.

Interpretation of graphs

For a graph of a function $f(x)$:

• points on the graph satisfy $y = f(x)$
• points above the graph satisfy $y > f(x)$
• points below the graph satisfy $y < f(x)$.

Two or more functions on the same axes follow the same rules as above, with respect to vertical locations of the functions.

Zeros

The zeros of a function are where $y = f(x)$ meets the $x$-axis $y = 0$. Equivalently, finding the zeros of a function is equivalent to solving $f(x) = 0$. Why graphing is preferred over solver and how to increase speed of graphing. This relates to the idea that studying functions enhances the understanding of equations and how to solve them.

Extrema and concavity

See also calculus: extrema and concavity

The extrema of a function includes maxima and minima.

We typically say there is an extremum at $x = x_0$, or that $(x_0, y_0)$ is an extremum.

A global extremum is the highest or lowest point across the entire domain. In contrast, a local extremum is only the highest or lowest point if we restrict the domain to some interval.

For a continuous, smooth function, “concave up” is the $\cup$ shape near a minimum, and “concave down” is the $\cap$ shape near a maximum.

At a point of inflexion, the graph changes concavity from one to the other.

Asymptotes

Verticals asymptotes are where the function otherwise attempts to divide by zero. Both sides of the graph approach the vertical line (asymptote) without passing it. They can both approach $+\infty$ (positive infinity), $-\infty$ (negative infinity) or one approaches positive and other approaches negative infinity.

Horizontal asymptotes describe the constants that $f(x)$ approaches at very positive or very negative values of $x$. There can be a maximum of two horizontal asymptotes for a function (for very positive and very negative values of $x$). Horizontal asymptotes can be cross an infinite number of times, as long as the amplitude of the oscillation is decreasing.

Asymptotes are graphed using dotted or dashed lines.

HL: Non-function curves

On rare occasions, HL students may be asked to graph curves that are not relations. The best way is to split the relation into functions. Another way is to draw the relation (such as on TI-84 Plus) but these sketches have limited interactivity.