Investigating Features of Graphs and Sketching Graphs of Functions

Understanding the Basics of Graphs

A coordinate plane is a fundamental tool in algebra for plotting functions. It has two perpendicular lines namely, the horizontal x-axis and the vertical y-axis.
Points on a graph represent values of a function. They’re denoted as (x, y), where x is the input and y is the output of the function.
The origin is the point where the x and y axis cross, often denoted as (0, 0).

The Importance of Sketching Graphs

Sketching the graph of a function provides a visual representation of the function’s characteristics and behaviours.
The graph plot can help to identify whether the function is increasing or decreasing, its maximum or minimum value, along with any symmetries, intercepts, or asymptotes it may have.
Always label your diagrams correctly to indicate the function you are considering.

Features of Graphs

Graphs often start and end at intercepts. An x-intercept is a point where the graph crosses the x-axis, and y-intercept is where it crosses the y-axis.
The graph of a function may approach but never reach certain values, known as asymptotes. Asymptotes can be vertical, horizontal, or oblique/slanted.
Turning points (maxima, minima) are the points on the graph where it changes from increasing to decreasing or vice versa.
The slope or gradient of a function describes how steeply it rises or falls. It can be positive, negative, zero, or undefined.

Sketching Graphs of Functions

When sketching graphs, remember to plot clear points for all key features mentioned above including intercepts, turning points, and asymptotes.
The shape of a graph varies depending on the type of function, e.g., linear, quadratic, exponential, etc. Each type of function has its unique shape and key qualities to help in sketching.
Use transformations (translation, reflection, stretching) to draw variations of the basic function graphs. For example, the graph of y = f(x + a) will be a translation a units to the left of y = f(x).

Example

Sketch the graph of y = x^2: Start by finding the intercepts. As y = 0 when x = 0, the graph has a single intercept at the origin. This function is a basic quadratic function so its graph is a parabola opening upwards with its vertex (turning point) at the origin.

Note: Acquiring the skill to sketch and interpret different types of function graphs is essential for mastering algebra and progressing in more complex mathematical studies. Hence, it is always worth practising more.