Graphs

Plotting vs sketching

When making graphs, you can either do it by plotting or sketching. For plotting, you normally use graph paper and make a table of values for x and y coordinates and plot them as accurately as possible.

When sketching, you do not need to be as accurate. You need to draw the x and y axes and sketch the general shape of the curve, including the points where it intersects the x and y axes. In the case of a line graph, you only need a couple of points to draw the line crossing those two points. When you sketch the line graph for y = x, you only need one more point, as you know that the line crosses the origin (0, 0).

What are the different types of graphs?

Depending on the type of function that you are graphing, you will obtain different characteristic shapes for their curve. The main types of graphs are described below.

Linear graphs

Linear graphs are a straight line. They represent the graph of functions where the highest exponent in its equation is 1.

• Slopes and intercepts

In linear graphs, the slope is the line's rate of change in the vertical direction. The slope can be shallow or steep, depending on its value. The bigger the value of the slope, the steeper the line will be, and the smaller the value of the slope, the shallower the line will be. Also, you will need to remember that the slope of a horizontal line is zero, and the slope of a vertical line is undefined.

Any linear equation can be represented in the slope-intercept form, like this:

\[y = m x + b\]

x = independent variable

y = dependent variable

m = slope (how steep the line is)

b = y-intercept (y coordinate for the point where the line crosses the y axis)

The slope can be calculated using the formula:

\(m = \frac{rise}{run}\)

\(m = \frac{\text{change in y}}{\text{change in x}} = \frac{y_2-y_1}{x_2 - x_1}\)

where \((x_1, y_1)\) and \((x_2, y_2)\) are any two points on the line.

Reading Straight Line Graphs will expand your knowledge on this topic.

Quadratic graphs

If the function we want to graph is quadratic, generically represented as \(f(x) = ax^2 + bx + c\), then the shape of y = f (x) will be a parabola.

If the coefficient of \(x^2(a)\) is positive, then the parabola will be the right way up.

Quadratic graph with positive coefficient

If the coefficient of \(x^2(a)\) is negative, then the parabola will be upside down.

Quadratic graph with negative coefficient

Besides identifying if the parabola will be the right way up or upside down, to sketch a quadratic graph you need to proceed as follows:

• Substitute x = 0 in the function \(f(x) = ax^2 + bx +c\), to obtain the y-coordinate where the parabola crosses the y-axis, which is equal to c.

• Make the function \(f(x) = ax^2 + bx +c\) equal to zero, and find the roots of the function f(x). The roots will be the x-coordinates where the parabola crosses the x-axis. You can find the roots by factoring, completing the square, or using the quadratic formula.

• Find the turning point of the parabola (minimum or maximum), either by completing the square or using symmetry.

  • If you complete the square, then the turning point will be (-p, q) if \(f(x) = a(x + p)^2 + q\) .

  • If you use symmetry, the x-coordinate of the turning point will be in the middle of the two roots found in the previous step (add them together then divide by 2). After this, you need to substitute the resulting value of x into the original function to find the y-coordinate of the turning point.

• Sketch the graph.

Cubic graphs

If the function that you are graphing is cubic, generically represented as \(f(x) = ax^3 + bx^2 +cx +d\), then the shape of y = f (x) is shown below if the coefficient of \(x^3(a)\) is positive.

Cubic graph with a positive coefficient

If the coefficient of \(x^3(a)\) is negative, then the shape will be like this:

Cubic graph with a negative coefficient

To sketch the graph of cubic functions, you need to find the roots of the function.

Quartic graphs

If the function you are graphing is quartic, generically represented as \(f(x) = ax^4 +bx^3 +cx^2+dx+e\), then the shape of y = f (x) can have different forms depending on its roots. One of the possible shapes, if the coefficient of \(x^4(a)\) is positive, is shown below.

Quartic graph with positive coefficient

If the coefficient of (x^4(a)\) is negative, its curve can take the following shape:

Quartic graph with negative coefficient

Again, to sketch the graph of quartic functions, you need to find the roots of the function.

Please refer to the article about Polynomial Graphs for more details and examples about quadratic, cubic and quartic graphs.

Modulus function graphs

The modulus function, also known as the absolute value function, is generically represented. The modulus of a number x will be the same number but positive. The typical shape of a modulus function is shown below.

Modulus function graph

To sketch the graph of the modulus function \(y = |ax+b|\), you need to sketch \(y = ax+b\), and reflect the portion of the line that goes below the x-axis into the x-axis.

Read about Modulus Function to learn more about this type of graph.

Reciprocal graphs

Reciprocal functions are generally represented as \(y = \frac{a}{x}\), and \(y = \frac{a}{x^2}\). To sketch this type of graph, you need to consider its asymptotes. An asymptote is a line that the curve gets very close to, but it never touches it. The graph of reciprocal functions has asymptotes at x = 0 and y = 0. The shape of a reciprocal function where \(a = 1\), \(y = \frac{1}{x}\), is shown below.

Reciprocal graph with a positive coefficient

The shape of a reciprocal function where \(a = 1\), \(y = \frac{1}{x^2}\), is as follows.

Reciprocal graph squared

For more information and examples about this type of graph, read about Reciprocal Graphs.

Circle graphs

Another important type of graph that you will find in Coordinate Geometry is circle graphs. A circle is a set of points that are at the same distance from a fixed point called the centre. The equation of a circle with centre (0, 0) and radius r is \(x^2 + y^2 = r^2\). If the centre is (a, b), then the equation changes to \((x-a)^2 + (y-b)^2 = r^2\).

Circle graph

Read more about Circle Maths.