Graphs are visual representations of equations that can help us understand the relationship between two variables.
Using graphs for algebraic relationships
You can use the coordinate plane, which you can see below, to graph any algebraic relationship. The coordinate plane comprises a horizontal line (x-axis) and a vertical line (y-axis), and it is divided into four quadrants named using roman numbers (I, II, III and IV).
The different points in a graph have coordinates written as ordered pairs (pairs of numbers within parentheses separated by a comma). The first number in an ordered pair (x, y) represents the value of x, and the second one represents the value of y for a given point. For example, the middle point where the x and y axes meet is called the origin, and its coordinates are (0, 0).
Graphs help us analyse the behaviour of variables and can be used to make inferences about them and facilitate data interpretation.
Coordinate plane
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.
If you have the equation y = x, you can plot its graph like this:
Plotting a graph
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).
Sketching a graph
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.
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.
If you don't have the equation of the line graph, but you can identify two points on the line A = (2, 2) and B = (5, 5).
Calculating the slope
Then you can calculate the slope as follows:
\(m = \frac{y_2-y_1}{x_2 - x_1} = \frac{5-2}{5-2} = \frac{3}{3} = 1\)
m = 1
It doesn't matter which one you choose as point 1 and point 2. The resulting slope will be the same.
If you have the linear equation \(y = mx + b\), you don't need to calculate the slope. You can identify in the equation the value of m, which will be the slope of the line. Likewise, the value of b will be the y-intercept.
For the equation \(y = 2x + 3\), m = 2 and b = 3.
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.
Sketch the graph of \(f(x) = x^2 + 3x +2\), and find the coordinates of its turning point.
- The coefficient of \(x^2 (a)\) is positive, therefore the parabola will be the right way up, and it will have a minimum point.
- When x = 0, y = 2, therefore the parabola crosses the y-axis in the point (0, 2)
- \(x^2 + 3x +2 = 0\) Find the roots of the function by factoring
\((x+1)(x+2) = 0\)
The roots are x = -1 and x = -2
- Using symmetry to find the turning point:
\(x = \frac{-1 + (-2)}{2} = \frac{-3}{2}\)
\(y = \big(\frac{-3}{2} \big)^2 + 3\big(\frac{-3}{2} \big) + 2\)
Substitute x in the original equation
\(y = \frac{9}{4} + \frac{-9}{2} + 2\)
\(y = \frac{-1}{4}\)
The minimum point is \(\big( \frac{-3}{2} \frac{-1}{4}\big)\)
- Now you can sketch the graph:
Sketching a quadratic 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.
Sketch the curve for \(y=(x+1)(x+2)(x+3)\) showing the points where they cross the coordinate axes.
\((x+1)(x+2)(x+3) = 0\)
The roots are \(x = -1\), \(x = -2\), and \(x = -3\)
Therefore, the curve crosses the x-axis at (-1, 0), (-2, 0) and (-3, 0)
\(y = 1 \cdot 2 \cdot 3 = 6\)
The curve crosses the y-axis at (0, 6)
Sketching a cubic graph
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.
Sketch the curve for \(y = x(x-1)(x+3)(x-2)\) showing the points where they cross the coordinate axes.
\(x(x-1)(x+3)(x-2) = 0\)
The roots are x = 0, x = 1, x = -3 and x = 2
Therefore, the curve crosses the x-axis at (0, 0), (1, 0), (-3, 0), and (2, 0)
The curve crosses the y-axis at (0, 0)
Sketching a quartic graph
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
If you have an expression inside the modulus function, calculate the value inside, then find the positive version of the result.
If you have the function \(f(x) = |x-3| +1\) find \(f(-2))\)
\(f(-2) = |-2 -3| + 1 = |-5| +1 = 5+1 = 6\)
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.
Sketch the graph for \(y = |x-1|\) showing the points where they cross the coordinate axes.
Ignoring the modulus, you need to sketch the graph of \(y = |x-1|\)
The line crosses the x-axis at (1, 0)
The line crosses the y-axis at (0, -1)
- Sketch the graph for \(y = |x-1|\):
Sketching a modulus function graph
- For the negative values of y, reflect in the x-axis. In this case, (0, -1) becomes (0, 1)
Sketching a modulus function graph
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
Sketch the graph for \(y = \frac{4}{x}\)
Sketching a reciprocal graph
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
Write down the equation of the circle with centre (6, 5) and radius 3, then sketch its graph.
\((x-6)^2 + (y-5)^2 = 3^2\)
\((x-6)^2 + (y-5)^2 = 9\)
Sketching a circle graph
Read more about Circle Maths.
Graphs - key takeaways
Graphs are visual representations of equations that can help us understand the relationship between two variables.
When sketching, you do not need to be as accurate as when plotting; 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.
The slope of a line and the y-intercept can be used to help graph a linear equation.
To sketch quadratic, cubic and quartic equations the roots of the function must be identified, as well as the point where the curve crosses the y-axis.
To sketch the graph of the modulus function \(y = |ax+b|\), you need to sketch \(y = ax + b\), then reflect the portion of the line that goes below the x-axis into the x-axis.
In reciprocal function graphs, an asymptote is a line that the curve gets very close to, but it never touches it.
A circle is a set of points that are at the same distance from a fixed point called the centre.
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