Graphs of Common Functions

Remember that a function is a mathematical construct that takes x values as the input, and it outputs y values in a one-to-one or many-to-one correspondence. Functions represent the relationship between an independent variable x and a dependent variable y.

Representation of a function, Marilú García De Taylor - StudySmarter Originals

Types of function graphs

Some of the most common functions that you will find in Maths are listed below:

1. Constant: $f\left(x\right)=c$ where c is a constant. The shape of the graph of constant functions is a straight line parallel to the x-axis, that intercepts the y-axis, where y = c.

Constant function graph, Marilú García De Taylor - StudySmarter Originals

2. Linear (Identity) $f\left(x\right)=x$. The shape of linear Graphs is also a straight line. In this case, the line has a slope that can be shallow or steep depending on its value.

Identity function graph, Marilú García De Taylor - StudySmarter Originals

3. Quadratic: $f\left(x\right)=x^2$. The shape of the graph of quadratic functions is a parabola.

Quadratic function graph, Marilú García De Taylor - StudySmarter Originals

4. Cubic: $f\left(x\right)=x^3$. Cubic graphs are continuous and smooth lines that can have maximum or minimum points where they change direction in the middle section of the curve, and on either end of the curve they tend to go towards positive or negative infinity.

Cubic function graph, Marilú García De Taylor - StudySmarter Originals

5. Square Root: $f\left(x\right)=\sqrt{x}$. The graph of square root functions has a characteristic shape because of its restricted domain (id="2853741" role="math" $x\ge 0$). This is because the square root of a negative number has no real solution. Therefore, only positive numbers are used in this type of graph.

Square root function graph, Marilú García De Taylor - StudySmarter Originals

6. Cube Root: $f\left(x\right)=\sqrt[3]{x}$. Cube root graphs differ from square root graphs because the cube root of negative numbers has real solutions. Therefore, cube root functions do not have a restricted domain, x can take negative and positive values, which is shown on the shape of its graph.

Cube root function graph, Marilú García De Taylor - StudySmarter Originals

7. Modulus or absolute value: $f\left(x\right)=\left|x\right|$. The graph of modulus functions has a characteristic v-shape. It is the same as f (x) = x, but the negative values of y are reflected on the x-axis. This is because the modulus of a number x is the same number, but positive.

Modulus or absolute value function graph, Marilú García De Taylor - StudySmarter Originals

8. Reciprocal: $f\left(x\right)=\frac{1}{x}$. The graph of reciprocal functions has asymptotes, which are lines that the curve gets very close to, but it never touches them. The graph of $f\left(x\right)=\frac{1}{x}$ has asymptotes at x = 0 and y = 0.

Reciprocal function graph, Marilú García De Taylor - StudySmarter Originals

9. Reciprocal Squared: $f\left(x\right)=\frac{1}{x^2}$. The shape of the graph of the reciprocal squared function changes in comparison to the previous one, because having $x^2$ in the denominator means that all values of y will be positive.

x2Reciprocal squared function graph, Marilú García De Taylor - StudySmarter Originals

10. Exponential: $f\left(x\right)=e^x$. The graph of the exponential function has a horizontal asymptote at y = 0, and it intersects the y-axis at the point (0, 1). After that, it increases rapidly.

Exponential function graph, Marilú García De Taylor - StudySmarter Originals

11. Logarithmic: $f\left(x\right)=\ln \left(x\right)$. The logarithmic function is the inverse of the exponential function, therefore the graph of a logarithmic function will be the reflection of the exponential graph it relates to, over the line y = x. The logarithmic function graph has a vertical asymptote at x = 0, and crosses the x-axis at the point (1, 0).

Logarithmic function graph, Marilú García De Taylor - StudySmarter Originals

12. Trigonometric Functions : The graphs of the trigonometric functions (sine, cosine and tangent) have a characteristic shape because they are periodic, which means that they repeat themselves after a specific interval.

a) Sine: $f\left(x\right)=\sin \left(x\right)$. The graph for sine has a maximum value of 1 and a minimum value of -1, and it repeats itself every 2π. The graph of sine crosses the y-axis at the origin (0, 0).

Trigonometric function graph - sin (x), Marilú García De Taylor - StudySmarter Originals

b) Cosine: $f\left(x\right)=\cos \left(x\right)$. The cosine graph also has a maximum value of 1 and a minimum value of -1, and repeats itself every 2π. You can differentiate it from the sine graph, because the cosine graph crosses the y-axis at the point (0, 1).

Trigonometric function graph - cos (x), Marilú García De Taylor - StudySmarter Originals

c) Tangent: $f\left(x\right)=\tan \left(x\right)$. The tangent graph does not have maximum or minimum points, and repeats itself every π. It therefore has vertical asymptotes at $\frac{-\pi }{2}$, $\frac{\pi }{2}$, $\frac{3\pi }{2}$, etc.

Trigonometric function graph - tan (x), Marilú García De Taylor - StudySmarter Originals

How do you find the common function of a graph?

In order to identify the common function of a graph, it is very useful to learn the shapes of their graphs. Focus on their features and formulas (for example, the shape of the curve) so that you can quickly identify what type of function a graph is representing. Memorising the previous list of common function graphs will help you acquire this important skill, in case you need them to solve specific problems.

Identifying if a graph is a function

If you are given a graph, and you are asked to check if the graph represents a function or not, then you can proceed as follows:

• Vertical line test: This test will tell you if the graph represents a function. You need to draw vertical lines intersecting the graph. If at any point, a vertical line intersects the graph more than once, then the graph is not a function (x has more than one output). For example:

Vertical line test example, Marilú García De Taylor - StudySmarter Originals

This graph is not a function because the vertical line intersects two points on the graph.

• Horizontal line test: This test tells you if a function is one-to-one or not. If you draw a horizontal line, and it intersects the graph more than once, then it is not a one-to-one function. For example:

Horizontal line test example, Marilú García De Taylor - StudySmarter Originals

This graph is a function because it passes the vertical line test, but it is not a one-to-one function, because the horizontal line intersects the graph twice.