Functions and graphs-G11-Maths-T4P1

Functions and graphs

A function describes a relationship between input values and output values. Graphs show this relationship visually on the Cartesian plane. When studying a graph, focus on its domain, range, intercepts, turning points, asymptotes, intervals and general shape.

Definition of a function

A function is a relation in which every input value has only one output value. Different inputs may have the same output, but one input may not have two different outputs.

Function notation is written as:

y = f (x)

The vertical line test determines whether a graph represents a function. If any vertical line cuts the graph more than once, the graph is not a function.

Example: The relation below is a function because each value of the input gives exactly one output.

f (x) = x^{2} - 3

For an input of two:

f (2) = {(2)}^{2} - 3

f (2) = 1

Domain and range

The domain is the set of all possible input values, usually the values of the independent variable on the horizontal axis. The range is the set of all possible output values, usually the values of the dependent variable on the vertical axis.

For the function below, any real number may be substituted for the input. The smallest output value is four.

f (x) = x^{2} + 4

Domain: x \in R

Range: y \geq 4

When reading domain and range from a graph, work from left to right for the domain and from bottom to top for the range. A solid endpoint is included, while an open endpoint is excluded.

y = mx + c

A straight-line function has the general form:

y = m x + c

The value of the gradient is represented by the coefficient of the input variable. The constant term gives the y-intercept.

The gradient between two points is:

m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}

A positive gradient means that the graph rises from left to right. A negative gradient means that it falls. A zero gradient gives a horizontal line.

Example: Determine the equation of the line through the points shown below.

A (1; 3) and B (4; 9)

m = \frac{9 - 3}{4 - 1}

m = 2

Substitute one point into the general equation.

3 = 2 (1) + c

c = 1

Therefore:

y = 2 x + 1

Parabolic functions

A parabola has the basic form:

y = a x^{2} + q

A translated parabola may be written as:

y = a {(x - p)}^{2} + q

Its turning point is:

(p; q)

Its axis of symmetry is:

x = p

If the leading coefficient is positive, the parabola opens upwards and has a minimum value. If it is negative, the parabola opens downwards and has a maximum value. A larger absolute value of the leading coefficient makes the graph narrower.

For a parabola in general form, the input coordinate of the turning point is:

x = - \frac{b}{2 a}

Hyperbolic functions

The standard form of a hyperbolic function is:

y = \frac{a}{x - p} + q

The graph has two separate branches. Its vertical and horizontal asymptotes are:

x = p

y = q

The domain and range are:

x \in R, x \neq p

y \in R, y \neq q

If the numerator is positive, the branches lie above-right and below-left relative to the point where the asymptotes meet. If it is negative, they lie above-left and below-right.

Example:

f (x) = \frac{2}{x - 3} - 1

The vertical asymptote is at an input of three, and the horizontal asymptote is at an output of negative one.

x = 3

y = - 1

Exponential functions

An exponential function may be written as:

y = a b^{x} + q

The base must be positive and may not equal one.

b > 0, b \neq 1

If the base is greater than one, the basic graph is increasing. If the base lies between zero and one, the basic graph is decreasing. The horizontal asymptote is:

y = q

The domain is normally all real numbers. If the coefficient is positive, the range lies above the asymptote.

Example:

f (x) = 2^{x} - 3

Domain: x \in R

Range: y > - 3

Horizontal asymptote: y = - 3

Intercepts

An x-intercept is where the graph crosses or touches the horizontal axis. At an x-intercept, set the output equal to zero.

A y-intercept is where the graph crosses the vertical axis. At a y-intercept, set the input equal to zero.

Example: Determine the intercepts of the following parabola.

f (x) = x^{2} - x - 6

For the x-intercepts:

0 = x^{2} - x - 6

0 = (x - 3) (x + 2)

x = 3 or x = - 2

The x-intercepts are therefore:

(3; 0) and (- 2; 0)

For the y-intercept:

f (0) = {(0)}^{2} - (0) - 6

f (0) = - 6

(0; - 6)

Turning points

A turning point is where a graph changes from increasing to decreasing, or from decreasing to increasing. Parabolas have one turning point.

Example: Determine the turning point of:

f (x) = x^{2} - 4 x + 1

x = - \frac{- 4}{2 (1)}

x = 2

f (2) = {(2)}^{2} - 4 (2) + 1

f (2) = - 3

The turning point is:

(2; - 3)

Maximum and minimum values

A maximum is the greatest output value reached by a graph. A minimum is the smallest output value.

For an upward-opening parabola, the output coordinate of the turning point is the minimum value. For a downward-opening parabola, it is the maximum value.

For the previous example, the parabola opens upwards. Therefore:

Minimum value = - 3

There is no maximum value because the graph continues upwards without limit.

Asymptotes

An asymptote is a line that a graph approaches but does not normally touch. Hyperbolic functions may have vertical and horizontal asymptotes. Exponential functions have horizontal asymptotes.

Asymptotes are drawn as broken lines when sketching a graph.

Important: An asymptote is not automatically an intercept. The graph approaches the asymptote as the input or output values increase or decrease.

Increasing and decreasing intervals

A function is increasing where its output values rise as the input values increase. It is decreasing where its output values fall as the input values increase.

For a parabola with turning point at an input of two that opens upwards:

Decreasing for x < 2

Increasing for x > 2

Intervals must always be read from left to right. Do not describe a graph as increasing or decreasing by reading it from right to left.

Average gradient / average rate of change

The average gradient measures the average change in output relative to the change in input between two points.

Average gradient = \frac{change in output}{change in input}

Average gradient = \frac{f (b) - f (a)}{b - a}

Example: Calculate the average gradient of the function below between an input of one and an input of three.

f (x) = x^{2} + 1

f (1) = {(1)}^{2} + 1 = 2

f (3) = {(3)}^{2} + 1 = 10

Average gradient = \frac{10 - 2}{3 - 1}

Average gradient = 4

This is the gradient of the secant line joining the two points on the graph.

Sketching graphs

Use the following examination method when sketching a graph:

Identify the type of function and its basic shape.
Determine any x-intercepts and the y-intercept.
Determine the turning point, if applicable.
Determine and draw any asymptotes.
Plot sufficient additional points.
Draw a smooth curve through the points.
Label intercepts, turning points and asymptotes clearly.
Label both axes and indicate the origin.

Do not draw a parabola with straight-line segments. A sketch should show the correct shape and relative positions of important points.

Reading information from graphs

A graph can be used to determine approximate or exact values, depending on the information provided. You may be asked to identify:

Function values for given inputs.
Inputs that produce a given output.
Intercepts and turning points.
Maximum or minimum values.
Intervals where the function is positive, negative, increasing or decreasing.
Points where two graphs intersect.

To solve an equation using graphs, identify the input coordinates of the intersection points. To solve an inequality, identify the intervals where one graph lies above or below the other.

Transformations of graphs

Transformations move, reflect, stretch or compress a graph.

A vertical translation is represented by:

y = f (x) + q

A positive value moves the graph upwards, while a negative value moves it downwards.

A horizontal translation is represented by:

y = f (x - p)

A positive value of the parameter moves the graph to the right.

Reflection in the x-axis is represented by:

y = - f (x)

Reflection in the y-axis is represented by:

y = f (- x)

A vertical stretch or compression is represented by:

y = a f (x)

Exam Tip: Horizontal transformations work in the opposite direction to the sign inside the brackets.

Inverse functions

An inverse function reverses the action of the original function. If a function maps an input to an output, its inverse maps that output back to the original input.

f^{- 1} (f (x)) = x

The notation for an inverse does not mean the reciprocal of the function.

f^{- 1} (x) \neq \frac{1}{f (x)}

The domain of a function becomes the range of its inverse, and the range of the function becomes the domain of its inverse.

Restrictions on domain for inverse functions

An inverse is a function only if the original function is one-to-one. A one-to-one function gives different outputs for different inputs and passes the horizontal line test.

A complete parabola is not one-to-one because many horizontal lines cut it twice. Its domain must therefore be restricted to one side of the turning point.

For the function below:

f (x) = x^{2}

A suitable restriction is:

x \geq 0

Alternatively, the domain may be restricted to:

x \leq 0

The chosen restriction affects the equation and range of the inverse.

Determining and sketching inverse graphs

To determine an inverse algebraically:

Write the function using the output variable.
Interchange the input and output variables.
Make the new output variable the subject.
Write the answer using inverse notation.

Example: Determine the inverse of:

f (x) = 2 x - 5

y = 2 x - 5

Interchange the variables.

x = 2 y - 5

Make the output variable the subject.

x + 5 = 2 y

y = \frac{x + 5}{2}

Therefore:

f^{- 1} (x) = \frac{x + 5}{2}

The graphs of a function and its inverse are reflections of each other in the line:

y = x

Their coordinates are interchanged. For example:

(a; b) ⟶ (b; a)

When sketching an inverse, draw the original graph and the reflection line, interchange important coordinates, and reflect the shape accurately. Points lying on the reflection line remain unchanged.

Study Guide For Grade

11

Mathematics