What is a one-to-one mapping?

A one-to-one mapping is a transformation where each input is mapped to exactly one unique output , and no two inputs are mapped to the same output .

True or False? A many-to-one mapping can be a function .

True. A many-to-one mapping can be a function .

What is the vertical line test ?

The vertical line test is a method to determine if a graph represents a function . If a graph does represent a function , then any vertical line should intersect with the graph once at most .

What is the domain of a function?

The domain of a function is the set of values that are used as inputs for the function.

What is the range of a function?

The range of a function is the set of values that are given as outputs by the function.

True or False? An inverse function reverses the effect of a function .

True. An inverse function reverses the effect of a function .

True or False? Only a many-to-one function has an inverse function.

False. A many-to-one function does not have an inverse function. Only one-to-one functions have inverses .

What is the horizontal line test ?

The horizontal line test is a method to determine if a function has an inverse . If a function has an inverse , then any horizontal line should intersect with the graph once at most .

What is the relationship between the domain of a function and its inverse function ?

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

What are piecewise functions ?

Piecewise functions are functions defined by different equations depending on which interval the input is in .

True or False? The intervals for individual functions in a piecewise function can overlap .

False. The intervals for individual functions in a piecewise function can not overlap .

What is the difference between the command terms ' draw ' and ' sketch ' when graphing?

To sketch : show the general shape of the graph and label key points and axes. To draw : use a pencil and ruler to draw the graph to scale, plot points accurately, join points with a smooth curve or a straight line, and label key points and axes.

An asymptote is a line which the graph will get closer and closer to but not touch .

True or False? Most GDC models will automatically plot asymptotes .

False. Most GDC makes/models will not plot or show asymptotes just from inputting a function.

Define local minimum ( maximum ).

A local minimum ( maximum ) is a point at which the graph reaches the minimum ( maximum ) value that it takes in the immediate vicinity of the point. The graph may reach lower (higher) values further away from the point. It is also called a turning point .

True or False? A local minimum / maximum is always the global minimum / maximum of a function.

False. A local minimum / maximum is not necessarily the global minimum / maximum (the minimum/maximum of the whole graph).

What is the vertex of a quadratic function?

The vertex of a quadratic function, is the turning point of the parabola, lying on the axis of symmetry .

How many x -intercepts can a cubic function have?

The number of x -intercepts that a cubic function can have is 1 , 2 or 3 . It is useful to be able to graph the function to see the number of times that the curve crosses the x -axis .

True or False? An exponential function can have up to 2 roots .

False. An exponential function can have a maximum of 1 root .

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What is a one-to-one mapping?
A one-to-one mapping is a transformation where each input is mapped to exactly one unique output, and no two inputs are mapped to the same output.
True or False?
A many-to-one mapping can be a function.
True.
A many-to-one mapping can be a function.
What is the vertical line test?
The vertical line test is a method to determine if a graph represents a function.
If a graph does represent a function, then any vertical line should intersect with the graph once at most.
What is the domain of a function?
The domain of a function is the set of values that are used as inputs for the function.
What is the range of a function?
The range of a function is the set of values that are given as outputs by the function.
What symbol is used to represent the set of all real numbers?
$ℝ$ represents the set of all real numbers (i.e. all the numbers that can be placed on a number line).
What does $ℚ$ represent?
$ℚ$ represents the set of all rational numbers, $\frac{a}{b}$ , where $a$ and $b$ are integers and $b \neq 0$ .
True or False?
An inverse function reverses the effect of a function.
True.
An inverse function reverses the effect of a function.
True or False?
Only a many-to-one function has an inverse function.
False.
A many-to-one function does not have an inverse function.
Only one-to-one functions have inverses.
What is the horizontal line test?
The horizontal line test is a method to determine if a function has an inverse.
If a function has an inverse, then any horizontal line should intersect with the graph once at most.
What is the relationship between the domain of a function and its inverse function?
The domain of a function becomes the range of its inverse function.
State the relationship between the graphs of $f (x)$ and $f^{- 1} (x)$ .
The graph $y = f^{- 1} (x)$ is a reflection of the graph $y = f (x)$ in the line $y = x$ .
What are piecewise functions?
Piecewise functions are functions defined by different equations depending on which interval the input is in.
True or False?
The intervals for individual functions in a piecewise function can overlap.
False.
The intervals for individual functions in a piecewise function can not overlap.
How do you evaluate a piecewise function for a particular value $x = k$ ?
To evaluate a piecewise function for $x = k$ , find which interval includes $k$ and substitute $x = k$ into the corresponding function.
What notation is commonly used to define piecewise functions?
Piecewise functions are commonly defined using curly brackets with different functions listed for different intervals.
E.g. $f (x) = \{\begin{cases} 3 x - 1 & 0 < x \leq 4 \\ x^{2} - 5 & x > 4 \end{cases}$
How is continuity determined in piecewise functions?
Continuity in piecewise functions is determined by checking if the function values match at the boundaries between intervals.
E.g. The piecewise function $f (x) = \{\begin{cases} x + 8 & 2 < x \leq 5 \\ 4 x - 7 & 5 < x \leq 10 \end{cases}$ is continuous, because at the boundary between the two intervals, $x = 5$ , the output is the same: $(5) + 8 = 13$ and $4 (5) - 7 = 13$ .
How do you determine if a point $(a, b)$ lies on the graph $y = f (x)$ ?
A point $(a, b)$ lies on the graph $y = f (x)$ if $f (a) = b$ .
What is the difference between the command terms 'draw' and 'sketch' when graphing?
To sketch: show the general shape of the graph and label key points and axes.
To draw: use a pencil and ruler to draw the graph to scale, plot points accurately, join points with a smooth curve or a straight line, and label key points and axes.
Define asymptote.
An asymptote is a line which the graph will get closer and closer to but not touch.
True or False?
Most GDC models will automatically plot asymptotes.
False.
Most GDC makes/models will not plot or show asymptotes just from inputting a function.
How can you use graphs to solve the equation $f (x) = a$ ?
Plot the graphs $y = f (x)$ and $y = a$ on your GDC, and find the points of intersection.
The x-coordinates are the solutions of the equation.
Define local minimum (maximum).
A local minimum (maximum) is a point at which the graph reaches the minimum (maximum) value that it takes in the immediate vicinity of the point. The graph may reach lower (higher) values further away from the point.
It is also called a turning point.
True or False?
A local minimum/maximum is always the global minimum/maximum of a function.
False.
A local minimum/maximum is not necessarily the global minimum/maximum (the minimum/maximum of the whole graph).
What is the general form of a quadratic function?
The general form of a quadratic function is $f (x) = a x^{2} + b x + c$ , where $a \neq 0$ .
This is not given in your exam formula booklet.
How does the equation of quadratic affect the shape of the graph?
When the equation of a quadratic is given in its general form, $f (x) = a x^{2} + b x + c$ , the coefficient of the $x^{2}$ term, $a$ , determines the shape of the graph.
- If $a > 0$ , then the equation is positive and the curve is a 'u' shape.
- If $a < 0$ , then the equation is negative and the curve is an 'n' shape.
What is the vertex of a quadratic function?
The vertex of a quadratic function, is the turning point of the parabola, lying on the axis of symmetry.
What is the formula for the axis of symmetry of a quadratic function?
The formula for the axis of symmetry of a quadratic function $f (x) = a x^{2} + b x + c$ is $x = - \frac{b}{2 a}$
Where:
- $a$ is the coefficient of the $x^{2}$ term for the quadratic in its general form
- $b$ is the coefficient of the $x$ term for the quadratic in its general form
This is given in your exam formula booklet.
How can you find the y-coordinate of the vertex of a quadratic function?
Because the vertex of a quadratic function $f (x) = a x^{2} + b x + c$ lies on the axis of symmetry, its x-coordinate is $- \frac{b}{2 a}$ .
To find the corresponding y-coordinate, you can substitute this x-coordinate back into the equation of the original quadratic function.
What is the general form of a cubic function?
The general form of a cubic function is $f (x) = a x^{3} + b x^{2} + c x + d$ , where $a \neq 0$ .
This is not given in your exam formula booklet.
How many x-intercepts can a cubic function have?
The number of x-intercepts that a cubic function can have is 1, 2 or 3.
It is useful to be able to graph the function to see the number of times that the curve crosses the x-axis.
What are the common forms of an exponential function?
The common forms of an exponential function are
- $f (x) = k a^{x} + c$ ,
- or $f (x) = k a^{- x} + c$ ,
where $a > 0$ .
Note that an exponential can also be written in terms of the mathematical constant $e$ , in the form $f (x) = k e^{r x} + c$ .
These are not given in your exam formula booklet.
What part of the equation of an exponential, of the form $f (x) = k a^{r x} + c$ , determines whether its graph is increasing or decreasing?
To determine whether an exponential of the form $f (x) = k a^{r x} + c$ is increasing or decreasing, look at the coefficient of $x$ (i.e. $r$ ) and the value of $k$ .
- If they have the same sign, then the graph is increasing.
- If they have different signs, then the graph is decreasing.
True or False?
An exponential function can have up to 2 roots.
False.
An exponential function can have a maximum of 1 root.
What is the equation of the horizontal asymptote for an exponential function of the form $f (x) = k a^{x} + c$ ?
An exponential function of the form $f (x) = k a^{x} + c$ has a horizontal asymptote with equation $y = c$ .
What are the coordinates of the y-intercept of an exponential function of the form $f (x) = k a^{x} + c$ ?
An exponential function of the form $f (x) = k a^{x} + c$ has a y-intercept with coordinates $(0, k + c)$ .
What is the general form of a sinusoidal function, in terms of the sine function?
The general form of a sinusoidal function in terms of the sine function is $f (x) = a \sin (b x) + d$ .
A sinusoidal function can also be written in terms of the cosine function, $f (x) = a \cos (b x) + d$ .
What is the principal axis, in the context of a sinusoidal function?
The principal axis, in the context of a sinusoidal function, is the horizontal line about which a sinusoidal function fluctuates.
It is halfway between the maximum and minimum values of the curve.
The equation of the principal axis is $y = d$ from the general equation $f (x) = a \sin (b x) + d$ or $f (x) = a \cos (b x) + d$ .
True or False?
The coordinates of the y-intercept for the graph $y = a \cos (b x) + d$ are $(0, d)$ .
False.
The coordinates of the y-intercept for the graph $y = a \cos (b x) + d$ are not $(0, d)$ . This is the y-intercept for the graph $y = a \sin (b x) + d$ .
The coordinates of the y-intercept for the graph $y = a \cos (b x) + d$ are $(0, a + d)$ . This is because $\cos (0) = 1$ .
Define amplitude, in the context of sinusoidal functions.
The amplitude of a sinusoidal function is the distance between the principal axis and the maximum value or the distance between the principal axis and the minimum value.
This is $a$ in the general equation $f (x) = a \sin (b x) + d$ or $f (x) = a \cos (b x) + d$ .
Define period, in the context of a sinusoidal function.
The period, in the context of a sinusoidal function, is the length of the interval for a complete cycle, e.g. the length of the interval between two maximum points on the graph of a sinusoidal function.
This is $\frac{360 º}{b}$ in the general equation $f (x) = a \sin (b x) + d$ or $f (x) = a \cos (b x) + d$ .