What does the domain represent in mathematical modelling ?

In mathematical modelling , the domain represents the reasonable range of input values , considering the real-life context of the situation being modelled.

How can unknown parameters be found in mathematical models ?

Unknown parameters in mathematical models can be found by forming equations , substituting given values and solving the equations simultaneously. You will frequently use a graphing calculator (GDC) to help with this.

What is extrapolation in mathematical modelling ?

Extrapolation is making predictions outside the range of the data.

True or False? Extrapolation is an accurate method for predicting new values in mathematical modelling.

False. Extrapolation is not considered to be an accurate method for predicting new values in mathematical modelling.

What is a key strategy used to overcome the limitation of a model only being accurate for a portion of the real-life situation ?

A key strategy used to overcome the limitation of a model only being accurate for a portion of the real-life situation is to restrict the domain appropriately.

True or False? Exponential models are commonly used for arithmetic sequences .

False. Exponential models are not used for arithmetic sequences . Linear models are used for arithmetic sequences, while exponential models are used for geometric sequences.

What type of model is typically used for projectile motion ?

A quadratic model is typically used to model projectile motion.

How can you estimate outputs using a model?

You can estimate outputs by substituting values for the input variable into the model.

What does substituting t = 0 into a time-based model typically give you?

Substituting t = 0 into a time-based model typically gives you the initial value according to the model.

IBMathsDPAnalysis & Approaches (AA)SLFlashcards2. FunctionsFurther Functions & Graphs

Further Functions & Graphs (DP IB Analysis & Approaches (AA))

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FrontReciprocal & Rational Functions
What does the graph of the function $f (x) = \frac{1}{x}, x \neq 0$ look like?

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What does the graph of the function $f (x) = \frac{1}{x}, x \neq 0$ look like?
The graph of $f (x) = \frac{1}{x}, x \neq 0$ is shown below.
What is the range of an exponential function $f (x) = a^{x}$ , where $a > 0$ ?
The range of an exponential function $f (x) = a^{x}$ , where $a > 0$ , is $f (x) > 0$ .
True or False?
The inverse of the reciprocal function, $f (x) = \frac{1}{x}$ , is itself.
True.
The inverse of the reciprocal function, $f (x) = \frac{1}{x}$ , is itself.
The reciprocal function is a self-inverse function.
True or False?
The domain of $f (x) = a^{x}$ , where $a > 0$ , is $x \in ℝ$ .
True.
The domain of $f (x) = a^{x}$ , where $a > 0$ , is $x \in ℝ$ .
What are the equations of the asymptotes of the function $f (x) = \frac{1}{x}, x \neq 0$ ?
The equations of the asymptotes of the function $f (x) = \frac{1}{x}, x \neq 0$ are:
- $x = 0$
- $y = 0$
True or False?
$a^{x} = e^{a \ln x}$ .
False.
$a^{x} = e^{x \ln a}$
This is given in the exam formula booklet.
What is the range of the reciprocal function, $f (x) = \frac{1}{x}, x \neq 0$ ?
The range of the reciprocal function, $f (x) = \frac{1}{x}, x \neq 0$ is all real number values except zero: $f (x) \in ℝ, f (x) \neq 0$ .
What does the graph of $f (x) = a^{x}$ look like when $a > 1$ ?
$f (x) = a^{x}$ is an increasing function when $a > 1$ .
How do you find the equation of the vertical asymptote of the rational function $f (x) = \frac{a x + b}{c x + d}$ ?
To find the equation of the vertical asymptote of the rational function $f (x) = \frac{a x + b}{c x + d}$ , you set the denominator equal to zero.
The equation is $c x + d = 0$ which can be written as $x = - \frac{d}{c}$ .
The equation of the graph below is $y = a^{x}$ . What are the possible values for $a$ ?
The exponential graph is decreasing, therefore $0 < a < 1$ .
How do you find the equation of the horizontal asymptote of the rational function $f (x) = \frac{a x + b}{c x + d}$ ?
To find the equation of the horizontal asymptote of the rational function $f (x) = \frac{a x + b}{c x + d}$ , you look at the coefficients of the $x$ terms.
The equation is $y = \frac{a}{c}$ .
Every exponential graph of the form $f (x) = a^{x}$ where $a > 0$ passes through which point?
Every exponential graph of the form $f (x) = a^{x}$ where $a > 0$ passes through the point (0, 1).
True or False?
Any rational function of the form $f (x) = \frac{a x + b}{c x + d}$ , will always have exactly one real root.
False.
If $a = 0$ , the rational function will not have any real roots as the horizontal asymptote is $y = 0$ .
Any rational function of the form $f (x) = \frac{a x + b}{c x + d}$ , will always have exactly one real root provided $a \neq 0$ .
What is the equation of the asymptote for every exponential graph of the form $f (x) = a^{x}$ where $a > 0$ ?
The equation of the asymptote for every exponential graph of the form $f (x) = a^{x}$ where $a > 0$ is $y = 0$ .
If you are asked to sketch a rational function of the form, $f (x) = \frac{a x + b}{c x + d}$ , what should you include in the sketch?
If you are asked to sketch a rational function of the form, $f (x) = \frac{a x + b}{c x + d}$ , you should include:
- the horizontal asymptote and its equation,
- the vertical asymptote and its equation,
- the coordinates of any intercepts with the axes.
What is the inverse function of $f (x) = a^{x}$ where $a > 0$ ?
The inverse function of $f (x) = a^{x}$ where $a > 0$ is $f^{- 1} (x) = \log_{a} x$ .
True or False?
The inverse of any rational function of the form $f (x) = \frac{a x + b}{c x + d}$ is itself.
False.
The inverse of any rational function of the form $f (x) = \frac{a x + b}{c x + d}$ is not always itself.
What is the domain of the logarithmic function $f (x) = \log_{a} x$ where $a > 0$ ?
The domain of the logarithmic function $f (x) = \log_{a} x$ where $a > 0$ is $x > 0$ .
True or False?
The graph of $y = \log_{a} x$ where $a > 0$ crosses the coordinate axes at the point (0, 1).
False.
The graph of $y = \log_{a} x$ where $a > 0$ crosses the the coordinate axes at the point (1, 0) not (0, 1).
What does $\log_{a} (a^{x})$ simplify to when $a > 0$ ?
$\log_{a} (a^{x})$ simplifies to $x$ when $a > 0$ .
This is because the logarithmic function and exponential function are inverses.
What is the equation of the asymptote for every logarithmic graph of the form $f (x) = \log_{a} x$ where $a > 0$ ?
The equation of the asymptote for every logarithmic graph of the form $f (x) = \log_{a} x$ where $a > 0$ is $x = 0$ .
What does the graph of $f (x) = \log_{a} x$ look like when $a > 1$ ?
The graph of $f (x) = \log_{a} x$ is increasing when $a > 1$ .
True or False?
$\sqrt{x} = x - 2$ has the same solutions as $x = {(x - 2)}^{2}$ .
False.
$\sqrt{x} = x - 2$ does not have the same solutions as $x = {(x - 2)}^{2}$ .
When you square both sides of an equation, you can introduce additional solutions.
$\sqrt{x} = x - 2$ has one solution, $x = 4$ , whereas $x = {(x - 2)}^{2}$ has two solutions, $x = 1$ or $x = 4$ .
What substitution could you use to turn $2 x^{6} + 3 x^{3} - 4 = 0$ into a quadratic equation?
You could use the substitution $y = x^{3}$ to turn $2 x^{6} + 3 x^{3} - 4 = 0$ into the quadratic equation $2 y^{2} + 3 y - 4 = 0$ .
True or False?
The first step to solving the equation $e^{x} + e^{- x} = 2$ , without a GDC, is to take natural logarithms of each term, $\ln (e^{x}) + \ln (e^{- x}) = \ln (2)$ .
False.
You can not take natural logarithms of each term of an equation, you can only take natural logarithms of both sides of the equation.
The first step to solving the equation $e^{x} + e^{- x} = 2$ is to transform it into a quadratic equation using the substitution $y = e^{x}$ .
How could you use a graph to solve the equation $f (x) = g (x)$ ?
To solve the equation $f (x) = g (x)$ , you could:
- sketch the graphs of $y = f (x)$ and $y = g (x)$ on your GDC,
- find the points of intersection,
- the $x$ -coordinates are the solutions to the equation.
How could you use the graph of $y = f (x) - g (x)$ to find the solutions to the equation $f (x) = g (x)$ ?
To find the solutions to the equation $f (x) = g (x)$ , you could find the x-intercepts (roots) of the graph of $y = f (x) - g (x)$ .
True or False?
The solutions to the equation $(x - 1) (x^{2} + 2) = (x - 1) (3 x + 6)$ are the same as the solutions to the equation $x^{2} + 2 = 3 x + 6$ .
False.
The solutions to the equation $(x - 1) (x^{2} + 2) = (x - 1) (3 x + 6)$ are not the same as the solutions to the equation $x^{2} + 2 = 3 x + 6$ .
The first equation also has a solution of $x = 1$ .
Dividing both sides of an equation by something that could equal zero can cause you to lose solutions.
What would be the first step to solving an equation such as $\log_{2} (x - 1) = 2 - \log_{2} (x + 2)$ without a GDC?
The first step to solving an equation such as $\log_{2} (x - 1) = 2 - \log_{2} (x + 2)$ would be to rearrange to get all the logarithms on one side of the equation such as $\log_{2} (x - 1) + \log_{2} (x + 2) = 2$ .
Once the equation is in that form, laws of logarithms can be used.
What does the domain represent in mathematical modelling?
In mathematical modelling, the domain represents the reasonable range of input values, considering the real-life context of the situation being modelled.
How can unknown parameters be found in mathematical models?
Unknown parameters in mathematical models can be found by forming equations, substituting given values and solving the equations simultaneously.
You will frequently use a graphing calculator (GDC) to help with this.
What is extrapolation in mathematical modelling?
Extrapolation is making predictions outside the range of the data.
True or False?
Extrapolation is an accurate method for predicting new values in mathematical modelling.
False.
Extrapolation is not considered to be an accurate method for predicting new values in mathematical modelling.
What is a key strategy used to overcome the limitation of a model only being accurate for a portion of the real-life situation?
A key strategy used to overcome the limitation of a model only being accurate for a portion of the real-life situation is to restrict the domain appropriately.
True or False?
Exponential models are commonly used for arithmetic sequences.
False.
Exponential models are not used for arithmetic sequences.
Linear models are used for arithmetic sequences, while exponential models are used for geometric sequences.
What type of model is typically used for projectile motion?
A quadratic model is typically used to model projectile motion.
How can you estimate outputs using a model?
You can estimate outputs by substituting values for the input variable into the model.
What does substituting t = 0 into a time-based model typically give you?
Substituting t = 0 into a time-based model typically gives you the initial value according to the model.