Do you need the exact turning points in order to sketch a polynomial?

No, a sketch only needs them roughly placed, so that the overall shape is right. Exact coordinates require differentiation , which is a separate technique.

True or False? A sketch reliably tells you how many times two graphs intersect.

False. A sketch can easily miss a crossing, or suggest one that is not really there, especially where two curves pass close together. Only working algebraically settles the number for certain.

Why use graphs and algebra together rather than either alone?

The algebra gives the exact values, while the graph shows what the situation looks like and whether an answer is plausible. A sketch on its own is not precise enough, and algebra on its own gives no picture at all.

A graph is given to you rather than asked for. How do you use it to solve the equations?

Read the coordinates of the intersection points off the axes. Values read from a graph are approximate, so give them only to the accuracy the scale can support.

Define mathematical model .

A mathematical model simplifies a real-world situation so that it can be described using mathematics, and then used to make predictions. The path of a stone thrown from a cliff top and the number of toys a factory can produce in a day are both things a model can describe.

Why are assumptions made when a real-world situation is turned into a mathematical model?

To simplify the situation enough for the mathematics to be manageable. Ignoring air resistance on a thrown stone, or treating a factory's machines as producing at a constant rate, is what makes a workable equation possible.

True or False? When a reciprocal function models a real situation, both branches of its graph are usually relevant.

False. A modelled quantity such as a time, a length or a volume cannot be negative, so usually only the positive branch describes the situation. The model therefore comes with a restriction on the values its variable is allowed to take.

What does it mean to refine a mathematical model?

It means improving it, either because further information has become available or because its predictions have been compared with real-world data. A thrown-stone model might be refined by taking the stone's mass into account, and a factory model by allowing thirty minutes a day for machine repairs.

Where should you look in a model to find a value of the variable for which it breaks down?

At the extremes : what the model does at or near zero, and what it does for very large values of the variable. It is also worth checking whether negative values appear where every quantity in the situation should be positive.

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Graphs of Functions (Edexcel International A Level (IAL) Maths: Pure 1): Flashcards

Exam code: YMA01

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How many $x$ -axis intercepts must an odd-degree polynomial have?

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Cards in this collection (31)

How many $x$ -axis intercepts must an odd-degree polynomial have?
At least one.
Its two ends head off in opposite directions, so the curve has to cross the axis somewhere between them.
True or False?
Every polynomial graph crosses the $x$ -axis at least once.
False.
An even-degree polynomial need not meet the $x$ -axis at all, in the same way that some quadratics have no real roots.
A positive cubic graph starts in the $______$ left and ends in the $______$ right.
A positive cubic graph starts in the bottom left and ends in the top right.
As $x$ becomes large and negative so does $y$ , and as $x$ becomes large and positive so does $y$ .
What does a repeated root tell you about a polynomial's graph?
The curve touches the $x$ -axis there rather than crossing it.
The exception is a point of inflection, where the curve flattens out and still passes through.
Do you need the exact turning points in order to sketch a polynomial?
No, a sketch only needs them roughly placed, so that the overall shape is right.
Exact coordinates require differentiation, which is a separate technique.
Why is a polynomial graph always drawn as one smooth curve?
Because a polynomial is defined for every value of $x$ , with no gaps and no sharp corners.
That is what separates it from a graph such as $y = \frac{1}{x}$ , which breaks at $x = 0$ .
What are the two basic reciprocal graphs to know?
$y = \frac{1}{x}$ and $y = \frac{1}{x^{2}}$ .
The second is always positive, so both of its branches sit above the $x$ -axis.
What are the asymptotes of $y = \frac{a}{x}$ ?
The two axes: $x = 0$ vertically and $y = 0$ horizontally.
They are drawn with dotted lines, because the curve approaches them without ever reaching them.
True or False?
The graph of $y = \frac{a}{x}$ never crosses either axis.
True.
No value of $x$ makes $y$ equal to zero, and $x = 0$ gives no value of $y$ at all.
A transformation of the graph can move it so that it does cross an axis.
How does the sign of $a$ affect the graph of $y = \frac{a}{x}$ ?
With $a > 0$ the two branches sit in the top-right and bottom-left.
With $a < 0$ they sit in the top-left and bottom-right instead.
How does the size of $a$ affect the graph of $y = \frac{a}{x}$ ?
It controls how steep the curves are.
The closer $a$ is to zero, the more tightly the curve hugs the axes and the more L-shaped it looks.
When sketching a reciprocal graph, label the points where $x =______$ and $x =______$ to give a sense of scale.
When sketching a reciprocal graph, label the points where $x = 1$ and $x = - 1$ to give a sense of scale.
Without a labelled point the sketch shows the shape but says nothing about size.
How can you use graphs to solve the equation $x^{2} + 3 x + 1 = 2 x + 1$ ?
Draw $y = x^{2} + 3 x + 1$ and $y = 2 x + 1$ on the same axes.
The $x$ -coordinates of the points where the curve and the line cross are the solutions of the equation.
True or False?
A sketch reliably tells you how many times two graphs intersect.
False.
A sketch can easily miss a crossing, or suggest one that is not really there, especially where two curves pass close together.
Only working algebraically settles the number for certain.
Why use graphs and algebra together rather than either alone?
The algebra gives the exact values, while the graph shows what the situation looks like and whether an answer is plausible.
A sketch on its own is not precise enough, and algebra on its own gives no picture at all.
The graphs of $y = x^{2} + 3 x + 1$ and $y = 2 x + 1$ meet where $x^{2} + 3 x + 1 = 2 x + 1$ , which simplifies to:
$x^{2} +______= 0$
$x^{2} + x = 0$
Factorising gives $x (x + 1) = 0$ , so the curve and the line meet where $x = 0$ and $x = - 1$ .
A graph is given to you rather than asked for. How do you use it to solve the equations?
Read the coordinates of the intersection points off the axes.
Values read from a graph are approximate, so give them only to the accuracy the scale can support.
Define constant of proportionality.
The constant of proportionality is the fixed number $k$ that turns a proportionality statement into an equation.
Once found, it holds for every pair of values in the relationship, which is what lets the relationship be used to predict.
Complete the two standard proportional forms, where $k$ is a constant:
$y \propto x \Rightarrow y =______$
$y \propto \frac{1}{x} \Rightarrow y =______$
The completed forms are:
$y \propto x \Rightarrow y = k x$
$y \propto \frac{1}{x} \Rightarrow y = \frac{k}{x}$
In the first, the two variables rise and fall together; in the second, one rises as the other falls.
$y$ is inversely proportional to the cube of $x$ . How do you write that as an equation?
It becomes $y \propto \frac{1}{x^{3}}$ and then $y = \frac{k}{x^{3}}$ .
The word inversely is what puts the expression in the denominator, and the cube of $x$ is what makes that expression $x^{3}$ .
True or False?
If $y$ is inversely proportional to $x$ , then $x y$ is the same for every pair of values.
True.
The constant of proportionality is exactly that product, so multiplying the two variables together always returns it.
Direct proportion works the other way round: there it is the ratio $\frac{y}{x}$ that stays the same.
A relationship is modelled by $y = \frac{k}{x^{2}}$ , and $y = 3$ when $x = 2$ . How do you find $k$ , and what is its value?
Substitute the known pair of values into the equation and solve: $3 = \frac{k}{2^{2}}$ .
That gives $k = 3 \times 4 = 12$ , so the relationship is $y = \frac{12}{x^{2}}$ .
If $y$ is proportional to $x + 4$ , what is the equation of the relationship?
$y = k (x + 4)$ , with the whole bracket multiplied by the constant.
Whatever is named after proportional to is treated as a single quantity, so the bracket cannot be dropped.
A graph of $y$ against $x$ is a straight line. What extra condition makes $y$ proportional to $x$ ?
The line must pass through the origin.
For $y = m x + c$ it is $y - c$ that is directly proportional to $x$ , so $y$ itself is proportional to $x$ only when $c = 0$ .
Define mathematical model.
A mathematical model simplifies a real-world situation so that it can be described using mathematics, and then used to make predictions.
The path of a stone thrown from a cliff top and the number of toys a factory can produce in a day are both things a model can describe.
Why are assumptions made when a real-world situation is turned into a mathematical model?
To simplify the situation enough for the mathematics to be manageable.
Ignoring air resistance on a thrown stone, or treating a factory's machines as producing at a constant rate, is what makes a workable equation possible.
True or False?
When a reciprocal function models a real situation, both branches of its graph are usually relevant.
False.
A modelled quantity such as a time, a length or a volume cannot be negative, so usually only the positive branch describes the situation.
The model therefore comes with a restriction on the values its variable is allowed to take.
What does it mean to refine a mathematical model?
It means improving it, either because further information has become available or because its predictions have been compared with real-world data.
A thrown-stone model might be refined by taking the stone's mass into account, and a factory model by allowing thirty minutes a day for machine repairs.
A model is $f (d) = \frac{L}{d + 1}$ for $d \geq 0$ , where $d$ is the number of days since the start. What does the constant $L$ represent?
The value of the modelled quantity at the start, when $d = 0$ .
Substituting $d = 0$ gives $f (0) = \frac{L}{1} = L$ , which is how you interpret any constant in a model: put in the starting value of the variable.
A gas model gives the pressure $P$ pascals in a container of volume $V$ cubic metres as $P = \frac{1750000}{V}$ . If $P$ must stay below $120000$ , what happens to the inequality sign when you rearrange to find $V$ ?
It reverses: $\frac{1750000}{V} < 120000$ becomes $V > \frac{1750000}{120000}$ .
Making the pressure smaller means making the volume larger, so an upper limit on $P$ sets a lower limit on $V$ .
Where should you look in a model to find a value of the variable for which it breaks down?
At the extremes: what the model does at or near zero, and what it does for very large values of the variable.
It is also worth checking whether negative values appear where every quantity in the situation should be positive.

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