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.

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

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

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Cards in this collection (24)

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.
Direct proportion is written $y \propto x$ , giving $y =______$ . Inverse proportion is written $y \propto \frac{1}{x}$ , giving $y =______$ .
Direct proportion gives $y = k x$ , and inverse proportion gives $y = \frac{k}{x}$ .
The symbol $\propto$ means "is proportional to", and replacing it with $= k$ is always the first move.
Define constant of proportionality.
The fixed number $k$ that turns a statement of proportion into an equation.
Writing $y \propto x$ says only that the relationship holds; $y = k x$ says by how much.
For $y = k x$ and $y = \frac{k}{x}$ , what quantity stays constant in each case?
Under direct proportion the ratio $\frac{y}{x}$ is constant.
Under inverse proportion it is the product $x y$ that is constant.
How do you write " $y$ is inversely proportional to the square of $x$ " as an equation?
As $y = \frac{k}{x^{2}}$ , since $y \propto \frac{1}{x^{2}}$ .
Be precise about what $y$ is proportional to: "the square of $x$ " gives $x^{2}$ , while " $x$ plus two" would give $(x + 2)$ .
What do the graphs of direct and inverse proportion look like?
Direct proportion gives a straight line through the origin, with $k$ as its gradient.
Inverse proportion gives a reciprocal curve.
How do you find the constant of proportionality?
Substitute a known pair of values into the equation and solve for $k$ .
Once $k$ is known, the equation gives you any other pair you need.
True or False?
If $y = m x + c$ with $c \neq 0$ , then $y$ is directly proportional to $x$ .
False.
Direct proportion means $y = k x$ , whose graph passes through the origin.
Rearranging to $y - c = m x$ shows that it is $(y - c)$ , not $y$ , that is directly proportional to $x$ .

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