What is the difference between a sketch of a graph and a drawing of one?

A sketch shows the correct general shape and the key points, but need not be to scale or plotted precisely. A drawing is accurate: points are plotted exactly, straight lines are ruled, and the whole curve is to scale.

Define local maximum and local minimum .

A local maximum is a point higher than everything immediately around it, and a local minimum is a point lower than everything around it. The word local matters: neither has to be the highest or lowest point on the whole graph.

An asymptote is a line that a graph gets steadily closer to without ever meeting it. Asymptotes may be horizontal or vertical , and are usually shown as dashed lines on a sketch.

True or False? A horizontal line can cross a linear rational graph at most once.

True. A linear rational function takes each of its values just once, so no horizontal line can meet the curve twice. The single horizontal line that misses the graph entirely is its horizontal asymptote.

When would you solve an equation graphically, and what is the cost?

When the equation cannot be rearranged into anything algebra can solve, which happens readily once logarithms and exponentials appear together. The cost is accuracy: the answer is only an estimate , no better than the drawing and the reading of the scale.

True or False? A horizontal line is always the right line to draw when solving an equation from a graph.

False. A horizontal line only works when the curve already plotted is of the whole expression you are setting equal to a number. Otherwise the equation has to be rearranged first, and the line that comes out of that is usually a sloping one.

How do you work out which straight line to draw on a given graph?

Rearrange the equation you are solving until one whole side is exactly the function already plotted. Whatever is left on the other side is the line to draw, and the crossing points then give the solutions you want.

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Graphs (Edexcel IGCSE Further Pure Maths): Flashcards

Exam code: 4PM1

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FrontGraphing Functions
Complete the condition for the point $(a, b)$ to lie on the graph of $y = f (x)$ :
$f (______) =______$

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Cards in this collection (27)

Complete the condition for the point $(a, b)$ to lie on the graph of $y = f (x)$ :
$f (______) =______$
The completed condition is:
$f (a) = b$
The first coordinate is the input and the second the output, which is why the horizontal axis carries the domain and the vertical axis the range.
What is the difference between a sketch of a graph and a drawing of one?
A sketch shows the correct general shape and the key points, but need not be to scale or plotted precisely.
A drawing is accurate: points are plotted exactly, straight lines are ruled, and the whole curve is to scale.
How do you find the intercepts of $y = f (x)$ with the two axes?
Put $x = 0$ to get the $y$ -intercept, and set $y = 0$ and solve to get the $x$ -intercepts.
The $x$ -intercepts are also called the roots of the function, and there may be none, one or many of them.
Define local maximum and local minimum.
A local maximum is a point higher than everything immediately around it, and a local minimum is a point lower than everything around it.
The word local matters: neither has to be the highest or lowest point on the whole graph.
Where on the graph of $y = f (x)$ do turning points occur?
Wherever the gradient is zero, that is where $f' (x) = 0$ .
They are the peaks and valleys of the curve, and a graph may have several of them or none at all.
Define asymptote.
An asymptote is a line that a graph gets steadily closer to without ever meeting it.
Asymptotes may be horizontal or vertical, and are usually shown as dashed lines on a sketch.
True or False?
Every graph with a vertical asymptote also has a horizontal one.
False.
A logarithmic graph has a vertical asymptote at the $y$ -axis but climbs without limit, so it has no horizontal asymptote at all.
The two kinds are independent: a curve may have one, the other, both, or neither.
Where is the axis of symmetry of $y = a x^{2} + b x + c$ ?
It is the vertical line $x = - \frac{b}{2 a}$ , which also passes through the vertex.
When the curve has two roots $p$ and $q$ , that line sits exactly halfway between them, at $x = \frac{p + q}{2}$ .
How does the sign of $a$ affect the graph of $y = a x^{2} + b x + c$ ?
A positive $a$ gives a U-shaped parabola opening upwards, and a negative $a$ turns it upside down.
The size of $a$ also stretches the curve vertically, making it narrower as $a$ gets larger.
True or False?
The vertex of a quadratic graph is always its $y$ -intercept.
False.
The $y$ -intercept is at $(0, c)$ while the vertex sits on the line $x = - \frac{b}{2 a}$ , and the two agree only when $b = 0$ .
In that special case the curve is symmetric about the $y$ -axis, so the two points really do coincide.
Which form of a quadratic would you use if you know its two roots?
The factorised form $y = a (x - p) (x - q)$ , which shows the roots $p$ and $q$ directly.
One further point on the curve is still needed to pin down $a$ , since every value of $a$ gives that same pair of roots.
How does the sign of $a$ affect the ends of the graph of $y = a x^{3} + b x^{2} + c x + d$ ?
A positive cubic comes up from the bottom left and goes off towards the top right.
A negative cubic does the reverse, starting at the top left and falling away to the bottom right.
For $y = a x^{3} + b x^{2} + c x + d$ , complete the equation that locates the turning points:
$______a x^{2} +______b x + c = 0$
The completed equation is:
$3 a x^{2} + 2 b x + c = 0$
It is the derivative set equal to zero, so a cubic has two turning points when this quadratic has two unique solutions, and none otherwise — a repeated solution gives a point of inflection, not a turning point.
What is the first thing to find when sketching a cubic graph?
The $y$ -intercept, at $(0, d)$ , because putting $x = 0$ leaves only the constant term.
Then find the roots by setting $y = 0$ and factorising, and settle the direction of the ends from the sign of $a$ .
Define linear rational function.
A linear rational function has the form $f (x) = \frac{a x + b}{c x + d}$ , one linear expression divided by another.
The reciprocal function $f (x) = \frac{1}{x}$ is the simplest case of it.
Where is the vertical asymptote of $f (x) = \frac{a x + b}{c x + d}$ ?
At $x = - \frac{d}{c}$ , the value that makes the denominator zero.
That same value is the one number missing from the domain, since the function has no value there at all.
Where is the horizontal asymptote of $f (x) = \frac{a x + b}{c x + d}$ ?
At $y = \frac{a}{c}$ , the ratio of the two coefficients of $x$ .
Once $x$ is very large, positive or negative, the constants $b$ and $d$ barely matter and the function behaves like $\frac{a x}{c x}$ .
For $f (x) = \frac{a x + b}{c x + d}$ , complete the two axis intercepts:
$(0,______) and (______, 0)$
The completed intercepts are:
$(0, \frac{b}{d}) and (- \frac{b}{a}, 0)$
The first comes from putting $x = 0$ , the second from setting the numerator to zero, since a fraction is zero exactly when its top is.
Why can $\frac{a x + b}{c x + d}$ never equal $\frac{a}{c}$ ?
Because $y = \frac{a}{c}$ is the horizontal asymptote, which the curve approaches but never reaches.
So the range is every real number except $\frac{a}{c}$ , just as the domain leaves out one value.
True or False?
A horizontal line can cross a linear rational graph at most once.
True.
A linear rational function takes each of its values just once, so no horizontal line can meet the curve twice.
The single horizontal line that misses the graph entirely is its horizontal asymptote.
Find both asymptotes of $f (x) = \frac{10 - 5 x}{x + 2}$ .
The denominator is zero at $x = - 2$ , so that is the vertical asymptote.
For large $x$ the function behaves like $\frac{- 5 x}{x} = - 5$ , so the horizontal asymptote is $y = - 5$ .
Complete the rule for solving $f (x) = g (x)$ from a graph:
Draw both curves on one set of axes; the solutions are the $______$ coordinates of the points of $______$ between them.
The completed rule is:
Draw both curves on one set of axes; the solutions are the $x$ coordinates of the points of intersection between them.
At an intersection the two expressions take the same value, which is exactly what the equation is asking for.
When would you solve an equation graphically, and what is the cost?
When the equation cannot be rearranged into anything algebra can solve, which happens readily once logarithms and exponentials appear together.
The cost is accuracy: the answer is only an estimate, no better than the drawing and the reading of the scale.
Given a graph of $y = f (x)$ , how do you solve $f (x) = k$ ?
Draw the horizontal line $y = k$ and read off the $x$ coordinates wherever it meets the curve.
Finding roots is the special case $k = 0$ , where that horizontal line is the $x$ -axis itself.
True or False?
A horizontal line is always the right line to draw when solving an equation from a graph.
False.
A horizontal line only works when the curve already plotted is of the whole expression you are setting equal to a number.
Otherwise the equation has to be rearranged first, and the line that comes out of that is usually a sloping one.
How do you work out which straight line to draw on a given graph?
Rearrange the equation you are solving until one whole side is exactly the function already plotted.
Whatever is left on the other side is the line to draw, and the crossing points then give the solutions you want.
Which line should be drawn on the graph of $y = 2^{(\frac{x}{3} + 1)} - 1$ to solve $\log_{2} {(3 x - 1)}^{2} - \frac{2}{3} x = 2$ ?
Rearranging gives $\log_{2} (3 x - 1) = \frac{x}{3} + 1$ , then $3 x - 1 = 2^{(\frac{x}{3} + 1)}$ , and finally $3 x - 2 = 2^{(\frac{x}{3} + 1)} - 1$ .
The right-hand side is now the plotted curve, so the line to draw is $y = 3 x - 2$ .

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