True or False? A curved graph drawn from a table of values should be joined up using a ruler.

False. A curve must be drawn as a single smooth freehand curve passing through all the plotted points. A ruler is only for straight-line graphs, and using one on a curve produces a series of straight segments instead.

How accurately should each point be plotted on the grid?

Mark each point with a cross , positioned to within half of the smallest square on the grid. A point plotted more roughly than that can pull the whole curve out of shape.

Define a root of an equation.

A root is a value of the unknown that makes both sides of the equation equal. It is another word for a solution .

Why must both graphs be drawn on the same axes before you can use their intersection?

Because an intersection is a point lying on both graphs, and that only means anything if the two share the same scales and origin. On separate axes the same pair of coordinates would refer to two different positions.

True or False? A curve drawn from one equation can be used to solve other equations as well.

True. The curve shows every value that its expression takes, so any equation asking when that expression equals something can be answered by drawing the matching line. Only the line changes, never the curve.

Why can a solution read off a graph only ever be an estimate?

Because the answer is judged by eye against a printed scale rather than calculated exactly. Round it to an accuracy that suits the scale, typically no more than two decimal places.

IGCSEMathsCambridge (CIE)CoreFlashcardsCoordinate Geometry & GraphsFurther Graphs

Further Graphs (Cambridge (CIE) IGCSE Maths: Core): Flashcards

Exam code: 0580 & 0980

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FrontTypes of Graphs
Define a parabola.

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Cards in this collection (23)

Define a parabola.
A parabola is the smooth symmetrical curve of a quadratic graph, $y = a x^{2} + b x + c$ .
It is the shape you get whether the curve opens upwards or downwards.
Which part of an equation tells you whether its graph is a straight line, a parabola or a reciprocal curve?
Look at how $x$ appears: on its own it gives a straight line, squared it gives a quadratic curve, and in the denominator it gives a reciprocal graph.
So $y = 3 x + 2$ , $y = 2 x^{2} + 3 x + 4$ and $y = \frac{4}{x}$ are one of each.
Complete the two equation forms, putting the correct variable in each gap.
A horizontal line has an equation of the form $______= c$ and a vertical line has an equation of the form $______= k$ instead.
The completed forms are:
A horizontal line has an equation of the form $y = c$ and a vertical line has an equation of the form $x = k$ instead.
Every point on a horizontal line shares the same $y$ value, and every point on a vertical line shares the same $x$ value.
How can you tell from its equation whether a quadratic graph is a u-shape or an n-shape?
Look at the number in front of $x^{2}$ : positive gives a u-shape and negative gives an n-shape.
So $y = 2 x^{2} + 3 x + 4$ is a u-shape, while $y = - 3 x^{2} + 2 x + 4$ is an n-shape.
True or False?
Every quadratic graph crosses the $x$ -axis.
False.
A quadratic graph may cross the $x$ -axis twice, just touch it once, or miss it altogether, so it can have two roots, one root or none at all.
What kind of symmetry does every quadratic graph have?
A vertical line of symmetry down its middle, with an equation of the form $x = k$ .
The value of $k$ is the $x$ -coordinate of the point where the curve turns, and the roots sit symmetrically either side of it.
How many times does a quadratic graph cross the $y$ -axis?
Exactly once, so a quadratic always has one $y$ -intercept.
Substituting $x = 0$ gives exactly one value of $y$ , so there is only ever one crossing point.
What does the graph of $y = \frac{a}{x}$ look like?
It has two separate L-shaped branches, one in each of two opposite quadrants, and they never join up.
The L becomes more rectangular as $a$ gets smaller.
True or False?
Changing the sign of $a$ in $y = \frac{a}{x}$ moves both branches into the other pair of quadrants.
True.
For a positive $a$ both branches sit where $x$ and $y$ have the same sign, and for a negative $a$ they sit where the two signs are opposite.
How do you complete a table of values for the graph of $y = 10 - 8 x^{2}$ ?
Substitute each $x$ value from the table into the equation to work out the matching $y$ value.
For $x = - 1.5$ this gives $y = 10 - 8 \times 2.25 = - 8$ .
Complete two more values from the table for $y = 10 - 8 x^{2}$ :
When $x = 0$ , $y =______$ .
When $x = 1$ , $y =______$ .
The completed values are:
When $x = 0$ , $y = 10$ .
When $x = 1$ , $y = 2$ .
True or False?
A curved graph drawn from a table of values should be joined up using a ruler.
False.
A curve must be drawn as a single smooth freehand curve passing through all the plotted points.
A ruler is only for straight-line graphs, and using one on a curve produces a series of straight segments instead.
How accurately should each point be plotted on the grid?
Mark each point with a cross, positioned to within half of the smallest square on the grid.
A point plotted more roughly than that can pull the whole curve out of shape.
When making a table of values for $y = \frac{1}{x}$ , why is there no $y$ value to write in the $x = 0$ column?
Because $1$ divided by $0$ has no value: division by zero is not allowed, and a calculator returns an error.
Leave that column blank and plot no point there.
Define a root of an equation.
A root is a value of the unknown that makes both sides of the equation equal.
It is another word for a solution.
Why must both graphs be drawn on the same axes before you can use their intersection?
Because an intersection is a point lying on both graphs, and that only means anything if the two share the same scales and origin.
On separate axes the same pair of coordinates would refer to two different positions.
How do you use the graph of $y = x^{2} - 4 x - 2$ to solve $x^{2} - 4 x - 2 = 0$ ?
Read off the $x$ -coordinates where the curve crosses the horizontal axis.
Setting the expression equal to zero is the same as asking where $y = 0$ , which is exactly that axis.
Using the graph of $y = x^{2} - 4 x - 2$ , how do you solve $x^{2} - 4 x - 2 = 5$ ?
Draw the horizontal line $y = 5$ and read off the $x$ -coordinates where it crosses the curve.
Whatever number sits on the right-hand side, draw the horizontal line at that height.
True or False?
The solutions of $10 - 8 x^{2} = 8$ are the two points where the line $y = 8$ meets the curve.
False.
The solutions are the two $x$ -coordinates of those points rather than the points themselves, because the equation asks only for values of $x$ .
Here they are $x = - 0.5$ and $x = 0.5$ .
Using the graph of $y = x^{2} - 4 x - 2$ , how do you solve $x^{2} - 4 x - 2 = x + 1$ ?
Plot the straight line $y = x + 1$ on the same axes, then read off the $x$ -coordinates where it crosses the curve.
The right-hand side is no longer a constant, so the line you draw slopes instead of being horizontal.
The graph of $y = x^{2} - 4 x - 2$ has been drawn. Complete the rearrangement needed before you can use it to solve $x^{2} - 4 x + 3 = 1$ .
$x^{2} - 4 x + 3 = 1 \Rightarrow x^{2} - 4 x - 2 =______$
The completed rearrangement is:
$x^{2} - 4 x + 3 = 1 \Rightarrow x^{2} - 4 x - 2 = - 4$
Both sides have had $5$ subtracted, so the horizontal line to draw is $y = - 4$ .
True or False?
A curve drawn from one equation can be used to solve other equations as well.
True.
The curve shows every value that its expression takes, so any equation asking when that expression equals something can be answered by drawing the matching line.
Only the line changes, never the curve.
Why can a solution read off a graph only ever be an estimate?
Because the answer is judged by eye against a printed scale rather than calculated exactly.
Round it to an accuracy that suits the scale, typically no more than two decimal places.

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