What is a quadratic graph's shape called, and what is its turning point called?

The shape is a parabola , a smooth curve with a vertical line of symmetry . Its turning point is called the vertex , which is a minimum on a u-shape and a maximum on an n-shape.

Define an asymptote .

An asymptote is a line that a curve gets closer and closer to but never touches. Asymptotes may be horizontal or vertical lines.

True or False? You should always use a ruler when plotting the graph of a function.

False. You should only use a ruler if a graph is linear (and for drawing the axes if they are not given). For curves, draw a single smooth freehand curve .

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Graphs of Functions (Edexcel IGCSE Maths B): Flashcards

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What is a quadratic graph's shape called, and what is its turning point called?

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What is a quadratic graph's shape called, and what is its turning point called?
The shape is a parabola, a smooth curve with a vertical line of symmetry.
Its turning point is called the vertex, which is a minimum on a u-shape and a maximum on an n-shape.
True or False?
A quadratic with a negative number in front of $x^{2}$ gives a u-shaped curve.
False.
A negative number in front of $x^{2}$ gives an n-shaped curve, which has a maximum point.
It is a positive number in front of $x^{2}$ that gives the u-shape, with a minimum point.
How many times can a quadratic graph cross each axis?
It crosses the $y$ -axis exactly once, whatever the quadratic is.
It crosses the $x$ -axis twice, once, or not at all, and those crossing points are the roots.
Fill in the coordinates of the vertex of this curve.
$y = a {(x - p)}^{2} + q has vertex (______,______)$
The completed statement is:
$y = a {(x - p)}^{2} + q has vertex (p, q)$
Watch the sign change on the $x$ -coordinate: the curve $y = {(x + 3)}^{2} + 2$ has its vertex at $(- 3, 2)$ .
True or False?
The curves $y = {(x - 3)}^{2} + 2$ and $y = 5 {(x - 3)}^{2} + 2$ have the same vertex.
True.
The number in front of the bracket changes the shape of the curve, not the position of its vertex.
Both have their vertex at $(3, 2)$ , and both are u-shaped because that number is positive.
A quadratic graph has roots at $x = 2$ and $x = 3$ and passes through $(0, 24)$ . How do you find its equation?
Use the form $y = a (x - x_{1}) (x - x_{2})$ to write $y = a (x - 2) (x - 3)$ , then substitute the other point into it.
That gives $24 = 6 a$ , so $a = 4$ and the equation is $y = 4 (x - 2) (x - 3)$ .
A quadratic graph has vertex $(9, - 16)$ and passes through $(2, 82)$ . How do you find its equation?
Use the form $y = a {(x - p)}^{2} + q$ to write $y = a {(x - 9)}^{2} - 16$ , then substitute the other point into it.
That gives $82 = 49 a - 16$ , so $a = 2$ and the equation is $y = 2 {(x - 9)}^{2} - 16$ .
When is the vertex alone enough to find a quadratic's equation?
Only when you already know that $a = 1$ , so the equation is simply $y = {(x - p)}^{2} + q$ .
Otherwise a second point on the curve is needed as well, to pin down the value of $a$ .
Define a cubic.
A cubic is a function of the form $a x^{3} + b x^{2} + c x + d$ , which is a polynomial of degree 3.
The constants $b$ , $c$ and $d$ may each be zero, but $a$ cannot be zero.
True or False?
A cubic whose $x^{3}$ coefficient is negative starts in the bottom left and ends in the top right.
False.
That describes a positive cubic, one where the coefficient of $x^{3}$ is greater than zero.
A negative cubic runs the other way, starting in the top left and ending in the bottom right.
How many turning points can a cubic graph have?
Up to two, one maximum and one minimum.
Some cubics have none at all: $y = x^{3}$ and $y = - x^{3}$ have no maximum or minimum, and each meets the $x$ -axis only once, at $x = 0$ .
Why does a cubic need to be in factorised form before you can sketch it?
Because the factors give you the roots directly, and the roots fix where the curve meets the $x$ -axis.
Simple cubics factorise at once, such as $x^{3} - 4 x = x (x - 2) (x + 2)$ , while harder ones need the factor theorem and algebraic division.
What are the roots of $y = (x - 2) (x - 3) (x + 5)$ ?
The roots are $x = 2$ , $x = 3$ and $x = - 5$ .
A cubic written $(x - p) (x - q) (x - r)$ has roots at $p$ , $q$ and $r$ , so each one is read off by changing the sign inside its own bracket.
True or False?
The graph of $y = {(x - 2)}^{2} (x + 1)$ crosses the $x$ -axis at $x = 2$ .
False.
A repeated root makes the graph touch the $x$ -axis and turn back, rather than cross it, so at $x = 2$ it touches.
It does cross at $x = - 1$ , where the root is not repeated.
A positive cubic touches the $x$ -axis at $x = 3$ and crosses it at $x = \frac{1}{2}$ . Where must its two turning points lie?
One is a minimum at $x = 3$ itself, because the curve touches the axis there and turns back upwards.
The other is a maximum somewhere between the two roots, so between $x = \frac{1}{2}$ and $x = 3$ .
What form does a reciprocal graph take?
A reciprocal graph has the form $y = \frac{a}{x}$ or $y = \frac{a}{x^{2}}$ .
The two simplest are $y = \frac{1}{x}$ and $y = \frac{1}{x^{2}}$ , where $a = 1$ .
True or False?
The graph of $y = \frac{3}{x}$ never crosses either axis.
True.
It has no roots and no $y$ -intercept, so it meets neither axis anywhere.
Instead it approaches both axes without ever reaching them, which is what its two asymptotes describe.
Define an asymptote.
An asymptote is a line that a curve gets closer and closer to but never touches.
Asymptotes may be horizontal or vertical lines.
Why does $y = \frac{1}{x}$ have a vertical asymptote at $x = 0$ ?
Because $x = 0$ would mean dividing by zero, which has no value, so the curve can never reach that line.
The horizontal asymptote $y = 0$ arises the other way round: as $x$ grows large in either direction, $\frac{1}{x}$ shrinks towards zero without ever getting there.
Why do both branches of $y = \frac{1}{x^{2}}$ lie above the $x$ -axis?
Because $x^{2}$ is positive for every non-zero value of $x$ , whether $x$ itself is positive or negative.
That is what makes it differ from $y = \frac{1}{x}$ , whose two branches sit on opposite sides of the axis.
In $y = \frac{a}{x}$ , what do the sign and the size of $a$ each change?
The sign of $a$ decides where the two branches sit.
A positive $a$ puts them in the top-right and bottom-left, and a negative $a$ puts them in the top-left and bottom-right.
The size of $a$ decides how steep they are, and the closer $a$ is to zero the more L-shaped they become.
Fill in the equations of the two asymptotes of $y = \frac{5}{x} + 6$ .
$horizontal: y =______$
$vertical: x =______$
The completed equations are:
$horizontal: y = 6$
$vertical: x = 0$
Adding 6 shifts the whole curve up by 6 and takes the horizontal asymptote with it, while the vertical asymptote stays on the $y$ -axis.
True or False?
You should always use a ruler when plotting the graph of a function.
False.
You should only use a ruler if a graph is linear (and for drawing the axes if they are not given).
For curves, draw a single smooth freehand curve.
How would you find the y-intercept of a graph using its equation?
To find the y-intercept of a graph, you would substitute $x = 0$ into the equation.
True or False?
The solutions to $x^{3} - 4 x = 0$ are the value(s) where the graph of $y = x^{3} - 4 x$ crosses the y-axis.
False.
The solutions to $x^{3} - 4 x = 0$ are not the value(s) where the graph of $y = x^{3} - 4 x$ crosses the y-axis.
$x^{3} - 4 x = 0$ when $y = 0$ which is the x-axis. Therefore the solutions are the values where the graph crosses the x-axis.
The solutions of $x^{3} + x^{2} - 3 = x - 2$ are the x values of the intersections between $y = x^{3} + x^{2} - 3$ and which other graph?
The solutions of $x^{3} + x^{2} - 3 = x - 2$ are the x values of the intersections between $y = x^{3} + x^{2} - 3$ and $y = x - 2$ .
True or False?
The x values of the intersections of the two graphs $y = x + 1$ and $y = x^{2} + 5 x + 4$ are the solutions of $x^{2} + 4 x + 3 = 0$ .
True.
The x values of the intersections of the two graphs $y = x + 1$ and $y = x^{2} + 5 x + 4$ are the solutions of $x^{2} + 4 x + 3 = 0$ .
Set the equations equal to each other and rearrange: $x^{2} + 5 x + 4 = x + 1$ .

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