Converse

Clearly if a, b and c are the lengths of the sides of an equilateral triangle then a^2 + b^2 + c^2 = ab + bc + ca. Is the converse true?

Consecutive Squares

The squares of any 8 consecutive numbers can be arranged into two sets of four numbers with the same sum. True of false?

Parabolic Patterns

The illustration shows the graphs of fifteen functions. Two of them have equations y=x^2 and y=-(x-4)^2. Find the equations of all the other graphs.

Stage: 4 Challenge Level:

Muntej, from Wilson's School, sent us screenshots showing that for the first pair of functions, using the same input gave an output from $g$ which was twice the output from $f$.

Josh explained this by using  $f(x)$.

$f(x) = x^2 + 3x - 4$
$g(x) = 2x^2 +6x - 8 = 2(x^2 + 3x - 4)$
so $g(x) = 2f(x)$

He then went on to describe what happens to the graphs.

$2f(x)$ will be twice as large as $f(x)$ for any value of x including negative numbers, for example if $f(x) = -4$ then $2f(x) = -8$. However, the solutions for $y = 2f(x)$ will be the same as $y = f(x)$ because $2x0 = 0$. Thus, the graph $y = g(x)$ is $y=f(x)$ stretched in the $y$ axis by scale factor $2$.

Henry from Latymer School did the same for the second set of functions.

$f(x) = x^2 + 3x - 4$
$g(x) = (2x)^2 + 3(2x) - 4$
$x$ has essentially been doubled before being put into the function so $g(x) = f(2x)$.
$y=g(x)$ is $y=f(x)$ stretched by scale factor $0.5$ parallel to the $x$-axis.

Josh told us that you can work out the input numbers for which both machines give the same results by calculating $f(x) = g(x)$, which is the same as finding where the functions cross on a graph.He used algbra to find the solutions to the second functions.

The $x$ value for which $g(x) = f(x)$ is:

$4x^2 + 6x - 4 = x^2 + 3x - 4$
$3x^2 + 3x = 0$
$x^2 + x = 0$
$x(x+1) = 0$
so $x = 0$ or $x = -1$ because $0(0 +1) = 0$ and $-1(-1+1) = 0$

Ellen tried her own set of equations and used graphs to show the transformations.

$f(x) = x^2 +2, g(x) = 2f(x) = 2x^2 +4$

This is a stretch in the y-axis because the y coordinates are being affected.

$f(x) = x^2 - 4, g(x) = f(2x) = 4x^2 - 4$
This is a stretch in the x-axis because the x coordinates are being affected.

Now see if you can explain these findings further.
The text contains multiple mathematical questions and their corresponding answers. It discusses transformations of functions, specifically scaling in the y-axis and x-axis, and provides examples and solutions to illustrate these concepts.
multiple_questions_with_answers
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
Converse

Clearly if a, b and c are the lengths of the sides of an equilateral triangle then a^2 + b^2 + c^2 = ab + bc + ca. Is the converse true?

Consecutive Squares

The squares of any 8 consecutive numbers can be arranged into two sets of four numbers with the same sum. True of false?

Parabolic Patterns

The illustration shows the graphs of fifteen functions. Two of them have equations y=x^2 and y=-(x-4)^2. Find the equations of all the other graphs.

Stage: 4 Challenge Level:

Muntej, from Wilson's School, sent us screenshots showing that for the first pair of functions, using the same input gave an output from $g$ which was twice the output from $f$.

Josh explained this by using  $f(x)$.

$f(x) = x^2 + 3x - 4$
$g(x) = 2x^2 +6x - 8 = 2(x^2 + 3x - 4)$
so $g(x) = 2f(x)$

He then went on to describe what happens to the graphs.

$2f(x)$ will be twice as large as $f(x)$ for any value of x including negative numbers, for example if $f(x) = -4$ then $2f(x) = -8$. However, the solutions for $y = 2f(x)$ will be the same as $y = f(x)$ because $2x0 = 0$. Thus, the graph $y = g(x)$ is $y=f(x)$ stretched in the $y$ axis by scale factor $2$.

Henry from Latymer School did the same for the second set of functions.

$f(x) = x^2 + 3x - 4$
$g(x) = (2x)^2 + 3(2x) - 4$
$x$ has essentially been doubled before being put into the function so $g(x) = f(2x)$.
$y=g(x)$ is $y=f(x)$ stretched by scale factor $0.5$ parallel to the $x$-axis.

Josh told us that you can work out the input numbers for which both machines give the same results by calculating $f(x) = g(x)$, which is the same as finding where the functions cross on a graph.He used algbra to find the solutions to the second functions.

The $x$ value for which $g(x) = f(x)$ is:

$4x^2 + 6x - 4 = x^2 + 3x - 4$
$3x^2 + 3x = 0$
$x^2 + x = 0$
$x(x+1) = 0$
so $x = 0$ or $x = -1$ because $0(0 +1) = 0$ and $-1(-1+1) = 0$

Ellen tried her own set of equations and used graphs to show the transformations.

$f(x) = x^2 +2, g(x) = 2f(x) = 2x^2 +4$

This is a stretch in the y-axis because the y coordinates are being affected.

$f(x) = x^2 - 4, g(x) = f(2x) = 4x^2 - 4$
This is a stretch in the x-axis because the x coordinates are being affected.

Now see if you can explain these findings further.
user
The text contains multiple mathematical questions and their corresponding answers. It discusses transformations of functions, specifically scaling in the y-axis and x-axis, and provides examples and solutions to illustrate these concepts.

TYPE: MULTIPLE_QUESTIONS_WITH_ANSWERS