text: 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.
distilabel_metadata.content: Evaluate the user given text extract and determine if it contains a math question and optionally and answer to this question.
  
  Output rules:
  
  1. Briefly describe the document and mention if it contains a question or a question and answer.
  2. Finalize the output with: Label: no-question,question,question-and-answer,multiple-question,multiple-question-and-answer
distilabel_metadata.role: system
distilabel_metadata.content: 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.
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The provided text extract contains multiple math questions and their corresponding answers. Here's a brief description:
  
  1. **Converse**: The text asks if the converse of the statement "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 true. It does not provide an answer but sets up the question.
  
  2. **Consecutive Squares**: The text asks whether the squares of any 8 consecutive numbers can be arranged into two sets of four numbers with the same sum. It does not provide an answer but sets up the question.
  
  3. **Parabolic Patterns**: The text asks for the equations of the other graphs given the equations \(y = x^2\) and \(y = -(x-4)^2\). It does not provide an answer but sets up the question.
  
  4. **Function Transformations**: The text provides a detailed explanation of the transformations of functions \(f(x)\) and \(g(x)\) and includes answers to specific problems:
     - It explains that \(g(x) = 2f(x)\) results in a vertical stretch by a factor of 2.
     - It explains that