Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "2021 Fall AMC 10A Problems/Problem 16"

(→‎Solution 2 (Graphing))
(→‎Solution 2 (Graphing))
Line 88: Line 88:
 
</asy>
 
</asy>
 
Note that <math>y_2</math> is a reflection of <math>y_1</math> about the <math>y</math>-axis, followed by a translation of <math>1</math> unit right. The graph of <math>y_2</math> is shown below:
 
Note that <math>y_2</math> is a reflection of <math>y_1</math> about the <math>y</math>-axis, followed by a translation of <math>1</math> unit right. The graph of <math>y_2</math> is shown below:
 +
<asy>
 +
/* Made by MRENTHUSIASM */
 +
size(250);
  
−
<b>DIAGRAM WILL BE READY VERY SOON. APPRECIATE IT IF THERE'S NO EDIT AT THIS PAGE WITHIN THE NEXT HOUR.</b>
+
int xMin = -10;
 +
int xMax = 10;
 +
int yMin = -10;
 +
int yMax = 10;
  
 +
//Draws the horizontal gridlines
 +
void horizontalLines()
 +
{
 +
  for (int i = yMin+1; i < yMax; ++i)
 +
  {
 +
    draw((xMin,i)--(xMax,i), mediumgray+linewidth(0.4));
 +
  }
 +
}
 +
 +
//Draws the vertical gridlines
 +
void verticalLines()
 +
{
 +
  for (int i = xMin+1; i < xMax; ++i)
 +
  {
 +
    draw((i,yMin)--(i,yMax), mediumgray+linewidth(0.4));
 +
  }
 +
}
 +
 +
//Draws the horizontal ticks
 +
void horizontalTicks()
 +
{
 +
  for (int i = yMin+1; i < yMax; ++i)
 +
  {
 +
    draw((-3/16,i)--(3/16,i), black+linewidth(1));
 +
  }
 +
}
 +
 +
//Draws the vertical ticks
 +
void verticalTicks()
 +
{
 +
  for (int i = xMin+1; i < xMax; ++i)
 +
  {
 +
    draw((i,-3/16)--(i,3/16), black+linewidth(1));
 +
  }
 +
}
 +
 +
//Draws and labels coordinate axes
 +
void drawLabelAxes()
 +
{
 +
draw((xMin,0)--(xMax,0),black+linewidth(1.5),EndArrow(5));
 +
draw((0,yMin)--(0,yMax),black+linewidth(1.5),EndArrow(5));
 +
label("$x$",(xMax,0),(2,0));
 +
label("$y$",(0,yMax),(0,2));
 +
}
 +
 +
horizontalLines();
 +
verticalLines();
 +
horizontalTicks();
 +
verticalTicks();
 +
drawLabelAxes();
 +
 +
path P[], Q[];
 +
for (int i = 0; i < 9; ++i) {
 +
P[i] = (i,i)--(i+1,i);
 +
    Q[i] = (-i,i+1)--(-i-1,i+1);
 +
}
 +
draw(P^^Q,heavygreen,"$y=|\lfloor x \rfloor|$");
 +
for (int i = 0; i < 9; ++i) {
 +
dot((i,i),heavygreen,UnFill);
 +
    dot((i+1,i),heavygreen);
 +
    dot((-i-1,i+1),heavygreen,UnFill);
 +
    dot((-i,i+1),heavygreen);
 +
}
 +
</asy>
 
Taking the difference, we graph <math>f(x) = |\lfloor x \rfloor| - |\lfloor 1 - x \rfloor|,</math> as shown below:
 
Taking the difference, we graph <math>f(x) = |\lfloor x \rfloor| - |\lfloor 1 - x \rfloor|,</math> as shown below:
  

Revision as of 14:24, 25 November 2021

Problem

The graph of \[f(x) = |\lfloor x \rfloor| - |\lfloor 1 - x \rfloor|\] is symmetric about which of the following? (Here $\lfloor x \rfloor$ is the greatest integer not exceeding $x$.)

$\textbf{(A) }\text{the }y\text{-axis}\qquad \textbf{(B) }\text{the line }x = 1\qquad \textbf{(C) }\text{the origin}\qquad \textbf{(D) }\text{ the point }\left(\dfrac12, 0\right)\qquad \textbf{(E) }\text{the point }(1,0)$

Solution 1 (Piecewise Function)

IN PROGRESS AND WILL FINISH SOON. NO EDIT PLEASE. A MILLION THANKS.

~MRENTHUSIASM

Solution 2 (Graphing)

Let $y_1=|\lfloor x \rfloor|$ and $y_2=|\lfloor 1 - x \rfloor|=|\lfloor -(x-1) \rfloor|.$

The graph of $y_1$ is shown below: [asy] /* Made by MRENTHUSIASM */ size(250); int xMin = -10; int xMax = 10; int yMin = -10; int yMax = 10; //Draws the horizontal gridlines void horizontalLines() { for (int i = yMin+1; i < yMax; ++i) { draw((xMin,i)--(xMax,i), mediumgray+linewidth(0.4)); } } //Draws the vertical gridlines void verticalLines() { for (int i = xMin+1; i < xMax; ++i) { draw((i,yMin)--(i,yMax), mediumgray+linewidth(0.4)); } } //Draws the horizontal ticks void horizontalTicks() { for (int i = yMin+1; i < yMax; ++i) { draw((-3/16,i)--(3/16,i), black+linewidth(1)); } } //Draws the vertical ticks void verticalTicks() { for (int i = xMin+1; i < xMax; ++i) { draw((i,-3/16)--(i,3/16), black+linewidth(1)); } } //Draws and labels coordinate axes void drawLabelAxes() { draw((xMin,0)--(xMax,0),black+linewidth(1.5),EndArrow(5)); draw((0,yMin)--(0,yMax),black+linewidth(1.5),EndArrow(5)); label("$x$",(xMax,0),(2,0)); label("$y$",(0,yMax),(0,2)); } horizontalLines(); verticalLines(); horizontalTicks(); verticalTicks(); drawLabelAxes(); path P[], Q[]; for (int i = 0; i < 9; ++i) { P[i] = (i,i)--(i+1,i); Q[i] = (-i,i+1)--(-i-1,i+1); } draw(P^^Q,red,"$y=|\lfloor x \rfloor|$"); for (int i = 0; i < 9; ++i) { dot((i,i),red); dot((i+1,i),red,UnFill); dot((-i-1,i+1),red); dot((-i,i+1),red,UnFill); } [/asy] Note that $y_2$ is a reflection of $y_1$ about the $y$-axis, followed by a translation of $1$ unit right. The graph of $y_2$ is shown below: [asy] /* Made by MRENTHUSIASM */ size(250); int xMin = -10; int xMax = 10; int yMin = -10; int yMax = 10; //Draws the horizontal gridlines void horizontalLines() { for (int i = yMin+1; i < yMax; ++i) { draw((xMin,i)--(xMax,i), mediumgray+linewidth(0.4)); } } //Draws the vertical gridlines void verticalLines() { for (int i = xMin+1; i < xMax; ++i) { draw((i,yMin)--(i,yMax), mediumgray+linewidth(0.4)); } } //Draws the horizontal ticks void horizontalTicks() { for (int i = yMin+1; i < yMax; ++i) { draw((-3/16,i)--(3/16,i), black+linewidth(1)); } } //Draws the vertical ticks void verticalTicks() { for (int i = xMin+1; i < xMax; ++i) { draw((i,-3/16)--(i,3/16), black+linewidth(1)); } } //Draws and labels coordinate axes void drawLabelAxes() { draw((xMin,0)--(xMax,0),black+linewidth(1.5),EndArrow(5)); draw((0,yMin)--(0,yMax),black+linewidth(1.5),EndArrow(5)); label("$x$",(xMax,0),(2,0)); label("$y$",(0,yMax),(0,2)); } horizontalLines(); verticalLines(); horizontalTicks(); verticalTicks(); drawLabelAxes(); path P[], Q[]; for (int i = 0; i < 9; ++i) { P[i] = (i,i)--(i+1,i); Q[i] = (-i,i+1)--(-i-1,i+1); } draw(P^^Q,heavygreen,"$y=|\lfloor x \rfloor|$"); for (int i = 0; i < 9; ++i) { dot((i,i),heavygreen,UnFill); dot((i+1,i),heavygreen); dot((-i-1,i+1),heavygreen,UnFill); dot((-i,i+1),heavygreen); } [/asy] Taking the difference, we graph $f(x) = |\lfloor x \rfloor| - |\lfloor 1 - x \rfloor|,$ as shown below:

DIAGRAM WILL BE READY VERY SOON. APPRECIATE IT IF THERE'S NO EDIT AT THIS PAGE WITHIN THE NEXT HOUR.

Therefore, the answer is $\boxed{\textbf{(D) }\text{ the point }\left(\dfrac12, 0\right)}.$

~MRENTHUSIASM

See Also

2021 Fall AMC 10A (Problems • Answer Key • Resources)
Preceded by
Problem 15 		Followed by
Problem 17
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 10 Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. AMC logo.png

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2021_Fall_AMC_10A_Problems/Problem_16&oldid=166319"