Question

you will create a graphic design by using (and building upon) your understanding of linear relationships
step 1. decide what type of design you want to create. it could be a design for a sticker, tee - shirt, tote bag, whatever you want! create a rough sketch on blank paper
your design must include at least:

• three lines with positive slopes
• three lines with negative slopes
• two vertical lines
• two horizontal lines
• two lines that are perpendicular to one another (what do you notice about these lines? tell me in your reflection!)

step 2. draw your graphic design more precisely on graph paper with a coordinate plane.
step 3. label each line with an equation. teacher check _____________
step 4. create your graphic design using the desmos graphing calculator at desmos.com
*you will need to learn how to restrict the domain and range of each line so that they dont extend infinitely.
step 5. write a reflection about your work on this project. i want to hear what you learned (or remembered) while working on this project. in other words, explain what you know about the graphs of linear relationships. i expect thorough, in - depth answers with examples and maybe even diagrams. if youd like, you may make a list, but be sure to explain each item on the list.
if youre not sure where to start, here are some questions to get you thinking:

• what did you need to know to complete this project? how does one do those things?
• what did you learn while working on this project?
• describe a misconception you had about equations/graphing and how your thinking changed.
• tell me about any questions/problems/wonderings that came up as you were working on this project.

Answer:

This is a project - based task related to linear relationships in mathematics (specifically in the sub - field of Algebra or Geometry as it involves linear equations and graphing of lines). Here is a step - by - step guide to complete it:

Step 1: Sketch the Design

• Explanation: Decide on the design theme (e.g., a geometric - style sticker). Then, on blank paper, draw a rough sketch. For the lines:
• Positive slopes: Draw three lines that go up from left to right. For example, a line that starts at the bottom left and goes to the top right.
• Negative slopes: Draw three lines that go down from left to right. Like a line starting at the top left and going to the bottom right.
• Vertical lines: Draw two lines that are straight up and down (parallel to the y - axis).
• Horizontal lines: Draw two lines that are straight left and right (parallel to the x - axis).
• Perpendicular lines: Draw two lines that intersect at a right angle. For example, one vertical and one horizontal line (we know that the slope of a vertical line is undefined and the slope of a horizontal line is 0, and they are perpendicular).
• Expression (Sketch): A rough hand - drawn sketch with the required lines.

Step 2: Draw on Graph Paper

• Explanation: Use graph paper with a coordinate plane (x - y axes). Transfer your rough sketch more precisely. Mark the coordinates of the points on each line. For example, if a horizontal line is at y = 3, mark several points (like (0,3), (1,3), (2,3)) on it.
• Expression (Graph): A more precise drawing on graph paper with labeled axes and points on lines.

Step 3: Label with Equations

• Explanation: For each line, write its equation.
• Positive slope lines: For a line with a positive slope, use the slope - intercept form $y=mx + b$, where $m>0$. For example, if a line has a slope of 2 and y - intercept of 1, its equation is $y = 2x+1$.
• Negative slope lines: Use $y=mx + b$ with $m < 0$. For a line with slope - 1 and y - intercept 4, the equation is $y=-x + 4$.
• Vertical lines: The equation of a vertical line is of the form $x = a$, where $a$ is a constant. For example, $x = 2$ and $x=-1$.
• Horizontal lines: The equation of a horizontal line is of the form $y = b$, where $b$ is a constant. For example, $y = 3$ and $y=-2$.
• Perpendicular lines: If one line is $y = 2x+1$ (slope $m_1 = 2$), a line perpendicular to it will have a slope $m_2=-\frac{1}{2}$ (since for perpendicular lines $m_1\times m_2=-1$). So an equation could be $y =-\frac{1}{2}x + 2$.
• Expression (Equations): A list of equations next to each line on the graph. For example:
• Positive slope: $y = x+1$, $y = 2x$, $y=3x - 2$
• Negative slope: $y=-x + 5$, $y=-2x+3$, $y =-\frac{1}{2}x+1$
• Vertical: $x = 1$, $x=-3$
• Horizontal: $y = 2$, $y=-4$
• Perpendicular: $y = 2x$ and $y=-\frac{1}{2}x$

Step 4: Use Desmos

• Explanation: Go to desmos.com. Enter each equation into the graphing calculator. To restrict the domain and range (so lines don't extend infinitely), use the curly braces. For example, for the line $y = 2x+1$, if you want to restrict it to $-5\leq x\leq5$ and $-10\leq y\leq10$, enter $y = 2x + 1\{-5\leq x\leq5,-10\leq y\leq10\}$.
• Expression (Desmos Input): A list of equations with domain and range restrictions entered into Desmos.

Step 5: Write the Reflection

• Explanation: Answer the reflection questions.
• What you needed to know: You needed to know the slope - intercept form of a line ($y = mx + b$), the concept of slope (positive, negative, zero, undefined), and how to determine if two lines are perpendicular (product of slopes is - 1 for non - vertical/non - horizontal lines, vertical and horizontal lines are perpendicular).
• What you learned: You learned how to apply the properties of linear equations to create a design, how to restrict the domain and range in Desmos, and a deeper understanding of the relationship between the equation of a line and its graph.
• Misconception: Maybe you thought that all lines with positive slopes have the same steepness, but you learned that the value of $m$ in $y = mx + b$ determines the steepness (a larger $|m|$ means a steeper line).
• Questions/Problems: Maybe you had trouble initially restricting the domain and range in Desmos, but you learned how to use the curly braces to do it.
• Expression (Reflection): A written reflection (e.g., a paragraph or a list with explanations) addressing the reflection questions.

If you follow these steps, you will be able to complete the graphic design project based on linear relationships.