--- title: "What c does to a line" subtitle: "Linear relations · Lines and their rules · Lesson 7" --- # What c does to a line **Name:** {{field-24}} **Date:** {{field-15}} **Class:** {{field-15}} > **Key question:** Two phone plans with the same price per gigabyte but different monthly fees never meet: their graphs are parallel. The fee is c. :::teacher-note **In one sentence:** Students vary c in y = 2x + c on a graph board, find that c moves the line up or down and is where it crosses the y-axis, and that lines with the same m are parallel (AC9M8A04, node 2). **Students should be able to say:** I can describe what happens to the line y = mx + c when c changes. I can read where a line crosses the y-axis from its rule. I can tell from their rules whether two lines are parallel. **Width:** the unmarked parts are the floor every student does (predict, run, investigate and make). Typical adds the modify practice; wider adds the real-world contexts. They add contexts, not difficulty. Students work in the online tutorial alongside this sheet. ::: ## Predict $$y = 2x + 1 \quad\longrightarrow\quad y = 2x + 4$$ Work out $y$ when $x = 0$ for each rule. Then think about what happens at $x = 1$. :::keep-together **The line $y = 2x + 1$ has $c = 1$. If $c$ becomes 4, so the rule is $y = 2x + 4$, what happens to the line?** ☐ It crosses the $x$-axis at 4 ☐ It moves up by 3, and is just as steep ☐ It moves 3 to the right ☐ It gets steeper My prediction, and why: {{lines-3}} ::: :::teacher-note **What happens:** It moves up by 3 and is just as steep. Every value of y is 3 more than before (2x + 4 instead of 2x + 1), so the whole line moves up 3. m is still 2, so the steepness is the same. ::: ## Run Try it on the board in the tutorial. Was your prediction right? What was different? {{lines-2}} ## Reading c :::keep-together ### Where it crosses Here is how to read $c$ from a rule. 1. $y = 2x + 4$ — Where does this line cross the $y$-axis, and how does it compare with $y = 2x + 1$? 2. $x = 0: \; y = 2(0) + 4 = 4$ — The $y$-axis is where $x = 0$. Put 0 in place of $x$. 3. $c = 4$ — $2x$ is 0 there, so $y$ is just $c$: the line crosses at 4. 4. $x = 1: \; 2(1) + 4 = 6, \; 2(1) + 1 = 3$ — Compare the two rules at $x = 1$. 5. $6 - 3 = 3$ — $y$ is 3 more, and it is 3 more at every $x$, because only $c$ changed. 6. $\text{same } m \Rightarrow \text{parallel}$ — The line moved up 3 without turning: lines with the same $m$ are parallel. **Where does the line $y = 3x - 5$ cross the $y$-axis? Give the value of $y$.** Answer: {{field-15}} ::: :::teacher-note **Answer:** -5 −5. When x = 0, y = 3 × 0 − 5 = −5, which is c. ::: :::keep-together ### Find the mistake **Ana decides whether two lines are parallel. Click the line where her reasoning goes wrong.** ``` 1 Are y = 2x + 5 and y = 5x + 2 parallel? 2 Both rules use a 2 and a 5. 3 So they have the same numbers. 4 So the lines are parallel. ``` Line: {{field-15}} Why is it wrong? {{lines-2}} ::: :::teacher-note **Answer:** line 4 Line 4: parallel lines need the same m, the number in front of x. Here m is 2 and 5, so one is steeper than the other: they are not parallel. ::: :::keep-together ### Parallel **Which line is parallel to $y = 3x + 1$?** ☐ $y = x + 3$ ☐ $y = 3x - 4$ ☐ $y = -3x + 1$ ::: :::teacher-note **Answer:** $y = 3x - 4$ y = 3x − 4 has the same m, 3, so it is just as steep: it is y = 3x + 1 moved down 5. ::: :::typical ## Change c on purpose **Below the axis.** Each task puts a point $T$ on the board. Set $c$ so the line passes through $T(0, -2)$. **Through the origin.** Now take the line through $T(2, 4)$. What is $c$? **Not on the axis.** $T(1, 6)$ is not on the $y$-axis. Work out $2x$ at $T$ first. **Parallel.** Two sliders now. Make a line parallel to the dashed one, through $T(0, -1)$. Work in the tutorial. Plan first: what will you change, and how will you test it? {{box-4}} ::: ## Read the line :::keep-together ### From the rule **Where does $y = 5x + 7$ cross the $y$-axis? Give the value of $y$.** Answer: {{field-15}} ::: :::teacher-note **Answer:** 7 7: when x = 0, y = 5 × 0 + 7 = 7. ::: :::keep-together ### Parallel **Which two lines are parallel?** ☐ $y = 4x + 1$ and $y = 4x - 3$ ☐ $y = 4x + 1$ and $y = x + 4$ ☐ $y = 4x + 1$ and $y = 2x + 1$ ::: :::teacher-note **Answer:** $y = 4x + 1$ and $y = 4x - 3$ Both have m = 4, so they are equally steep. y = 4x + 1 and y = x + 4 use the same numbers but in different places; y = 4x + 1 and y = 2x + 1 cross at (0, 1). ::: :::keep-together ### From a point **The line $y = 2x + c$ passes through $(3, 10)$. What is $c$?** Answer: {{field-15}} ::: :::teacher-note **Answer:** 4 At x = 3, 2x = 6. 6 + c = 10, so c = 4. ::: :::keep-together ### Review **Review: how far does $y = -3x + 2$ go up for each step right? (A negative answer means it goes down.)** Answer: {{field-15}} ::: :::teacher-note **Answer:** -3 m = −3: down 3 for each step right. ::: ## Make: a parallel line Choose where $P$ goes, then find the $c$ that takes the line through it. Check: is your line parallel to the dashed one? Stretch: write the rule of the line parallel to $y = 2x + 1$ that passes through $(5, 0)$. Work in the tutorial. Plan first: what will you change, and how will you test it? {{box-4}} ## Exit ticket :::keep-together **Are y = 3x + 2 and y = 3x − 5 parallel? Where does each cross the y-axis?** {{lines-3}} ::: :::teacher-note **Answer:** Yes, both have m = 3. The first crosses at 2, the second at −5. :::