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# What c does to a line

Linear relations · Lines and their rules

<!-- Speaker note: 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). About 45 minutes (Predict and Run 8 min · Investigate 15 min · Modify 12 min · Practise and Make 10 min). -->

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## Today

**We are learning to** in y = mx + c, the number on its own moves the line up or down without changing its steepness, so lines with the same m are parallel.

- 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.

<!-- Speaker note: Before class: Do the Run and Modify boards yourself first. This lesson follows "What m does to a line": students should already read m as "up for each step right". Here m stays at 2 and only c changes, so every line is parallel to the dashed y = 2x + 1. The last Modify task is the first time both sliders appear: it is the stretch point of the lesson. The lesson says "where the line crosses the y-axis". The primer has the formal word, y-intercept, if you want to introduce it at the end. -->

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## Why it matters

> Two phone plans with the same price per gigabyte but different monthly fees never meet: their graphs are parallel. The fee is c.

<!-- Speaker note: In the last lesson $c$ stayed at 1 and you changed $m$, the steepness. Now $m$ stays at 2 and you change $c$, the number on its own. -->

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<span style="background:#dbeafe; color:#1a1a1a; border:1.5px solid #3b82f6; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">💬 Discussion</span>

## Predict

**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

<!-- Speaker 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. Take two or three predictions before anyone runs it. -->

<!-- Speaker note: Teacher: take a show of hands. The common wrong answer is 'it gets steeper': students expect any bigger number to steepen the line. Do not correct it yet. -->

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## Try it

{{embed: https://studio.learningwaypoints.com/tutorials/_playground-board/index.html?id=deck%2Fmaths%2Flinear-relations%2Fintercept%2Fstep-2&code=widget%3A+graph%0Ax%3A+-1%2C+5%0Ay%3A+-4%2C+14%0Aslider+c%3A+-3%2C+6%2C+1%0Aplot%3A+y+%3D+2x+%2B+c%0Aplot%3A+y+%3D+2x+%2B+1%2C+dashed%0Atable%3A+0%2C+1%2C+2%2C+3%0Astep%3A+1%0A&fresh=1}}

[Open this step in the tutorial](https://studio.learningwaypoints.com/tutorials/maths/linear-relations/intercept/?full=step-2#step-2)

<!-- Speaker note: Change $c$ and compare the solid line with the dashed line, $y = 2x + 1$. Watch the step, and the two rows of the table. -->

<!-- Speaker note: Teacher: ask students to compare the step on the solid line with the dashed line as they move c. It stays 'up 2'. Then read across a column of the table: each y is the same amount more. -->

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## What happened?

* 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.

<!-- Speaker note: Reveal after running. Ask who predicted it, and what made them think so. -->

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<span style="background:#dbeafe; color:#1a1a1a; border:1.5px solid #3b82f6; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">💬 Discussion</span>

## Where it crosses

Here is how to read $c$ from a rule.

$$y = 2x + 4$$

1) $x = 0: \; y = 2(0) + 4 = 4$ — The $y$-axis is where $x = 0$. Put 0 in place of $x$.
2) $c = 4$ — $2x$ is 0 there, so $y$ is just $c$: the line crosses at 4.
3) $x = 1: \; 2(1) + 4 = 6, \; 2(1) + 1 = 3$ — Compare the two rules at $x = 1$.
4) $6 - 3 = 3$ — $y$ is 3 more, and it is 3 more at every $x$, because only $c$ changed.
5) $\text{same } m \Rightarrow \text{parallel}$ — The line moved up 3 without turning: lines with the same $m$ are parallel.

<!-- Speaker note: Model this live on the board before revealing anything: write each line and say why, pausing to ask the class what comes next. Then click to reveal the lines and compare. Script. Start: write y = 2x + 4 and say "Where does this line cross the y-axis, and how does it compare with y = 2x + 1?" Line 1: write x = 0: y = 2(0) + 4 = 4 and say "The y-axis is where x = 0. Put 0 in place of x." Line 2: write c = 4 and say "2x is 0 there, so y is just c: the line crosses at 4." Line 3: write x = 1: 2(1) + 4 = 6, 2(1) + 1 = 3 and say "Compare the two rules at x = 1." Line 4: write 6 - 3 = 3 and say "y is 3 more, and it is 3 more at every x, because only c changed." Line 5: write same m parallel and say "The line moved up 3 without turning: lines with the same m are parallel." -->

<!-- Speaker note: Teacher: model the worked example live. Put x = 0 into y = 2x + 4 and say 'on the y-axis, x is 0, so 2x is 0 and y is just c'. Then compare both rules at x = 1. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Where it crosses

**Where does the line $y = 3x - 5$ cross the $y$-axis? Give the value of $y$.**

* Answer: -5

<!-- Speaker note: −5. When x = 0, y = 3 × 0 − 5 = −5, which is c. -->

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## 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.
```

* Answer: line 4

<!-- Speaker note: 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. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Parallel

**Which line is parallel to $y = 3x + 1$?**

- $y = x + 3$
- $y = 3x - 4$
- $y = -3x + 1$

* Answer: $y = 3x - 4$

<!-- Speaker note: y = 3x − 4 has the same m, 3, so it is just as steep: it is y = 3x + 1 moved down 5. -->

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## 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 $T(0, -1)$.

[Open this step](https://studio.learningwaypoints.com/tutorials/maths/linear-relations/intercept/?full=step-4#step-4)

<!-- Speaker note: What this step is for: One change at a time: a negative c, c = 0 (through the origin), a point that is not on the y-axis, then both numbers at once for a parallel line. Hints, in order: T is on the y-axis. Where must the line cross it? / Work out 2x at T: 2 × 2 = 4. / T is at y = 4, so how much is left for c? / At T, x = 1, so 2x is 2. / y at T is 6. What do you add to 2 to get 6? / Parallel means the same steepness. What is m for the dashed line? / Then set c so the line crosses the y-axis at T. For students who need more support: Keep the Run board open for the Investigate questions: set c to the number in the rule and read where the line crosses the y-axis. Use squared paper to draw the two parallel lines. For early finishers: Write the rule of the line parallel to y = 2x + 1 through (5, 0). Then: can two different lines with the same m ever cross? Why not? When students get stuck: A student says a bigger c makes the line steeper. Ask: How far up is the step on each line? Fix: Point at the two steps: both are 'up 2'. Only the starting height on the y-axis changed. / A student reads m as where the line crosses the y-axis. Ask: What is y when x is 0? Put 0 in the rule. Fix: Work it through: 3 × 0 − 5 = −5. The x term disappears at x = 0, so only c is left. / A student says lines with the same numbers are parallel (Ana's mistake). Ask: Which number sets the steepness? Fix: Compare m only: same m means same steepness, so parallel. Different m means the lines cross somewhere. / A student cannot find c when T is not on the y-axis. Ask: What is 2x at T? How much more is y? Fix: Write y = 2x + c with T's numbers in: 6 = 2 × 1 + c, so c = 4. (This is the first equation-solving step of the next lesson.) -->

<!-- Speaker note: Teacher: before 'Not on the axis', ask: why can't we just read c off the graph here? (T is not on the y-axis.) -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## From the rule

**Where does $y = 5x + 7$ cross the $y$-axis? Give the value of $y$.**

* Answer: 7

<!-- Speaker note: 7: when x = 0, y = 5 × 0 + 7 = 7. -->

<!-- Speaker note: Teacher: the parallel question plants the same numbers in different places. Ask a student who chose it to work out the steepness of each line. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Parallel

**Which two lines are parallel?**

- $y = 4x + 1$ and $y = 4x - 3$
- $y = 4x + 1$ and $y = x + 4$
- $y = 4x + 1$ and $y = 2x + 1$

* Answer: $y = 4x + 1$ and $y = 4x - 3$

<!-- Speaker note: 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). -->

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## From a point

**The line $y = 2x + c$ passes through $(3, 10)$. What is $c$?**

* Answer: 4

<!-- Speaker note: At x = 3, 2x = 6. 6 + c = 10, so c = 4. -->

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## Review

**Review: how far does $y = -3x + 2$ go up for each step right? (A negative answer means it goes down.)**

* Answer: -3

<!-- Speaker note: m = −3: down 3 for each step right. -->

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## 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 $(5, 0)$.

[Open this step](https://studio.learningwaypoints.com/tutorials/maths/linear-relations/intercept/?full=step-6#step-6)

<!-- Speaker note: What this step is for: Choosing a point and then finding the c that reaches it, with m fixed, reverses the lesson: from the picture to the rule. Hints, in order: Drag P first. Then work out 2x at P. / c is the y of P minus 2x. -->

<!-- Speaker note: Teacher: collect two students' answers and check them as a class: work out 2x at P and subtract from y. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Exit ticket

**Are y = 3x + 2 and y = 3x − 5 parallel? Where does each cross the y-axis?**

* Yes, both have m = 3. The first crosses at 2, the second at −5.

<!-- Speaker note: Students answer on paper or a mini whiteboard before the reveal. Anyone who cannot answer needs the main idea again next lesson. Where this leads: Next: solving equations with graphs. 2x + 3 = 11 becomes the point where y = 2x + 3 meets y = 11, which joins this unit to the Equations unit. Then modelling: two plans with different m and c, and where their lines cross. Where it leads in class: a fixed starting amount plus a rate (a call-out fee and an hourly charge, a deposit and weekly savings). c is the starting amount; m is the rate. -->