Slope and Linear Functions

📈 Slope & Linear Functions

Part 1 of 5 — What Slope Means

Topics in This Part

🔑 Key Concept: Slope measures how steep a line is — exactly how much it goes up (or down) every time you move one step to the right. A ramp, a staircase, and a hill all have slope.

Slope as Steepness

Picture walking up a wheelchair ramp. A gentle ramp barely rises — small slope. A steep ramp shoots up fast — big slope.

We measure slope as the ratio:

$\text{slope} = \frac{\text{rise}}{\text{run}} = \frac{\text{vertical change}}{\text{horizontal change}}$

• Rise = how far the line goes up (positive) or down (negative).
• Run = how far the line goes to the right (we always read left-to-right).

Example

A ramp goes up 2 feet for every 8 feet it stretches forward:

$\text{slope} = \frac{\text{rise}}{\text{run}} = \frac{2}{8} = \frac{1}{4}$

💡 Read it like a fraction. A slope of $\dfrac{1}{4}$ means: go up 1 every time you go right 4. The bigger the number, the steeper the line.

Concept Check 🎯

Four Kinds of Slope

As you read a line from left to right, it can do one of four things:

⚠️ A horizontal line has slope $0$ (no rise). A vertical line has an undefined slope (the run is $0$, and we can't divide by zero). Don't mix these two up!

Name That Slope 🔽

For each description, pick the kind of slope.

Counting Rise and Run

When a line is on a grid, you can find its slope by counting boxes: pick two points on the line, count how many squares you go up or down (rise), then how many you go right (run).

Example

From one point to another, a line goes up 3 and right 4:

$\text{slope} = \frac{\text{rise}}{\text{run}} = \frac{3}{4}$

If instead it goes down 3 and right 4, the rise is $-3$, so the slope is $-\dfrac{3}{4}$.

💡 Going down counts as a negative rise. The run (right) is almost always positive because we read left-to-right.

Count It Out 🧮

Find each slope as $\dfrac{\text{rise}}{\text{run}}$. Enter a whole number or a fraction like $3/4$.

1) up $4$, right $2$ → slope $= \,?$ 2) down $5$, right $5$ → slope $= \,?$ 3) up $1$, right $3$ → slope $= \,?$

Wrapping Up Part 1

You now know what slope is: a number that captures steepness and direction, found as $\dfrac{\text{rise}}{\text{run}}$.

In Part 2 you'll learn the slope formula — a quick way to get the slope straight from two points, without ever drawing the picture.

🔑 Remember: rise is up/down, run is left/right, and we always read a line from left to right.

📈 Slope & Linear Functions

Part 2 of 5 — The Slope Formula

🔑 The Idea: Give me any two points on a line and I can find the slope with one formula — no graph needed.

The Slope Formula

For two points $(x_1, y_1)$ and $(x_2, y_2)$:

$m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{\text{change in } y}{\text{change in } x}$

This is just rise over run in disguise: the top is how much $y$ changed (rise), the bottom is how much $x$ changed (run).

Worked Example: $(1, 2)$ and $(4, 8)$

Label them: $x_1 = 1,\ y_1 = 2,\ x_2 = 4,\ y_2 = 8$.

$m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{8 - 2}{4 - 1} = \frac{6}{3} = 2$

The slope is $2$.

⚠️ Stay consistent! Whatever point you call "second" for the $y$'s, use the same point's $x$ as "second." Mixing them up flips the sign.

Worked Example with a Negative Slope: $(2, 5)$ and $(6, 1)$

$m = \frac{1 - 5}{6 - 2} = \frac{-4}{4} = -1$

The slope is $-1$ — the line goes downhill, falling 1 unit for every 1 to the right. That matches: as $x$ grew, $y$ shrank.

A Subtraction Reminder

Watch out for negatives inside the formula. For $(-3, 4)$ and $(1, -2)$:

$m = \frac{-2 - 4}{1 - (-3)} = \frac{-6}{1 + 3} = \frac{-6}{4} = -\frac{3}{2}$

💡 $1 - (-3)$ becomes $1 + 3 = 4$. Subtracting a negative adds.

Concept Check 🎯

Use the Formula 🧮

Find the slope through each pair of points. Enter a whole number or a fraction like $-3/2$.

1) $(2, 3)$ and $(5, 12)$ → $m = \,?$ 2) $(1, 8)$ and $(4, 2)$ → $m = \,?$ 3) $(-1, 2)$ and $(3, 4)$ → $m = \,?$

Horizontal & Vertical 🔽

Use the formula to spot special lines.

📈 Slope & Linear Functions

Part 3 of 5 — Slope-Intercept Form $y = mx + b$

🔑 The Power Form: Every straight (non-vertical) line can be written as $y = mx + b$. Just by looking at it, you can read off the slope and where it crosses the $y$-axis.

Reading $y = mx + b$

$y = \underbrace{m}_{\text{slope}}\,x + \underbrace{b}_{y\text{-intercept}}$

• $m$ is the slope — the number multiplied by $x$.
• $b$ is the $y$-intercept — the $y$-value where the line crosses the $y$-axis (this is the point $(0, b)$).

Example: $y = 3x + 4$

• Slope $m = 3$ → up 3, right 1.
• $y$-intercept $b = 4$ → the line passes through $(0, 4)$.

Example: $y = -\dfrac{1}{2}x - 5$

• Slope $m = -\dfrac{1}{2}$ → down 1, right 2.
• $y$-intercept $b = -5$ → passes through $(0, -5)$.

💡 Watch the signs. In $y = 2x - 7$, the $y$-intercept is $-7$, not $7$, because $y = 2x + (-7)$.

Read the Equation 🔽

Identify the slope and $y$-intercept of each line.

When It's Not Already Solved for $y$

Sometimes a line is written like $2x + y = 10$. To read the slope and intercept, solve for $y$ first.

Example: $2x + y = 10$

Subtract $2x$ from both sides:

$y = -2x + 10$

Now it's clear: slope $m = -2$, $y$-intercept $b = 10$.

Example: $4x + 2y = 6$

Subtract $4x$: $\;2y = -4x + 6$. Then divide every term by $2$:

$y = -2x + 3$

So $m = -2$ and $b = 3$.

⚠️ Divide every term. When you divide $2y = -4x + 6$ by $2$, all three pieces get divided: $y = -2x + 3$.

Concept Check 🎯

Slope & Intercept 🧮

Rewrite in $y = mx + b$ form if needed, then enter the values.

1) $y = 6x - 11$. Slope $m = \,?$ 2) Same line: $y$-intercept $b = \,?$ 3) Solve $x + y = 4$ for $y$, then give the slope $m = \,?$

📈 Slope & Linear Functions

Part 4 of 5 — Graphing Lines & Writing Equations

🔑 Two Skills, One Form: With $y = mx + b$ you can draw any line from its equation, and write the equation of any line you can see.

Graphing from $y = mx + b$

There's a simple recipe:

1. Plot $b$ — put a point at $(0, b)$ on the $y$-axis.
2. Use the slope as $\dfrac{\text{rise}}{\text{run}}$ to step to a second point.
3. Connect the points with a straight line.

Example: $y = 2x + 1$

1. $b = 1$ → start at $(0, 1)$.
2. $m = 2 = \dfrac{2}{1}$ → from $(0,1)$ go up 2, right 1 to $(1, 3)$.
3. Draw the line through $(0,1)$ and $(1,3)$.

Example: $y = -\dfrac{2}{3}x + 4$

1. $b = 4$ → start at $(0, 4)$.
2. $m = -\dfrac{2}{3}$ → go down 2, right 3 to $(3, 2)$.

💡 A negative slope like $-\dfrac{2}{3}$ means down 2 and right 3 (or, equally, up 2 and left 3). Either way, the line falls.

Concept Check 🎯

Writing an Equation from a Graph (or Two Points)

To build $y = mx + b$, you need two things: the slope $m$ and the intercept $b$.

Example: a line through $(0, 2)$ and $(3, 8)$

1. Slope: $m = \dfrac{8 - 2}{3 - 0} = \dfrac{6}{3} = 2$.
2. Intercept: one point is $(0, 2)$, so $b = 2$ (it's already on the $y$-axis).
3. Equation: $y = 2x + 2$.

Example: line through $(1, 5)$ with slope $3$

The intercept isn't given, so find $b$ by plugging the point into $y = mx + b$:

$5 = 3(1) + b \;\Rightarrow\; 5 = 3 + b \;\Rightarrow\; b = 2$

So the equation is $y = 3x + 2$.

🔑 The trick: once you know $m$, plug any point's $(x, y)$ into $y = mx + b$ and solve for $b$.

Write the Equation 🧮

Find the slope and intercept, then report them.

1) Line through $(0, -4)$ and $(2, 6)$. Slope $m = \,?$ 2) Same line: $y$-intercept $b = \,?$ 3) Line with slope $4$ passing through $(1, 9)$. Find $b = \,?$

Build the Line 🔽

A line has slope $-2$ and passes through $(0, 3)$.

📈 Slope & Linear Functions

Part 5 of 5 — Real-World Models & Mastery Check

The best part: linear functions describe real life. Pay-per-job, phone plans, savings, distance traveled — anything that changes at a steady rate is linear.

Slope = Rate, Intercept = Starting Amount

In a real-world line $y = mx + b$:

• $m$ (slope) is the rate of change — "per" something: dollars per hour, miles per gallon.
• $b$ (intercept) is the starting value — the amount when $x = 0$.

Example: a Plumber's Bill

A plumber charges a $50 service fee plus $30 per hour. The cost $y$ for $x$ hours is:

$y = 30x + 50$

• Slope $30$ → it costs $30 more per hour.
• Intercept $50$ → even for $0$ hours, the bill starts at $50 (the fee).

For a 4-hour job: $y = 30(4) + 50 = 120 + 50 = 170$, giving $170.

💡 The words tell you the equation: a fixed start is the intercept; a "per-unit" cost is the slope.

Translate the Story 🔽

A gym charges a $20 sign-up fee plus $15 each month. Let $y$ be the total cost after $x$ months.

Use the Model 🧮

A taxi charges $3 to start plus $2 per mile: $y = 2x + 3$, where $x$ is miles and $y$ is dollars.

1) What does a 5-mile ride cost? $\,?$ (dollars) 2) What does a 10-mile ride cost? $\,?$ (dollars) 3) If a ride cost $15, how many miles was it? $\,?$

Quick Reference

⚠️ Top traps: keep your two points in the same order in the slope formula, watch the sign of $b$, and remember horizontal lines have slope $0$ while vertical lines are undefined.

Mixed Practice 🎯

Exit Quiz ✅

Answer all three to finish the lesson.