Graphing a Positive Slope

Graph $y = 2x + 1$

Identify the y-intercept: In $y = 2x + 1$, the y-intercept is $b = 1$ = Start at $(0, 1)$

Identify the slope: The slope is $m = 2 = \frac{2}{1}$ = Rise 2, run 1

Plot the y-intercept: Mark the point $(0, 1)$ on the y-axis = First point plotted

Use slope to find second point: From $(0, 1)$: move up 2, right 1 to get $(1, 3)$ = Second point: $(1, 3)$

Draw the line: Connect the points and extend in both directions = Line complete

Answer: The line passes through $(0, 1)$ and $(1, 3)$, rising steeply from left to right.

Graphing a Negative Slope

Graph $y = -3x + 4$

Identify the y-intercept: In $y = -3x + 4$, the y-intercept is $b = 4$ = Start at $(0, 4)$

Identify the slope: The slope is $m = -3 = \frac{-3}{1}$ = Rise $-3$ (down 3), run 1

Plot the y-intercept: Mark the point $(0, 4)$ on the y-axis = First point plotted

Use slope to find second point: From $(0, 4)$: move down 3, right 1 to get $(1, 1)$ = Second point: $(1, 1)$

Draw the line: Connect the points and extend in both directions = Line falls from left to right

Answer: The line passes through $(0, 4)$ and $(1, 1)$, falling steeply from left to right.

Graphing a Fractional Slope

Graph $y = \frac{2}{3}x - 2$

Identify the y-intercept: In $y = \frac{2}{3}x - 2$, the y-intercept is $b = -2$ = Start at $(0, -2)$

Identify the slope: The slope is $m = \frac{2}{3}$ = Rise 2, run 3

Plot the y-intercept: Mark the point $(0, -2)$ on the y-axis = First point plotted

Use slope to find second point: From $(0, -2)$: move up 2, right 3 to get $(3, 0)$ = Second point: $(3, 0)$

Verify with a third point: From $(3, 0)$: move up 2, right 3 to get $(6, 2)$ = Third point confirms the line

Answer: The line passes through $(0, -2)$, $(3, 0)$, and $(6, 2)$, rising gently from left to right.

Converting to Slope-Intercept Form

Graph $2x + y = 6$

Solve for y: $2x + y = 6$ $y = -2x + 6$ = Now in slope-intercept form

Identify m and b: Slope $m = -2$, y-intercept $b = 6$ = Start at $(0, 6)$, slope $\frac{-2}{1}$

Plot the y-intercept: Mark the point $(0, 6)$ on the y-axis = First point plotted

Use slope to find second point: From $(0, 6)$: move down 2, right 1 to get $(1, 4)$ = Second point: $(1, 4)$

Find a third point: From $(1, 4)$: move down 2, right 1 to get $(2, 2)$ = Third point: $(2, 2)$

Answer: The line passes through $(0, 6)$, $(1, 4)$, and $(2, 2)$, falling from left to right.

Mistake: Confusing slope and y-intercept

Why: In $y = mx + b$, students sometimes think the first number is the y-intercept.

Correct: Remember: $m$ (multiplied by $x$) is the slope. $b$ (the constant) is the y-intercept. In $y = 3x + 5$, slope is 3, y-intercept is 5.

Mistake: Moving in the wrong direction for negative slope

Why: Students may go up instead of down when the slope is negative.

Correct: A negative slope means the line falls from left to right. For $m = -2$: go down 2, right 1 (or up 2, left 1).

Mistake: Inverting the slope fraction

Why: Students confuse rise/run with run/rise.

Correct: Slope is always $\frac{\text{rise}}{\text{run}}$ (vertical change over horizontal change). For $m = \frac{3}{4}$: rise 3, run 4.

Mistake: Starting at the origin instead of y-intercept

Why: Students default to starting at $(0, 0)$.

Correct: Always start at the y-intercept $(0, b)$. Only start at origin if $b = 0$.

Phone Plan Costs

Mobile phone plans often have a base monthly fee plus a per-minute or per-gigabyte charge.

A plan costs 15 euros per month plus 0.05 euros per text. The equation $y = 0.05x + 15$ models the monthly cost, where $x$ is the number of texts.

Temperature Conversion

The relationship between Celsius and Fahrenheit is linear.

The formula $F = 1.8C + 32$ converts Celsius to Fahrenheit. The slope 1.8 means each degree Celsius equals 1.8 degrees Fahrenheit.

Savings Growth

If you save a fixed amount regularly, your savings grow linearly.

Starting with 50 euros and saving 25 euros per week gives $y = 25x + 50$, where $x$ is weeks and $y$ is total savings.

Q: What if the slope is a whole number like 3?

A: Write it as a fraction: $3 = \frac{3}{1}$. This means rise 3, run 1. You can also use $\frac{6}{2}$ or $\frac{-3}{-1}$ to find more points.

Q: How do I graph a horizontal line like $y = 4$?

A: This is $y = 0x + 4$. The slope is 0 (no rise), so it's a flat horizontal line passing through all points where $y = 4$.

Q: What about vertical lines?

A: Vertical lines like $x = 3$ cannot be written in slope-intercept form because their slope is undefined. They pass through all points where $x$ equals that value.