IB Mathematics Analysis and Approaches SL 2.1—equations of Straight Lines, Parallel and Perpendicular Notes

Why Straight Lines Are The First Model You Reach For

1. A straight line is the simplest possible relationship between two quantities: as one changes, the other changes at a steady rate.
2. That steady rate is everywhere in the real world, from the fixed cost per extra guest at a party to the constant slope you climb on a ramp.
3. The graph of every such relationship is a linear function, so mastering lines is the foundation for every other function you will meet in this course.
4. Two numbers pin a line down completely: how steep it is, and where it sits, so once you can read and write those two numbers you can describe any line.
5. This subtopic gives you three ways to write that description, a formula for the steepness, and two quick tests for when lines run parallel or meet at a right angle.

Gradient Measures Steepness As Rise Over Run

1. The gradient $m$ of a line is the amount $y$ increases for every $1$ unit that $x$ increases.
2. Informally it is the $\textbf{rise}$ (change in $y$) divided by the $\textbf{run}$ (change in $x$).
3. For any two points $(x_1, y_1)$ and $(x_2, y_2)$ on the line, that ratio is fixed and equals: $$ m = \frac{y_2 - y_1}{x_2 - x_1} $$
4. Because a straight line has the same steepness everywhere, it does not matter which two points you pick, so the gradient is a single number for the whole line.
5. The sign and size of $m$ describe the line at a glance:
   1. a positive $m$ slopes uphill from left to right, and a negative $m$ slopes downhill.
   2. a larger $|m|$ is steeper, and $m = 0$ is a flat horizontal line $y = c$.
   3. a vertical line has an undefined gradient, because the run $x_2 - x_1$ is $0$ and you cannot divide by zero.

Gradients Of Real Inclines

1. Ramps and roads are lines seen side-on, so their steepness is a gradient you can measure directly.
2. Builders often quote an incline as a ratio such as $1:12$, meaning it rises $1$ unit for every $12$ units travelled horizontally, a gradient of $\tfrac{1}{12}$.
3. Road signs usually quote a percentage: a $8\%$ hill has gradient $0.08$, because percentage grade is just the gradient multiplied by $100$.
4. A steeper incline has a larger gradient, which is why an accessibility standard sets a maximum ramp gradient rather than a maximum height.

The Three Forms Of The Equation Of A Straight Line

1. The same line can be written three ways, and each form is built to make a different fact easy to read.
2. Gradient-intercept form: $$y = mx + c$$
   1. $m$ is the gradient and $c$ is the $y$-intercept, so the line meets the $y$-axis at $(0,\, c)$.
   2. This is the most useful form for sketching, because both numbers you need are visible.
3. Point-gradient form: $$y - y_1 = m(x - x_1)$$
   1. $(x_1,\, y_1)$ is any known point on the line and $m$ is the gradient.
   2. This is the form you reach for first whenever a question gives you a point and a gradient.
4. General form: $$ax + by + d = 0$$
   1. $a$, $b$ and $d$ are constants, usually written as integers with no fractions.
   2. Rearranging for $y$ recovers the gradient-intercept form, $y = -\tfrac{a}{b}x - \tfrac{d}{b}$, so the gradient is $-\tfrac{a}{b}$.

Converting Between The Forms

1. To go from point-gradient to gradient-intercept form, expand the bracket and make $y$ the subject.
2. To go from gradient-intercept to general form, move every term to one side and clear fractions so the coefficients are integers.
3. To go from general to gradient-intercept form, solve the equation for $y$.

Finding The x- And y-Intercepts

1. The y-intercept is where the line crosses the $y$-axis, found by setting $x = 0$.
2. The x-intercept is where the line crosses the $x$-axis, found by setting $y = 0$ and solving for $x$.
3. Both intercepts are just points where a coordinate is zero, and finding where two lines cross is covered separately in SL 2.4.

Parallel Lines Share A Gradient

1. Two lines are parallel when they never meet, which happens exactly when they rise at the same rate.
2. Rising at the same rate means having the same gradient, so the condition is: $$ m_1 = m_2 $$
3. The $y$-intercepts can be different: that is what slides one line above the other without changing its direction.

Finding A Parallel Line Through A Point

1. Read the gradient of the given line, since a parallel line must copy it exactly.
2. Substitute that gradient and the given point into point-gradient form, then tidy up.

Why Perpendicular Gradients Multiply To Minus One

1. Two lines are perpendicular when they cross at a right angle, and their gradients satisfy: $$ m_1 \times m_2 = -1 $$
2. Equivalently, each gradient is the negative reciprocal of the other, $m_2 = -\tfrac{1}{m_1}$.
3. This is not obvious, so here is why it must be true, using a rotation.
4. Represent the first line by a small arrow that runs $b$ across and rises $a$, so its gradient is $m_1 = \tfrac{a}{b}$.
5. Rotate that arrow $90^{\circ}$ anticlockwise about its start; the run and the rise swap roles, and one of them picks up a minus sign.
6. The rotated arrow now runs $-a$ across and rises $b$, so its gradient is $m_2 = \tfrac{b}{-a} = -\tfrac{b}{a}$.
7. Multiplying the two gradients gives the result: $$ m_1 \times m_2 = \frac{a}{b} \times \left(-\frac{b}{a}\right) = -1 $$
8. So rotating a line by a right angle turns its gradient into the negative reciprocal, which is exactly the condition $m_1 \times m_2 = -1$.

Finding A Perpendicular Line Through A Point

1. Take the negative reciprocal of the original gradient to get the new gradient.
2. Substitute the new gradient and the given point into point-gradient form, then tidy up.

What The Formula Booklet Gives You, And What It Does Not

1. The gradient formula $m = \tfrac{y_2 - y_1}{x_2 - x_1}$, together with the distance and midpoint formulas, sits in the $\textbf{Prior learning}$ section of the formula booklet, so you may look it up.
2. The three forms of a line are standard and expected, but the parallel and perpendicular conditions are $\textbf{not printed anywhere in the booklet}$.
3. That makes $m_1 = m_2$ and $m_1 \times m_2 = -1$ two facts you must carry in your own memory.