Revise Linear models and equations of straight lines for IB Mathematics: Applications and Interpretation (2.1)

The anatomy of a straight line

• $x$ and $y$ are the variables. In a model they are renamed for context - $d$ for distance, $t$ for time, $C$ for cost - but the structure is identical.

• $m$ is the gradient: the rate of change. For every one-unit increase in $x$, $y$ changes by $m$ units. A positive $m$ rises, a negative $m$ falls.

• $c$ is the $y$-intercept: the value of $y$ when $x = 0$. In a model this is the starting value - a fixed fee, an initial amount, a value at time zero.

• Gradient-intercept form $y = mx + c$ - best for reading off, or writing down, a rate and a starting value.

• Point-gradient form $y - y_1 = m(x - x_1)$ - best when you know the gradient and one point; no need to solve for $c$ separately.

• General form $ax + by + d = 0$ - a tidy way to present a final answer with integer coefficients, and the only form that also covers vertical lines.

Finding the gradient, and a line from two points

A line from a point and a gradient

• Parallel lines have equal gradients: $m_1 = m_2$. A line parallel to $y = 3x + 7$ also has gradient $3$.

• Perpendicular lines satisfy $m_1 m_2 = -1$, so each gradient is the negative reciprocal of the other: $m_2 = -\dfrac{1}{m_1}$. A line perpendicular to a line of gradient $3$ has gradient $-\dfrac{1}{3}$.

Interpreting a real-world linear model

• The gradient is a rate of change. It answers 'how much does the output change for each extra unit of input?' Its units are (units of $y$) per (unit of $x$): dollars per km, litres per minute, degrees per hour.

• The intercept is a starting value. It answers 'what is the output when the input is zero?' - a fixed fee, an initial amount, a value at time zero - again with units.

• The equation is a prediction machine. Substitute a value of the input to forecast the output. Predicting inside the data range (interpolation) is reasonably safe; outside it (extrapolation) is risky.

Model answer - marked the way our engine marks it

Common mistakes examiners penalise

• Interpreting the gradient without units - a rate of change is meaningless bare. Write 'the cost rises by $1.20 per km', not 'the cost rises by 1.20'. This is the single most common lost mark in AI interpretation questions.

• Confusing the gradient with the intercept - the gradient is the RATE of change (per unit), the intercept is the STARTING value (when the input is zero). Calling the fixed fee the 'rate', or the per-km charge the 'starting cost', reverses the meaning.

• Flipping the gradient fraction - use $\frac{y_2 - y_1}{x_2 - x_1}$ with the points in the SAME order top and bottom; swapping one but not the other changes the sign.

• Muddling the parallel and perpendicular conditions - parallel means EQUAL gradients ($m_1 = m_2$); perpendicular means the product is $-1$ ($m_1 m_2 = -1$). Using $m_1 = m_2$ for a perpendicular line (or vice versa) throws away the question.

• Doing only half the negative reciprocal - the perpendicular gradient both flips AND changes sign: perpendicular to $\frac{2}{3}$ is $-\frac{3}{2}$, not $-\frac{2}{3}$ or $\frac{3}{2}$.

• Not reading the general form correctly - rearrange $ax + by + d = 0$ fully to $y = mx + c$ before quoting the gradient; the gradient of $3x + 2y - 6 = 0$ is $-\frac{3}{2}$, not $3$.

• Trusting an extrapolation blindly - predicting far outside the data range assumes the straight-line trend continues; say when a prediction is an extrapolation and therefore less reliable.

• Over-rounding mid-calculation - carry full figures and round only the final answer, to 3 significant figures unless the context (like money) dictates otherwise.

Where this leads