Gradients and Intercepts

Gradients and Intercepts

The gradient and intercepts of a linear function carry precise mathematical meaning and, in real-world contexts, direct practical interpretation. Being able to identify them — both from a graph and from an equation in various forms — is essential for analysing linear models in science, economics and everyday situations.

Gradient as Rate of Change

The gradient of a line describes how steeply \( y \) changes with respect to \( x \): a gradient of 3 means \( y \) increases by 3 units for every 1-unit increase in \( x \). In a real-world graph, the gradient equals the rate of change of the quantity on the y-axis with respect to the quantity on the x-axis.

0 2 4 6 8 10 n 1 2 3 4 5 6 7 C Fixed cost: £2 Δn = 4 ΔC = 2 gradient = ΔC/Δn = 2/4 = 0.5 (£0.50 per item) C = 0.5n + 2

Finding Intercepts

The y-intercept is where the line crosses the y-axis. Set \( x = 0 \).

The x-intercept (root) is where the line crosses the x-axis. Set \( y = 0 \).

\[ 3x + 4y = 24: quad x\text{-intercept}: 3x=24 \Rightarrow x=8. quad y\text{-intercept}: 4y=24 \Rightarrow y=6. \]

Rearranging to \( y = mx + c \)

Equations in other forms (e.g. \( ax + by = k \)) must be rearranged before reading off gradient and intercept directly.

\[ 4x - 2y + 6 = 0 \implies 2y = 4x+6 \implies y = 2x+3 quad (m=2, c=3) \]

Worked Examples

Rearrange \( 5x - 2y + 8 = 0 \) into \( y = mx + c \) form. State the gradient and y-intercept.
\[ 2y = 5x+8 \implies y = \frac{5}{2}x + 4 quad m = \frac{5}{2}, c = 4 \]
Find the x-intercept and y-intercept of \( 4x + 3y = 24 \). Use these to sketch the line.

y-intercept (\(x=0\)): \( 3y=24 \Rightarrow y=8 \). Point \((0,8)\).

x-intercept (\(y=0\)): \( 4x=24 \Rightarrow x=6 \). Point \((6,0)\).

Plot \((0,8)\) and \((6,0)\) and draw the straight line through them.

A taxi charges a fixed fee of £3 plus £2 per km. Write the cost equation \( C = 2d + 3 \) and interpret the gradient and y-intercept.

Gradient = 2: the cost increases by £2 for each additional kilometre — the rate of charge per km.

y-intercept = 3: the fixed starting charge (£3), which is paid regardless of distance.

These two components together fully describe the pricing structure of the journey.

 Key Takeaways

  • Gradient = rate of change of \( y \) per unit increase in \( x \). In applied contexts it carries the units of the y-quantity per unit of the x-quantity.
  • y-intercept: set \( x = 0 \). x-intercept (root): set \( y = 0 \).
  • Always rearrange to \( y = mx + c \) form before reading off \( m \) and \( c \) — equations like \( 2x + 3y = 12 \) have \( m = -\frac{2}{3} \), not 2.
  • In real-world linear models: gradient = rate (cost per item, speed, etc.); y-intercept = initial/fixed value when \( x = 0 \).