Gradients and Intercepts of Linear Graphs
Understanding straight-line graphs is key to making sense of real-world data, from taxi fares to fitness tracking. This guide will show you how to interpret the two most important features of any straight line: its gradient and its y-intercept.
The Equation of a Straight Line: $y = mx + c$
Every straight line can be described by this powerful equation. Each letter tells you something specific about the line’s properties.
The Two Key Parts
$m$ is the Gradient: This tells you the steepness of the line.
- A positive $m$ means the line slopes upwards (uphill).
- A negative $m$ means the line slopes downwards (downhill).
$c$ is the Y-intercept: This is the point where the line crosses the vertical y-axis. It’s the starting value when $x = 0$.
Worked Examples
Example 1: From a Graph
Find the gradient and y-intercept of the line on the graph.
- Find the y-intercept ($c$): The line crosses the y-axis at 2. So $c = 2$.
- Find the gradient ($m$): Pick two clear points, e.g., $(0, 2)$ and $(3, 1)$.
Rise $= 1 – 2 = -1$.
Run $= 3 – 0 = 3$.
Gradient $= \frac{\text{Rise}}{\text{Run}} = \frac{-1}{3}$.
Equation: $y = -\frac{1}{3}x + 2$
Example 2: From an Equation
Find the gradient and y-intercept of $2y + 6x = 10$.
- Rearrange into the form $y = mx + c$.
- Subtract $6x$ from both sides: $2y = -6x + 10$.
- Divide all terms by 2: $y = -3x + 5$.
Answer: Gradient ($m$) = −3, Y-intercept ($c$) = 5.
Example 3: Interpreting in Context
A plumber charges a £40 call-out fee plus £25 per hour. Write an equation for the total cost ($C$) for ($h$) hours and interpret the gradient and y-intercept.
The equation is set up just like $y = mx + c$.
- The “per hour” charge is the rate of change, so it’s the gradient: $m = 25$.
- The fixed “call-out fee” is the starting value, so it’s the y-intercept: $c = 40$.
- The gradient (25) represents the cost per hour (£25/hr).
- The y-intercept (40) represents the fixed call-out fee (£40).
Tutor Insights
🤔 Common Misunderstandings
- Mixing up $m$ and $c$: Remember, $m$ is always the number multiplied by $x$.
- Negative Gradients: Students might calculate a positive gradient for a line that is clearly sloping downwards. Always do a visual check!
- Rise over Run: Accidentally calculating run over rise. Always put the change in $y$ on top.
📝 Common Exam Mistakes
- Failing to Rearrange: Trying to identify $m$ and $c$ from an equation like $2y = 4x + 6$ without first dividing everything by 2.
- Not showing working for the gradient when calculating from two points.
- Forgetting units when interpreting the gradient and y-intercept in a real-world problem.
Practice Questions
- For the equation $y = 3x + 1$, state the gradient and the y-intercept.
- Find the gradient and y-intercept of the line given by the equation $2y + 6x = 10$.
- A line has a gradient of 3 and crosses the y-axis at $(0, -2)$. Write down the equation of this line.
Show Answers
- Gradient ($m$) = 3. Y-intercept ($c$) = 1.
- Working: $2y = -6x + 10 \implies y = -3x + 5$.
Answer: Gradient ($m$) = −3, Y-intercept ($c$) = 5. - Answer: $y = 3x – 2$.
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