Graphing Linear Functions

Overview

In this second lesson of the unit “Advanced Functions Exploration”, students learn to graph a straight-line relationship using slope-intercept form and then interpret the gradient (slope) and intercepts from both the equation and the graph. They will build on prior knowledge of Cartesian coordinates and basic plotting skills to model real relationships.

Learning intentions

Students will be able to:

• Construct a straight-line graph from a linear function of the form (y = mx + c).
• Identify gradient (m) and y-intercept (c) from the equation, and explain what they mean on the graph.
• Determine the x-intercept and y-intercept from both an equation and a graph.
• Use a practical interpretation of slope and intercepts in context.

Success criteria

Students can:

• Plot at least two correct points and draw a straight line that matches the given equation.
• State the gradient as “rise/run” (or equivalent) and link it to whether the line rises or falls.
• Correctly read and label the x-intercept and y-intercept from a graph.
• Describe the meaning of slope and intercepts in a given situation (in words, not just numbers).

Curriculum links

• Straight-line graphs: construct a straight-line graph using (y = mx + c), where (m) is slope (gradient) and (c) is y-intercept.
• Slope-intercept form: understand and use (y = mx + c).
• Intercepts: determine slope, x-intercept and y-intercept from both equation and graph.
• Practical modelling: construct and analyse a straight-line graph to model a given linear relationship (e.g. cost against litres).

Lesson structure (60 minutes)

1. 0–8 min · Quick review + hook. Teacher shows two small graphs (one rising, one falling) without equations and asks: “How can we tell what the gradient is just by looking?” Students discuss what changes from left to right and identify which line has the greater increase.

2. 8–18 min · Direct teach: slope-intercept plotting method. Teacher demonstrates using one linear equation in the form (y = mx + c), showing step-by-step: find the y-intercept (where (x=0)), then use the gradient to find a second point, then draw the straight line. Students follow along and complete a “plotting scaffold” for a second equation using the same steps.

3. 18–30 min · Guided practice: Cartesian coordinate system accuracy. Teacher gives a blank coordinate grid and two equations (e.g. one with positive gradient, one with negative gradient) and models how to read/write points accurately, including consistent scale and correct ordering ((x, y)). Students plot points, label the axes, and draw the line; teacher checks for common errors (swapping x and y, inconsistent scaling, drawing curved lines).

4. 30–42 min · Intercepts from equation and graph. Teacher runs a worked example showing how to find intercepts from the equation:

• y-intercept by substituting (x=0).
• x-intercept by setting (y=0) and solving for (x). Teacher then shows the graph of the same function and highlights where those intercepts appear. Students answer: “What does the intercept mean in context?” using one sentence per intercept.

5. 42–55 min · Applied modelling mini-task (fuel/cost example). Teacher presents a practical scenario where cost depends linearly on litres (or a similar everyday context) and provides the corresponding equation in slope-intercept form. Students:

• construct the graph,
• interpret the gradient as “change in cost per litre”,
• interpret the y-intercept as the starting cost,
• state the x-intercept meaning (e.g. when cost would be zero, if meaningful). Teacher prompts students to keep interpretations tied to the graph.

6. 55–60 min · Exit ticket (quick check). Students complete two short prompts on paper:

• “Given (y=2x-3), state gradient and y-intercept.”
• “From a provided small graph, identify x-intercept and whether the line rises or falls.”

Resources

• Graph paper or printed coordinate grids (multiple sizes)
• Straight-edge, coloured pencils/markers
• Small “plotting scaffold” worksheet: given (y=mx+c), find y-intercept, use gradient to find second point, join points
• Example equations and two intercept-focused worked examples
• Scenario card for the practical modelling task (fuel/cost or similar linear context)
• Exit ticket sheet
• Whiteboard/interactive board for teacher demonstrations
• Calculator (optional) for checking algebra, not for plotting

Assessment

• Teacher observation during plotting: accuracy of points, correct line drawing, and correct coordinate order ((x,y)).
• Targeted questioning during guided practice: “What is the y-intercept here?” “How do you know your second point is correct?”
• Exit ticket checking gradient, intercept identification from equation and from a graph, and correct direction of change (rising/falling).

Differentiation

• Support: provide a partially completed plotting scaffold (y-intercept already marked, remaining point steps prompted); include a reference “rise/run” example on the worksheet.
• Support for language: sentence starters for interpretation (e.g. “The gradient tells us that for each 1 increase in x, y changes by …”).
• Extension: if a student finishes early, ask them to generate a second equation that would produce the same y-intercept but a different gradient, then sketch the line and state the intercepts again.
• SEN/EAL considerations: allow working with a partner or using verbal explanation; use consistent notation (x) and (y) and model reading the graph slowly with finger tracing along grid lines.