Mathematics • Year 8 • Unit 2 • Lesson 14
Sketching Linear Graphs
Build fluency with the three sketching methods: two-point, gradient-intercept and intercept. One worked example, one guided fill-in, then eight independent sketches.
1. I do, fully worked example
Sketch y = 2x + 1 using the gradient-intercept method (fastest when the equation is in y = mx + c form).
Problem. Sketch y = 2x + 1 on a set of axes.
Step 1, Read m and c.
m = 2, c = 1.
Reason: in gradient-intercept form, the coefficient of x is m and the constant is c.
Step 2, Plot the y-intercept first.
Plot the point (0, 1) on the y-axis.
Step 3, Use the gradient to find a second point.
m = 2 = 2/1. From (0, 1): run 1 right, rise 2 up → (1, 3).
Reason: gradient = rise/run. Treat m = 2 as 2/1.
Step 4, Join the two points with a straight line and label.
Draw a line through (0, 1) and (1, 3). Label it y = 2x + 1.
Check: at x = 2, y = 5. The point (2, 5) should lie on your line. ✓
2. We do, fill in the missing steps
Sketch 2x + 3y = 6 using the intercept method. Fill in each blank. 4 marks
Step 1, y-intercept: set x = 0.
2( ____ ) + 3y = 6 → 3y = ______ → y = ______ . Point: ( ______, ______ )
Step 2, x-intercept: set y = 0.
2x + 3( ____ ) = 6 → 2x = ______ → x = ______ . Point: ( ______, ______ )
Step 3, Plot both intercepts and join with a straight line.
Mark ( ______, ______ ) on the y-axis and ( ______, ______ ) on the x-axis, then draw.
Step 4, Check with another point. At x = ____ (try a value between the intercepts), y should sit on the line. Pick one and verify.
3. You do, independent practice
For each, choose the fastest method, list your two points, and describe the sketch. (You don't have to draw a perfect graph, state the two points clearly and identify the type of line.)
Foundation, gradient-intercept form
3.1 Sketch y = x + 2. State the y-intercept and one other point. 1 mark
3.2 Sketch y = 3x − 2. State the y-intercept and one other point. 1 mark
3.3 Sketch y = −x + 4. State the y-intercept and one other point. 1 mark
3.4 Sketch y = −½x + 3 starting from the y-intercept. (Use m = −1/2: run 2 right, rise 1 down.) 1 mark
Standard, intercept method
3.5 Sketch x + y = 5 using the intercept method. State both intercepts. 2 marks
3.6 Sketch 3x + 4y = 12 using the intercept method. State both intercepts. 2 marks
Extension, special lines
3.7 Sketch y = −3 and describe the line in one sentence. Then sketch x = 2 and describe it in one sentence. 2 marks
3.8 Sketch y = 2x using the two-point method (pick x = 0 and x = 2). Why is this both a line through the origin AND a special case? 2 marks
How did this worksheet feel?
What I'll revisit before next class:
Section 2, We do (2x + 3y = 6)
Step 1: 2(0) + 3y = 6 → 3y = 6 → y = 2. Point: (0, 2).
Step 2: 2x + 3(0) = 6 → 2x = 6 → x = 3. Point: (3, 0).
Step 3: plot (0, 2) and (3, 0); draw line through both.
Step 4: e.g. at x = 1.5: 2(1.5) + 3y = 6 → 3 + 3y = 6 → 3y = 3 → y = 1. Point (1.5, 1) sits exactly on the line ✓.
3.1, y = x + 2
y-intercept (0, 2). Use m = 1: run 1, rise 1 → second point (1, 3). Line goes uphill through both points.
3.2, y = 3x − 2
y-intercept (0, −2). Use m = 3: run 1, rise 3 → second point (1, 1). Steep uphill line.
3.3, y = −x + 4
y-intercept (0, 4). Use m = −1: run 1, rise −1 → second point (1, 3). Gentle downhill line; x-intercept (4, 0).
3.4, y = −½x + 3
y-intercept (0, 3). Use m = −1/2: run 2 right, rise 1 down → second point (2, 2). x-intercept (6, 0).
3.5, x + y = 5
y-int (x = 0): y = 5 → (0, 5). x-int (y = 0): x = 5 → (5, 0). Plot both and join. Downhill line of gradient −1.
3.6-3x + 4y = 12
y-int: 4y = 12 → y = 3 → (0, 3). x-int: 3x = 12 → x = 4 → (4, 0). Join both; the line falls gently from upper-left to lower-right.
3.7, y = −3 and x = 2
y = −3: horizontal line (gradient 0) crossing the y-axis at (0, −3), every point has y = −3.
x = 2: vertical line crossing the x-axis at (2, 0), every point has x = 2. (Vertical lines are NOT in y = mx + c form because their gradient is undefined.)
3.8, y = 2x via two points
At x = 0: y = 0 → (0, 0). At x = 2: y = 4 → (2, 4). Line passes through the origin with gradient 2. Special case: c = 0, so the y-intercept and x-intercept are both the origin (0, 0).