Graphs Of Functions
Drawing and reading graphs accurately
In this note
- 2.10Complete a table, plot points accurately and draw a smooth curve.
- 2.10Read values from a graph: y for a given x, and every x for a given y.
- 2.10Use the graph to solve \(f(x)=k\) and to answer questions in context.
Before you start: Graph shapes. Next: lines on curves in Graphical solutions.
Formula bank
Drawing a curve that earns full marks
General methodHow to draw a graph from a table
- Complete the table with a calculator, to the accuracy asked (often 2 d.p.). The values should change smoothly.
- Work out what one small square is worth on each axis.
- Plot each point with a small cross, accurate to half a small square.
- Draw ONE smooth curve freehand through every point - no ruler, no straight bits, no double lines.
- At a turning point, let the curve go a little beyond the highest/lowest plotted point if the shape needs it.
Reading values
Reading x for a given y: the line \(A=16\) meets the curve twice, at \(x=2\) and \(x=8\). The maximum area is 25 at \(x=5\).
General methodHow to read a graph
- y for a given x: go UP from the x-value to the curve, then ACROSS to the y-axis.
- x for a given y: go ACROSS from the y-value; mark EVERY point where you meet the curve, then go down.
- Draw these lines on the graph (they show your method) and read to the nearest small square.
Example The graph of \(y=x^{3}-3x+1\) is drawn for \(-2\le x\le2\). Use it to solve \(x^{3}-3x+1=0\).
Solutions of y = 0 are where the curve crosses the x-axis.
- The curve crosses the x-axis three times
- read: about −1.9, 0.3 and 1.5
Answer x ≈ −1.9, 0.3, 1.5
Try it yourself
Show answer
Answer 21
Show answer
Answer 4, 6
Ways they test it
Type 1 Complete the table
"Complete the table of values for \(y=x^{3}-3x+1\)." (3 marks)
Spot it a table with gaps.
- Substitute carefully; brackets for negatives.
- Round as asked.
For example \(x=-1.5\)
- \(-3.375+4.5+1\)
Answer 2.13 (2.125)
Type 2 Draw the curve
"On the grid, draw the graph ..." (4 marks)
Spot it a grid after the table.
- Plot to ½ small square.
- One smooth curve.
Type 3 Use the graph
"Use your graph to solve \(x^{3}-3x+1=0\)" or "... = 2".
Spot it 'use your graph'.
- Draw y = k across.
- Read every x.
Type 4 Context
"Find the maximum area" / "the two widths that give an area of 16 cm²".
Spot it a real quantity on the axes.
- Read the turning point or the crossing points.
- Answer with units.
Mistakes to avoid
- ✗ Joining points with a ruler✓ one smooth freehand curveCurves are curved.
- ✗ Giving only one x for a given y✓ every crossing pointA curve can be met more than once.
- ✗ Reading y-values from the table for 'solve f(x) = 0'✓ read the x-values where y = 0Solutions are x-values.
Summary
- Know what one small square is worth.
- Plot to ½ small square; one smooth curve.
- Read up-and-across, or across-and-down; give ALL the x-values.
- Show your reading lines on the graph.
If you see …
- "use your graph to solve f(x) = k"draw y = k and read where it meets the curve.
- a maximum or minimum in contextread the turning point.
Practice
Practice: easy, medium, hard
Easy
Show answer
Answer 0.4
Show answer
Answer 3
Show answer
Answer 25
Medium
Guide me
What is it asking? a value
What do we know? \(0.125-1.5+1\)
Plan round
Show working
- \(0.125-1.5+1=-0.375\)
- −0.38
Answer −0.38
Guide me
What is it asking? a value
What do we know? \(3.375-4.5+1\)
Plan round
Show working
- \(3.375-4.5+1=-0.125\)
Answer −0.13
Guide me
What is it asking? x-values
What do we know? line A = 16
Plan two crossings
Show working
- x = 2 and x = 8
Answer 2, 8
Hard
Guide me
What is it asking? three x-values
What do we know? where y = 0
Plan the curve crosses the x-axis three times
Show working
- between −2 and −1.5: about −1.9
- between 0 and 0.5: about 0.3
- between 1.5 and 2: about 1.5
Answer −1.9, 0.3, 1.5
| Part | Answer / working | Marks | Guidance |
|---|---|---|---|
| −1.9 | B1 | ||
| 0.3 to 0.4 | B1 | ||
| 1.5 | B1 | ||
| Total: 3 marks | |||
Guide me
What is it asking? three x-values
What do we know? draw y = 2
Plan three crossings
Show working
- the line y = 2 meets the curve at about −1.5, −0.3 and 1.9
Answer −1.5, −0.3, 1.9
| Part | Answer / working | Marks | Guidance |
|---|---|---|---|
| line y = 2 | M1 | ||
| −1.5, −0.3, 1.9 | A2 | ||
| Total: 3 marks | |||
Guide me
What is it asking? a width
What do we know? A = x(10 − x)
Plan the top of the ∩
Show working
- maximum at x = 5 (a square)
Answer 5 cm
| Part | Answer / working | Marks | Guidance |
|---|---|---|---|
| 5 | B1 | ||
| Total: 1 mark | |||
Can I do this?
- I can complete a table and draw a smooth, accurate curve.
- I can read values from a graph in both directions.
- I can use a graph to solve equations and answer questions in context.
Anything unticked: revise that section first. All ticked: take the quiz.
Quiz
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