Sketching curves
Section: Algebra and graphs 2 | Syllabus: Cambridge IGCSE Mathematics (0580)
Sketching Quadratic Graphs from Factorised Form
A quadratic graph can be sketched directly from its equation, without plotting a table of values, by identifying its roots, y-intercept, and overall shape.
- If the equation is given in factorised form, e.g. y=(x-p)(x-q), the graph crosses the x-axis at x = p and x = q
- The y-intercept is found by substituting x = 0 into the equation
- The sign of the x² coefficient determines the shape: positive gives a u-shaped curve (minimum), negative gives an n-shaped curve (maximum)
- A sketch does not need to be to scale, but the key features (roots, intercepts, turning point) should be clearly shown and labelled
Worked Example: Sketching a Quadratic from Factorised Form
- Question: On a diagram, sketch the graph of y=(x+2)(x-4), indicating the values where the graph crosses the axes.
- Step 1: The leading term is x² with a positive coefficient, so the curve is u-shaped
- Step 2: Set y = 0: (x+2)(x-4)=0 x=-2 or x=4
- Step 3: Find the y-intercept by substituting x = 0: y=(2)(-4)=-8
- Answer: u-shaped curve crossing the x-axis at x = -2 and x = 4, and the y-axis at y = -8
Common Mistakes
MistakeSketching a quadratic crossing the x-axis at the values inside the brackets with their sign unchanged, e.g. reading (x+2)(x-4)=0 as x=2 and x=4
Fixsolve each bracket equal to zero separately - x + 2 = 0 gives x = -2, so the sign flips from the value written inside the bracket
MistakeForgetting to find the y-intercept, or substituting the wrong value to find it
Fixalways substitute x = 0 (not a root, or any other value) into the original equation to find where the curve crosses the y-axis
Sketching Cubic Graphs and Repeated Roots
A cubic graph can also be sketched directly from its factorised equation. A repeated factor produces a different feature at the x-axis than a single factor.
- A single (non-repeated) factor, e.g. (x-p), means the curve crosses straight through the x-axis at x = p
- A repeated factor, e.g. (x-p)^2, means the curve touches the x-axis at x = p and turns back, rather than crossing through - a turning point sitting on the axis
- The sign of the x³ coefficient determines the overall direction: positive means the curve rises overall from bottom-left to top-right; negative means it falls overall from top-left to bottom-right
- As with quadratics, the y-intercept is found by substituting x = 0
Worked Example: Sketching a Cubic with a Repeated Root
- Question: On a diagram, sketch the graph of y=(x+1)(x-2)^2, indicating the values where the graph crosses the axes.
- Step 1: The leading term is x³ with a positive coefficient, so the curve rises overall from bottom-left to top-right
- Step 2: Set y = 0: (x+1)(x-2)^2=0 x=-1 (single root) or x=2 (repeated root)
- Step 3: Find the y-intercept by substituting x = 0: y=(1)(-2)^2=(1)(4)=4
- Answer: positive cubic crossing the x-axis at x = -1, touching it at x = 2, and crossing the y-axis at y = 4
Common Mistakes
MistakeDrawing the curve crossing straight through the x-axis at a repeated root, the same way it does at a single root
Fixa repeated factor means the curve only touches the axis and turns back at that point, rather than crossing through it
MistakeGetting the overall left-to-right direction of the cubic wrong by ignoring the sign of the leading coefficient
Fixcheck the sign of the x³ term first - it fixes which corner the curve starts and ends in
Interpreting a Sketch Graph
Sketch graphs are also used the other way round - reading the key features already marked on a sketch to answer questions about the function, without needing the full equation to start with.
- The x-values where the curve crosses or touches the x-axis are the roots of the function
- A point where the curve just touches the x-axis (rather than crossing it) indicates a repeated root at that value
- The value where the curve crosses the y-axis is the value of the function when x = 0
- The overall shape (u, n, S, or reversed S) confirms whether the function is quadratic or cubic, and the sign of its leading coefficient
Worked Example: Reading Information from a Sketch
- Question: A sketch shows a u-shaped quadratic curve crossing the x-axis at x = -4 and x = 2, and crossing the y-axis at y = -8. Write down a possible equation for the curve in factorised form.
- Step 1: Roots at x = -4 and x = 2 suggest the factors (x+4) and (x-2)
- Step 2: Check the y-intercept this gives: substituting x = 0 into y=(x+4)(x-2) gives y=(4)(-2)=-8
- Step 3: This already matches the y-intercept shown on the sketch, so no extra scale factor is needed
- Answer: y = (x + 4)(x − 2)
Common Mistakes
MistakeAssuming the basic factorised form from the roots alone must automatically give the correct y-intercept, without checking
Fixalways check the y-intercept the basic factorised form gives, and compare it with the value shown on the sketch, to see whether a scale factor is needed in front of the brackets
MistakeConfusing a curve that touches the x-axis with one that crosses it, when writing down the roots
Fixtouching the axis means a repeated root (a squared factor); crossing straight through means a single, non-repeated root
Sketching Exponential and Trigonometric Graphs
Extended Only
Exponential and trigonometric graphs can also be sketched by recognising their key features, without needing to plot a table of values.
- For an exponential graph y=a^x with a > 1, sketch a curve passing through (0, 1), lying just above the x-axis for negative x, and rising steeply for positive x
- For y = sin x or y = cos x over 0° to 360°, sketch a smooth repeating wave between y = -1 and y = 1, marking the maximum, minimum, and any points where the curve crosses the x-axis
- A sketch of a trigonometric or exponential graph should still show its key features clearly labelled, even though it is not drawn from a table of values
Worked Example: Sketching an Exponential Graph
- Question: On a diagram, sketch the graph of y=2^x, indicating the value where the graph crosses the y-axis.
- Step 1: Since 2 > 1, the curve rises steeply for positive x, and lies just above the x-axis (never touching it) for negative x
- Step 2: Find the y-intercept by substituting x = 0: y=2^0=1
- Answer: exponential growth curve passing through (0, 1), with the x-axis as an asymptote
Common Mistakes
MistakeSketching an exponential curve that crosses the x-axis
Fixy=a^x is never zero or negative - the x-axis is an asymptote, so the curve should only ever approach it, never touch or cross it
MistakeSketching a sine or cosine wave that doesn't return to its starting height after 360°
Fixboth graphs are periodic - the shape from 0° to 360° repeats exactly if the sketch is extended further
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