Graphs of functions
Section: Algebra and graphs 2 | Syllabus: Cambridge IGCSE Mathematics (0580)
Plotting Graphs from a Table of Values
To draw the graph of a curve, first complete a table of values by substituting each x-value into the equation, then plot the resulting points and join them with a smooth curve.
- Substitute each given x-value into the equation to find the matching y-value
- Plot each pair of values as a coordinate point
- Join the points with a single smooth curve (not straight line segments), unless the graph is known to be a straight line
- Use a sensible scale so that every point fits on the grid provided
Worked Example: Completing a Table of Values
- Question: Complete the table of values for y=x^2-2x for -2 x3.
- Step 1: Substitute each x-value into y=x^2-2x
- Step 2: Complete the table:
x -2 -1 0 1 2 3 y 8 3 0 -1 0 3 - Answer: table completed as above, ready to plot and join with a smooth curve
Common Mistakes
MistakeJoining plotted points with straight line segments instead of a smooth curve
Fixfor curved graphs (anything other than y = mx + c), draw a single smooth curve through the points, not a series of straight lines
MistakeMaking a sign error when substituting a negative x-value, e.g. treating (-2)² as -4 instead of 4
Fixwork out the squared (or cubed) term first, keeping track of the sign carefully, before combining it with the rest of the expression
Quadratic and Cubic Graphs
The graph of a quadratic function y=ax^2+bx+c is a smooth curve called a parabola. The graph of a cubic function y=ax^3+bx^2+cx+d has a distinctive S-shape.
- A quadratic graph (parabola) has a single turning point - a minimum if a > 0 (u-shaped) or a maximum if a < 0 (n-shaped)
- A quadratic graph is symmetrical about a vertical line through its turning point
- A cubic graph typically has two turning points (a local maximum and a local minimum) and an S-shape, though it can also have no turning points at all
- The overall shape depends on the sign of the leading coefficient: positive gives a curve rising overall from left to right; negative gives one falling overall from left to right
Worked Example: Finding the Turning Point of a Quadratic Graph
- Question: The graph of y=x^2-4x+3 is drawn. Find the coordinates of its turning point.
- Step 1: The turning point lies on the line of symmetry, x=(-b/2a)=(4/2)=2
- Step 2: Substitute x = 2 into the equation: y=(2)^2-4(2)+3=-1
- Answer: Turning point at (2, -1) - a minimum, since a = 1 is positive
Common Mistakes
MistakeAssuming every quadratic graph opens upward, even when the coefficient of x² is negative
Fixcheck the sign of a - positive gives a minimum (u-shape), negative gives a maximum (n-shape)
MistakeForgetting that a cubic graph does not always have an S-shape with two turning points
Fixsome cubic graphs rise (or fall) smoothly throughout, with no turning points at all - check the actual shape rather than assuming
Reciprocal Graphs
The graph of a reciprocal function y=(a/x) consists of two separate curved branches, one in each of two opposite quadrants, and never touches either axis.
- A reciprocal graph has two branches, symmetrical about the origin
- As x gets very large (positive or negative), y gets closer and closer to 0, but never reaches it
- As x gets very close to 0, y becomes very large (positive or negative) - x = 0 itself is never reached, since the function is undefined there
- A line that a curve approaches but never touches (here, the x-axis and y-axis) is called an asymptote
Worked Example: Completing a Table for a Reciprocal Graph
- Question: Complete the table of values for y=(12/x) for x = -6, -4, -3, -2, 2, 3, 4, 6.
- Step 1: Substitute each x-value into y=(12/x)
- Step 2: Complete the table:
x -6 -4 -3 -2 2 3 4 6 y -2 -3 -4 -6 6 4 3 2 - Answer: table completed as above - note that x = 0 is never included, since y=(12/x) is undefined there
Common Mistakes
MistakeIncluding x = 0 in the table of values, or trying to plot a point there
Fixa reciprocal function is undefined at x = 0 - it must always be excluded from the table
MistakeJoining the two branches of a reciprocal graph across the axes with a single curve
Fixthe two branches never meet - draw them as two completely separate curves
Using Graphs to Solve Equations
Once a graph has been drawn, it can be used to solve equations - reading values directly from the curve avoids the need for further algebra.
- The roots (solutions) of y = 0 are the x-values where the curve crosses the x-axis
- To solve f(x) = k for a constant k, draw the horizontal line y = k and read off the x-values where it crosses the curve
- To solve two equations simultaneously, draw both graphs on the same grid - the coordinates of the points where they intersect are the solutions
Worked Example: Solving an Equation Graphically
- Question: The graph of y=x^2-x-6 crosses the x-axis at two points. Use the graph to write down the solutions of x^2-x-6=0.
- Step 1: The solutions are the x-values where the curve crosses the x-axis (y = 0)
- Step 2: From the graph, the curve crosses the x-axis at x = -2 and x = 3
- Answer: x = -2 or x = 3
Common Mistakes
MistakeReading a y-coordinate instead of an x-coordinate when finding roots from a graph
Fixthe roots of an equation are always x-values - read where the curve crosses the x-axis, not any other point
MistakeOnly finding one solution when a curve crosses the x-axis at two (or more) points
Fixcheck the whole graph across its full domain - a quadratic can have two roots, and other curves can have even more
Exponential Graphs
Extended Only
The graph of an exponential function y=a^x curves increasingly steeply for a > 1, always passes through (0, 1), and always lies above the x-axis, no matter how far left it is drawn.
- For y=a^x with a > 1, the graph rises steeply as x increases (exponential growth), and gets closer and closer to the x-axis (but never reaches it) as x decreases
- Every graph of the form y=a^x passes through (0, 1), since a^0=1 for any a
- The x-axis is an asymptote - the curve never crosses or touches it, since a^x>0 for every value of x
Worked Example: Completing a Table for an Exponential Graph
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