Tables, function graphs and graphical solutions
Plot ordered pairs with suitable scales and use intersections to solve an equation approximately.
Core: ax + b, ±x² + ax + b and a/x with integer constants. Extended also includes rational coefficients, cubic, reciprocal-square, square-root, inverse-square-root and exponential forms and their sums.
Before you begin
- Substitute negative values into expressions with brackets.
- Plot ordered pairs using labelled axes and consistent scales.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What is x² − 2 at x = −2?
What you will learn
- Construct tables for linear, quadratic and reciprocal functions.
- Recognise Extended power and exponential graph families and their domains.
- Read roots and line–curve intersections as graphical solutions.
Make the table before drawing
For y = x² − 2, inputs −2, −1, 0, 1 and 2 give outputs 2, −1, −2, −1 and 2. Plot the pairs, including their signs, on labelled axes. Draw a smooth curve through the points. A quadratic is not a chain of straight segments merely because the table has a few sampled inputs.
For y = 4/x, x = 0 is undefined and must not receive a plotted point. Inputs −4, −2, −1, 1, 2 and 4 produce −1, −2, −4, 4, 2 and 1. Plot two separate branches rather than joining through the forbidden input. Use extra points where curvature is stronger and select a scale that shows the requested interval.
Extended: identify the formula's domain and shape
Positive cubic powers x³ allow positive and negative inputs and have an S-shaped basic graph. The reciprocal square 1/x² is positive on both sides of zero and is undefined at zero. The square root √x has real domain x ≥ 0; the inverse square root 1/√x requires x > 0. Negative and fractional powers therefore need domain checks before a table is formed.
An exponential such as y = 2ˣ has outputs 1/4, 1/2, 1, 2 and 4 at x = −2, −1, 0, 1 and 2. It stays positive. A decay model such as y = 80(1/2)ˣ halves for each unit input increase. A shifted or combined expression must be evaluated as a whole, for example y = 2x + 3/x² for x ≠ 0.
An intersection satisfies both expressions
The roots of y = f(x) are the x-coordinates where the curve meets y = 0. To solve f(x) = g(x), draw both graphs on the same axes and read the x-coordinates of their intersections. The y-coordinates are useful for checking the points but are not themselves the solutions for x.
For x² − 2 = x, draw the parabola y = x² − 2 and the line y = x. Their intersections are (−1, −1) and (2, 2), so the solutions are x = −1 and 2. On a hand-drawn graph a reading is generally approximate; state the accuracy that the scale supports. Algebra can check a reading, but use the requested graphical method when the question requires it.
Pause and explain
When solving f(x) = g(x) from two graphs, which coordinates give x?
Worked example
Use a table and a graph to solve x² − 2 = x.
Show the worked solution
- For x = −2, −1, 0, 1, 2, the parabola's y values are 2, −1, −2, −1, 2.
- Plot the smooth parabola and the straight line y = x on the same axes.
- Read the intersections (−1, −1) and (2, 2); their x-coordinates solve the equation.
Answer x = −1 and 2
From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
Make a table for y = −x² + 4 at x = −2, −1, 0, 1, 2 and state its roots.
Give me a hint
Square the bracketed input before applying the outside negative sign.
Compare my reasoning
- The outputs are 0, 3, 4, 3 and 0.
- Plot a downward-opening smooth curve with symmetry about x = 0.
- It meets the horizontal axis at x = −2 and x = 2.
y values 0, 3, 4, 3, 0; roots −2 and 2
Look for these in your work
- Distinguished −x² from (−x)².
- Read roots from zero outputs.
For y = 6/x, find y at x = −3, −2, 2, 3 and explain why x = 0 cannot be plotted.
Give me a hint
Evaluate each quotient and keep the branches separate.
Compare my reasoning
- The negative inputs give y = −2 and −3.
- The positive inputs give y = 3 and 2.
- At zero the denominator vanishes, so there is no graph point and no line joining the two branches through it.
−2, −3, 3, 2 respectively; undefined at x = 0
Look for these in your work
- Kept input and output signs paired.
- Excluded division by zero.
Extended: compare model A, y = 4x + 4, and model B, y = 2ˣ, at integer x from 0 to 5. Between which consecutive inputs do their graphs switch order?
Give me a hint
Build both tables on the same input values.
Compare my reasoning
- A gives 4, 8, 12, 16, 20, 24; B gives 1, 2, 4, 8, 16, 32.
- At x = 4, A exceeds B; at x = 5, B exceeds A.
- The continuous graphs cross between four and five; the table alone does not give an exact intersection.
Between x = 4 and x = 5
Look for these in your work
- Compared models at matching inputs.
- Separated a bracketed interval from an exact graphical reading.
Common mistakes
- Joining reciprocal branches across zero.
- Drawing a curved function as a polygon without instruction.
- Reporting intersection y values when solving for x.
What can you explain now?
State the roots of y = x² − 9.
Compare with the explanation
x = −3 and 3
The curve has zero output when x² = 9, giving both square-root signs.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
Plot a quadratic and a reciprocal using your own tables. Extended: also compare x³, 1/x², √x, 1/√x and 2ˣ, stating each real domain.
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.
Original StudyForA teaching content · AI-assisted checks · Human teacher review pending.
Number contains 19 sequenced lessons; Algebra and graphs contains 21; Coordinate geometry contains 10 teaching lessons and a mixed checkpoint. Nine earlier overviews support selected topics across the syllabus. The other six syllabus areas, full cumulative assessment and human teacher review are not yet complete. Written lesson practice is self-checked, not automatically graded. The checkpoint samples skills and does not save an exam grade or certify mastery.
Check the official syllabus 2025–2027 Open related practice and resources