To solve f(x) = g(x) from a graph, draw y = f(x) and y = g(x) on the same axes. The x-values of the points where they cross are the solutions. Because you read them from a drawing, the answers are approximate, and a substitution check shows how close you are.
This is the last lesson in graphs and transformations, and it links graphs back to equations. It builds on plotting accurately and on the ideas in translating and reflecting.
How do you use intersections?
First, write the equation as two functions, one on each side. To solve x² = x + 3, draw y = x² and y = x + 3.
Second, draw both carefully on the same axes with a suitable scale, using a table of values for any curve.
Third, read the x-coordinate at each crossing. Give the answer to the accuracy requested and check it in the original equation.
Worked example
Use graphs to solve x² − x − 3 = 0, giving each answer to 1 decimal place.
Step 1, rearrange: x² = x + 3. Draw y = x² and the line y = x + 3.
Step 2, table for y = x²:
| x | −2 | −1 | 0 | 1 | 2 | 3 |
|---|---|---|---|---|---|---|
| y | 4 | 1 | 0 | 1 | 4 | 9 |
The line y = x + 3 passes through (−2, 1), (0, 3) and (2, 5).
Step 3, read: the graphs cross near x = −1.3 and x = 2.3.
Step 4, check: at x = 2.3, x² = 5.29 and x + 3 = 5.3, a difference of only 0.01. At x = −1.3, x² = 1.69 and x + 3 = 1.7, again a difference of 0.01. Both readings are accurate.
The solutions are x ≈ −1.3 and x ≈ 2.3. The formula would give 2.303 and −1.303, which agrees.
The mistake to watch for
A common slip is to give the coordinates of the crossing, or only one of the crossings.
Mistaken answer: “The solution is (2.3, 5.3).”
The student gave a point. The equation asks for values of x, and there are two crossings.
The correction is to read only the x-coordinates: x ≈ −1.3 and x ≈ 2.3. The y-value 5.3 is useful for a substitution check, but it is not a solution. Always count the crossings before writing the answer.
Check yourself
Try these without a calculator, then open each answer.
1. The graphs of y = 2x + 1 and y = x² − 2 cross. Find the x-values of the crossings exactly, and check them.
Show answer
Set x² − 2 = 2x + 1, so x² − 2x − 3 = 0, which factorises as (x − 3)(x + 1) = 0. Then x = 3 or x = −1. Check: at x = 3, 2(3) + 1 = 7 and 9 − 2 = 7. At x = −1, 2(−1) + 1 = −1 and 1 − 2 = −1.
2. Which two graphs would you draw to solve x² = 2x + 1, and what are the solutions to 1 decimal place?
Show answer
Draw y = x² and y = 2x + 1. The equation becomes x² − 2x − 1 = 0, with x = 1 ± √2, which gives x ≈ 2.4 and x ≈ −0.4. Check: 2.4² = 5.76 and 2 × 2.4 + 1 = 5.8. Also (−0.4)² = 0.16 and 2(−0.4) + 1 = 0.2.
3. The curve y = x² − 3x is drawn. Which horizontal line do you add to solve x² − 3x = 4, and what are the solutions?
Show answer
Add y = 4. The equation becomes x² − 3x − 4 = 0, so (x − 4)(x + 1) = 0 and x = 4 or x = −1. Check: 16 − 12 = 4 and 1 + 3 = 4.
Where this leads next
Put everything together in the graphs and transformations practice set. The non-calculator working trainer helps when you need to check a substitution by hand.
A graph can give you a good approximation, yet some students still lose marks on the reading or the wording. A teacher in online one-to-one Mathematics tuition can see whether the issue is scale, reading or algebra.