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Solving equations with graphs
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Some equations, like 2x = x + 3, cannot be solved with algebra. A graph shows how many solutions there are and roughly where they are.
Intersections
The solutions of f(x) = g(x) are the x-coordinates of the points where y = f(x) and y = g(x) cross.
The number of crossings is the number of real solutions.
The number of crossings is the number of real solutions.
Using a graph you already have
You have the graph of y = x2 − 4x + 2. To solve x2 − 3x − 1 = 0:
x2 − 4x + 2 − (x2 − 3x − 1) = 3 − x, so when x2 − 3x − 1 = 0, x2 − 4x + 2 = 3 − x
So draw the line y = 3 − x and read off the x-coordinates where it crosses.
x2 − 4x + 2 − (x2 − 3x − 1) = 3 − x, so when x2 − 3x − 1 = 0, x2 − 4x + 2 = 3 − x
So draw the line y = 3 − x and read off the x-coordinates where it crosses.
Transcendental equations
Equations mixing powers like 2x with x, or sin x with x, are solved graphically.
2x = x + 3: sketch y = 2x and y = x + 3. They cross twice, once between −3 and −2, once between 2 and 3.
A table of values can locate a root: f(x) = 2x − x − 3 changes sign between x = 2 (f = −1) and x = 3 (f = 2).
2x = x + 3: sketch y = 2x and y = x + 3. They cross twice, once between −3 and −2, once between 2 and 3.
A table of values can locate a root: f(x) = 2x − x − 3 changes sign between x = 2 (f = −1) and x = 3 (f = 2).
The graph of y = x3 is drawn. Which line should be added to solve x3 − 2x − 1 = 0?
- Rearrange: x3 = 2x + 1
- The left side is the curve already drawn
Answer: Draw y = 2x + 1 and read the x-coordinates of the crossings
Solutions of f(x) = g(x) are x-coordinates of intersections. Rearrange so one side is the graph you already have; the other side is the line to draw.
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