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Desmos graphs and tables

Use the built-in calculator to read answers off a graph or a table when the algebra is long.

When to use it

  • The question asks where two graphs meet, how many solutions an equation has, or a maximum or minimum.
  • The equation has an absolute value, a square root or a rational part that makes algebra fiddly.
  • You need many values of a function or a sequence quickly (use a table).

How it works

  1. Type each side of the equation as its own graph (y = left side, y = right side). Intersection points are the solutions.
  2. Tap an intersection, a vertex or an intercept to read its exact coordinates. Zoom or use the “fit” button if you cannot see the graph.
  3. For a count of solutions, count where one graph crosses the x-axis, or where the two graphs cross each other.
  4. For repeated calculations, make a table: type the rule in the first row, then read the values.
Watch out: Reading a screen that is zoomed in on the wrong region. If you see no intersection, zoom out before deciding there is none. And a decimal you read off the graph still has to match a choice.

Time: Typing takes 20–30 seconds. It pays off on absolute-value, quadratic-extremum and intersection questions. It does not pay off on simple linear equations.

Worked examples

Example 1

Two graphs, two meeting points: type both and read them.

The graphs of y = x² − 4 and y = 2x − 1 intersect at two points. What is the sum of their x-coordinates?

  1. A−3
  2. B2Correct
  3. C3
  4. D4

Using the technique

  1. Type y = x² − 4 and y = 2x − 1.

  2. Tap the two intersections. Desmos shows (−1, −3) and (3, 5).

  3. The x-coordinates are −1 and 3, and −1 + 3 = 2.

Practice this set

Example 2

An absolute value inside a quadratic makes algebra messy, but the graph is easy to read.

How many real solutions does x² − |x| − 6 = 0 have?

  1. A1
  2. B2Correct
  3. C3
  4. D4

Using the technique

  1. Type y = x² − abs(x) − 6. Solutions of the equation are the places where this graph meets the x-axis.

  2. The graph crosses the x-axis twice, at x = −3 and x = 3. (Check: 9 − 3 − 6 = 0.)

  3. Two real solutions. The graph is symmetric because of |x|, so crossings come in pairs.

Practice this set

Example 3

A maximum value is the top of the parabola.

What is the maximum value of f(x) = −2x² + 8x − 1?

  1. A3
  2. B7Correct
  3. C8
  4. D15

Using the technique

  1. Type y = −2x² + 8x − 1. The parabola opens downward, so it has a highest point.

  2. Tap the top: Desmos shows the vertex (2, 7).

  3. The maximum value is the y-coordinate, 7. (Check: −2(4) + 16 − 1 = 7.)

Practice this set

Practice with a hard-only Math set (calculator on)

Related lessons: Quadratic functions and parabolas · Radical and absolute-value equations · Function notation