Graphing Basics
Plot any equation, find intersections and zeros in a single click, and use coordinate labels to read exact values.
Step-by-step
- Click the Desmos icon in Bluebook's question toolbar. The calculator opens in a side panel; the question stays visible.
- Type the equation into any expression line. For $y = 2x + 3$ just type
y=2x+3, Desmos auto-converts * implicit multiplication and renders exponents.
- Read the graph. Desmos auto-fits the viewport to show interesting features. Use two-finger scroll on trackpad (or pinch zoom on tablet) to zoom.
- Click anywhere interesting, an intersection, an x- or y-intercept, a vertex, and Desmos shows the exact coordinate in a grey label.
- Add a second equation on a new line to overlay. The intersection(s) appear as colored dots; click each for coordinates.
Worked example: linear equation
Problem: The graph of $y = 3x - 5$ crosses the x-axis at what value of $x$?
- Type
y=3x-5. Graph draws instantly.
- Click the single point where the line crosses the x-axis. Label shows $(1.667, 0)$.
- Desmos can also show exact fractions, click the wrench → "Show labels as exact values" to see $\left(\tfrac{5}{3}, 0\right)$.
Answer: $x = \tfrac{5}{3}$. Zero pencil work.
Worked example: quadratic
Problem: What is the minimum value of $f(x) = x^2 - 6x + 7$?
- Type
y=x^2-6x+7.
- Click the vertex of the parabola. Label shows $(3, -2)$.
- The minimum value is the $y$-coordinate of the vertex: $-2$.
Pencil-and-paper: completing the square or using $x = -\tfrac{b}{2a}$ takes 60–90 seconds. Desmos takes 5.
Worked example: function intersection
Problem: At what positive value of $x$ does $y = x^2$ equal $y = 2x + 3$?
- Type each on its own line:
y=x^2, y=2x+3.
- Two intersection dots appear. Click the one with positive $x$. Label reads $(3, 9)$.
Answer: $x = 3$.
Common traps
Watch for these gotchas
- Decimal vs. exact. If the answer choices are fractions or radicals, turn on exact-value labels (wrench menu), otherwise you'll have to convert a decimal back by hand.
- Off-screen intersections. Desmos only auto-fits to what it can see. If a question mentions a large range, zoom out first or the intersection might not appear.
- Typing $\pi$ and $e$. Spell out
pi and e, Desmos converts automatically. Do not copy the symbols.
- Implicit vs. explicit functions. For $x^2 + y^2 = 25$ type it exactly like that, Desmos handles non-$y=$ equations fine. A missing equals sign shows nothing.
Try it yourself
Practice problem
The graph of $y = 2x^2 - 8x + 5$ has a minimum value at some point $(h, k)$. Use the Desmos calculator below to find the vertex. What is $k$?
Show answer
Steps: Type y=2x^2-8x+5, click the vertex, read the coordinates. Answer: vertex is $(2, -3)$, so $k = -3$.
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