How to use Desmos on the SAT

Example 1: a system with decimals

Problem Solve the system $0.4x + 1.3y = 7.3$ and $2.5x - 0.6y = 2$.

1. Type the first equation exactly as written on line 1, and the second on line 2. Desmos draws two lines.
2. Tap the point where they cross. Desmos shows $(2, 5)$.
3. Check in the first equation.
   $$0.4(2) + 1.3(5) = 7.3$$
4. Check in the second equation.
   $$2.5(2) - 0.6(5) = 2$$

Answer $(2, 5)$

Example 2: the minimum of a parabola

Problem What is the minimum value of $y = 3x^2 - 21x + 30$?

1. Type $y = 3x^2 - 21x + 30$ and tap the curve. Gray dots appear at the zeros, $(2, 0)$ and $(5, 0)$, and at the vertex, $(3.5, -6.75)$.
2. Check the vertex by hand: it sits halfway between the zeros, at $x = 3.5$.
   $$3(3.5)^2 - 21(3.5) + 30 = -6.75$$

Answer $-6.75$

Example 3: fit a table

Problem A table has the points $(0, 5)$, $(1, 7.5)$, $(2, 11.25)$ and $(3, 16.875)$. Find an equation for the data and predict y when $x = 5$.

1. Add a table in Desmos and enter the points in the $x_1$ and $y_1$ columns.
2. The y values multiply by 1.5 each step, so try an exponential fit: type $y_1 \sim ab^{x_1}$. Desmos shows $a = 5$ and $b = 1.5$.
3. Predict at $x = 5$.
   $$5(1.5)^5 = 37.96875$$

Answer $y = 5(1.5)^x$, and at $x = 5$, y is about 37.97

Example 4: check an equivalent expression

Problem Which expression is equivalent to $(x + 3)^2 - (x - 3)^2$?

1. Type $y = (x + 3)^2 - (x - 3)^2$. The graph is a straight line through the origin, and the point $(1, 12)$ is on it.
2. A line through the origin with slope 12 is $y = 12x$. Type it: it lies exactly on top. The algebra agrees:
   $$(x + 3)^2 - (x - 3)^2 = 12x$$

Answer $12x$