Curves and Equations

Try this

Choose “circle, center O, radius 5”. Drag $P$ to $(3, 4)$ , $(- 4, 3)$ , $(0, - 5)$ . Does the light come on each time? Now drag it inside the circle and outside: is $F (P)$ positive or negative?
Choose “two lines”. $x^{2} - y^{2} = 0$ is just one equation, so why does its graph consist of two lines? (Hint: $x^{2} - y^{2} = (x + y) (x - y)$ .)
Choose “a single point” and “nothing at all”. Search the whole plane: how many times can you make the light come on in each case?
Switch “Show curve” off. Drag only $P$ , watch only the light, and mark the points where it comes on, one by one. The more points you mark, the clearer the curve becomes. This is plotting by points.

In a rectangular coordinate system, if a curve $C$ and an equation $F (x, y) = 0$ satisfy both

① the coordinates of every point of the curve $C$ satisfy the equation $F (x, y) = 0$ , and
② every point whose coordinates satisfy $F (x, y) = 0$ lies on the curve $C$ ,

then $F (x, y) = 0$ is called the equation of the curve $C$ , and $C$ is called the curve (graph) of the equation. Condition ① says the equation does not “miss” any point of the curve; ② says it does not “add” any point off the curve.

Predict first

What is the graph of the equation $(x - 1)^{2} + (y + 2)^{2} = 0$ ?

A circleA single pointTwo linesNothing at all

Choose an answer first; then you can see the result.

Why: from a curve to its equation

Finding the equation of a curve means translating the geometric condition “the point is on the curve” into an equation in the coordinates $(x, y)$ . The usual steps: set up axes → let $P (x, y)$ be any point of the curve → write down the condition $P$ satisfies → express it in coordinates and simplify → check whether any points were gained or lost.

Example: the equation of the perpendicular bisector of A(−1, 3) and B(3, 1)

Step 1 / 4

Take a point, write the condition
The perpendicular bisector is the locus of points equidistant from $A$ and $B$ . Let $P (x, y)$ be any point on it; the condition is $∣ P A ∣ = ∣ P B ∣$ .

$∣ P A ∣ = ∣ P B ∣$
Write it in coordinates
Simplify
Check condition ②

If you only check half

Break it

Counterexample lab: remove one condition

Conclusion: $x^{2} + y^{2} - 25 = 0$ is the equation of the curve $C$

When $C$ is the whole circle, the two lights always come on together and go off together.

Reading the graph from the equation: symmetry

Given an equation, do not rush to plot points. Replace $y$ by $- y$ : if the equation does not change, then whenever $(x, y)$ is on the graph so is $(x, - y)$ , and the graph is symmetric about the $x$ -axis. Likewise, if replacing $x$ by $- x$ changes nothing, it is symmetric about the $y$ -axis; and if replacing both at once changes nothing, it is symmetric about the origin.

Pick a curve, then click the three symmetry buttons.

P^{'}

is the mirror image of

P

; a solid ring means it satisfies the equation too.

The cubic curve $y = x^{3}$ is symmetric only about the origin: replacing both gives $- y = - x^{3}$ , and multiplying by $- 1$ brings back the original equation. Note that if a curve has two of these symmetries, it automatically has the third; but a curve symmetric about the origin need not be symmetric about either axis.

Practice

Exercise 1Basic

The point $(a, 2)$ lies on the curve $x^{2} - 2 x y + 3 = 0$ . Find all possible values of $a$ (separate them with commas).

Exercise 2Basic

A moving point $P$ is twice as far from the origin as from the point $(3, 0)$ . Find the $x$ -coordinates of the points where the locus of $P$ meets the $x$ -axis (separate them with commas).

Exercise 3Advanced

The points whose distances to the $x$ -axis and to the $y$ -axis have product $4$ form a curve $C$ . How many points of $C$ have both coordinates integers?

Exercise 4Challenge

How many points does the graph of $(x + y - 1) x - 2 = 0$ have in common with the $y$ -axis?

Try this

Why: from a curve to its equation

Example: the equation of the perpendicular bisector of A(−1, 3) and B(3, 1)

If you only check half

Counterexample lab: remove one condition

Reading the graph from the equation: symmetry

Practice

Mistake Museum · Exhibits from this stop

Writing down only half of the locus

Squaring, and gaining points that do not belong