Combining Numbers and Figures

In one sentence

The same expression often has a second identity as a figure:

(x - a)^{2} + (y - b)^{2} \to distance, \frac{y - b}{x - a} \to slope, ∣ z - w ∣ \to distance

As algebra, such expressions need case analysis or derivatives; as figures, the answer is often visible at a glance: a straight segment is the shortest path, tangent lines bound the slope, and the point of a circle farthest from a fixed point lies on the line through the center. Combining numbers and figures means translating back and forth between the two: use the figure to find the way, use the numbers to make it rigorous.

Predict first

What is the minimum value of $f (x) = x^{2} + 4 + (x - 3)^{2} + 1$ ?

10

32

13 + 1

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

Example: the minimum of a sum of radicals

Step 1 / 5

Read the radicals as distances
$x^{2} + 4 = (x - 0)^{2} + (0 - 2)^{2}$ is the distance from the point $P (x, 0)$ to $A (0, 2)$ ; $(x - 3)^{2} + 1$ is the distance from $P$ to $B (3, 1)$ .

$f (x) = ∣ P A ∣ + ∣ P B ∣$
Move P around
Reflect: pull the broken line straight
Equality
Translate back

Stop 1 · Where the roots of a quadratic lie (Vol. 2)

“Where are the roots of the equation?” is a question about numbers; “where does the parabola cross the $x$ -axis?” is a question about a figure. The second needs only a few checks: which way it opens, the axis of symmetry, the values at the endpoints, and the discriminant.

Example: both real roots (possibly equal) of x² − 2ax + a + 2 = 0 lie in the interval (0, 3). Find the range of a

Step 1 / 4

Draw it
Let $f (x) = x^{2} - 2 a x + a + 2$ ; it opens upward. Both roots in $(0, 3)$ means, in the picture, that the points where the parabola meets (or touches) the $x$ -axis lie between $0$ and $3$ .

$f (x) = x^{2} - 2 a x + a + 2$
Read four conditions off the picture
Simplify each one
Intersect

All four conditions are needed. With only $f (0) > 0$ and $f (3) > 0$ , both roots could lie to the right of $3$ : draw a parabola with axis of symmetry $x = 5$ and you will see it.

Stop 2 · The modulus of a complex number (Vol. 4)

$∣ z - w ∣$ is the distance from the point $z$ to the point $w$ in the complex plane. So “the extreme values of a modulus” become “the extreme distances from a point on a circle to a fixed point”.

Example: |z − 3 − 4i| = 2. Find the maximum and minimum of |z|

Step 1 / 4

Translate
$∣ z - (3 + 4 i) ∣ = 2$ : the point $z$ lies on the circle with center $C (3, 4)$ and radius $2$ . $∣ z ∣$ is the distance from $z$ to the origin $O$ .

$∣ z - (3 + 4 i) ∣ = 2, ∣ z ∣ = ∣ O Z ∣$
From the center to the fixed point
The line through the center
Where equality holds

Stop 3 · The range of a trigonometric expression (Vol. 3)

$\frac{sin θ}{cos θ - 2}$ looks like a trigonometry problem, but it is exactly “the slope of the line joining the point $Q (cos θ, sin θ)$ on the unit circle to the fixed point $P (2, 0)$ ”.

Drag

Q

once around the unit circle. At which two positions does the slope reach its extremes? Tap “Draw the two tangents” to check.

One problem, several solutions

Find the range of $y = \frac{sin θ}{cos θ - 2}$ .

Read it as a slope

$y$ is the slope of $P Q$ , with $Q$ on the unit circle. A line $y = k (x - 2)$ through $P (2, 0)$ is tangent to the circle when the distance from the center to the line is $1$ : $\frac{∣2 k ∣}{k ^{2} + 1} = 1$ , so $k = \pm \frac{3}{3}$ . As $Q$ goes once around the circle, the slope varies continuously between those of the two tangents, so the range is $[- \frac{3}{3}, \frac{3}{3}]$ .

CostOne picture gives the answer and where the extremes are reached.

Auxiliary angle

$y cos θ - 2 y = sin θ$ , that is, $sin θ - y cos θ = - 2 y$ , and the left side $= 1 + y^{2} sin (θ - φ)$ . There is a solution if and only if $∣2 y ∣ \leq 1 + y^{2}$ , that is, $3 y^{2} \leq 1$ , which again gives $- \frac{3}{3} \leq y \leq \frac{3}{3}$ .

CostPure algebra, but y must first be moved to the other side.

Stop 4 · Curves and equations (Vol. 7)

How many roots does the equation have? Split it into “left side $=$ right side”, draw each side as a curve, and count the crossings.

Example: how many real roots does √(4 − x²) = x + m have?

Step 1 / 4

The left side is a semicircle
$y = 4 - x^{2}$ is equivalent to $x^{2} + y^{2} = 4$ with $y \geq 0$ : the upper semicircle with center at the origin and radius $2$ , with endpoints $(- 2, 0)$ and $(2, 0)$ .

$y = 4 - x^{2} ⟺ x^{2} + y^{2} = 4, y \geq 0$
The right side is a family of parallel lines
Find the key positions
Read off the answer

Practice

Exercise 1Basic

Find the minimum value of $x^{2} + 1 + (x - 4)^{2} + 4$ .

Exercise 2Basic

The complex number $z$ satisfies $∣ z ∣ = 1$ . Find the maximum of $∣ z - 3 - 4 i ∣$ .

Exercise 3Advanced

How many real roots does the equation $∣ x^{2} - 4 x + 3∣ = 1$ have?

Exercise 4Challenge

Real numbers $x, y$ satisfy $(x - 5)^{2} + y^{2} = 9$ . Find the maximum of $\frac{y}{x}$ .

In one sentence

Example: the minimum of a sum of radicals

Stop 1 · Where the roots of a quadratic lie (Vol. 2)

Example: both real roots (possibly equal) of x² − 2ax + a + 2 = 0 lie in the interval (0, 3). Find the range of a

Stop 2 · The modulus of a complex number (Vol. 4)

Example: |z − 3 − 4i| = 2. Find the maximum and minimum of |z|

Stop 3 · The range of a trigonometric expression (Vol. 3)

Read it as a slope

Auxiliary angle

Stop 4 · Curves and equations (Vol. 7)

Example: how many real roots does √(4 − x²) = x + m have?

Practice

Mistake Museum · Exhibits from this stop

Joining two points on the same side with a straight line

Treating a semicircle as a full circle

Stops on this line