Intercepts — x-intercept, y-intercept: Worked examples

Before we start, one word we will lean on: an intercept is the spot where a graph crosses an axis. On the flat x-axis every point has "up amount" $y = 0$ . On the up-down y-axis every point has "sideways amount" $x = 0$ . That is the whole engine.

The scenario matrix

Every problem this topic throws lives in one of these cells. The right column names the example that clears it.

#	Case class	What makes it different	Cleared by
A	Both intercepts positive	vanilla line, both crossings in the "nice" region	Example 1
B	A negative intercept	a sign flips — where does the crossing move?	Example 2
C	Line through the origin	both intercepts collapse to $(0, 0)$ — degenerate	Example 3
D	Horizontal line $y = k$	no x-intercept (unless $k = 0$ )	Example 4
E	Vertical line $x = h$	no y-intercept (unless $h = 0$ )	Example 4
F	Curve: parabola, two x-intercepts	roots = crossings, all signs	Example 5
G	Curve with no x-intercept	discriminant $< 0$ : it floats above the axis	Example 6
H	Read intercepts backwards to build the equation	given crossings, find the line	Example 7
I	Real-world word problem	intercepts carry units and meaning	Example 8
J	Exam twist — intercept depends on a parameter	solve for the value that forces a crossing	Example 9

The figures below show cells A, B, C, D/E, F, G so you can see every case.

Figure — Intercepts — x-intercept, y-intercept

The worked examples

Cell A — both intercepts positive

Link back to why this is the fast route: Graphing Straight Lines — two intercepts already give you the whole line.

Cell B — a negative intercept

Cell C — line through the origin (degenerate)

Worked example Example 3 — the line

$y = 2 x$ Forecast: what happens when you try intercept form $\frac{x}{a} + \frac{y}{b} = 1$ here? Predict before reading.

Step 1. x-intercept: set $y = 0$ : $0 = 2 x ⟹ x = 0$ . Point $(0, 0)$ . Why this step? Standard rule — but the answer is the origin itself.

Step 2. y-intercept: set $x = 0$ : $y = 2 (0) = 0$ . Point $(0, 0)$ again. Why this step? Same rule; both crossings land on the same point, the origin.

Step 3. So $a = 0$ and $b = 0$ . Intercept form would need $\frac{x}{0} + \frac{y}{0} = 1$ — division by zero. The intercept form is forbidden here. Why this step? This is the degenerate case: a line through the origin has its two crossings fused into one, so there are no two distinct axis points to build intercept form from. Use Slope-Intercept Form (y = mx + c) with $c = 0$ instead, i.e. just $y = 2 x$ .

Verify: the only common point of $y = 2 x$ with either axis is $(0, 0)$ (green line through the origin in the figure). Setting $y = 0$ and $x = 0$ both return the origin — consistent.

Cells D & E — horizontal and vertical lines (missing intercepts)

Worked example Example 4 — the lines

$y = 3$ (horizontal) and $x = - 4$ (vertical) Forecast: a flat line $y = 3$ never bends toward the x-axis. Does it have an x-intercept at all? Guess yes/no for each.

Step 1 (line $y = 3$ ). y-intercept: set $x = 0$ : $y = 3$ regardless, so point $(0, 3)$ . ✓ has one. Why this step? $y = 3$ means "always 3 high"; at $x = 0$ that is the point $(0, 3)$ .

Step 2 (line $y = 3$ ). x-intercept: set $y = 0$ : we'd need $3 = 0$ , which is false. No solution ⇒ no x-intercept. Why this step? A horizontal line at height $3$ stays $3$ above the x-axis forever; it is parallel to that axis, so they never meet.

Step 3 (line $x = - 4$ ). x-intercept: set $y = 0$ : point $(- 4, 0)$ . ✓ has one (a negative one). Why this step? $x = - 4$ is a wall $4$ units left; it crosses the flat x-axis at height $0$ , i.e. $(- 4, 0)$ .

Step 4 (line $x = - 4$ ). y-intercept: set $x = 0$ : we'd need $0 = - 4$ , false. No y-intercept. Why this step? This vertical wall is parallel to the y-axis, so it never touches it.

Verify: the red horizontal and gray vertical lines in the figure each pierce exactly ONE axis. Edge fact: only if $k = 0$ (line $y = 0$ , the x-axis itself) or $h = 0$ (line $x = 0$ , the y-axis itself) does the line lie along an axis, giving infinitely many intercepts.

Cell F — parabola with two x-intercepts (all signs)

Worked example Example 5 — the curve

$y = x^{2} + x - 6$ Forecast: how many times can a $\cup$ -shaped parabola cross the x-axis? Guess 0, 1, or 2, and whether the crossings can straddle the origin.

Step 1. x-intercepts: set $y = 0$ : $x^{2} + x - 6 = 0$ . Factor: $(x + 3) (x - 2) = 0 ⟹ x = - 3$ or $x = 2$ . Points $(- 3, 0)$ and $(2, 0)$ . Why this step? The x-crossings are exactly the real roots of $f (x) = 0$ — that is what a root means geometrically. See Roots of a Quadratic. Note one crossing is negative, one positive.

Step 2. y-intercept: set $x = 0$ : $y = 0 + 0 - 6 = - 6$ . Point $(0, - 6)$ . Why this step? Just plug $x = 0$ ; the constant term of a polynomial is always its y-intercept.

Verify: plug $x = - 3$ : $9 - 3 - 6 = 0$ ✓. Plug $x = 2$ : $4 + 2 - 6 = 0$ ✓. Plug $x = 0$ : $- 6$ ✓. The parabola in the figure dips below the axis between $x = - 3$ and $x = 2$ and cuts it at both.

Cell G — curve with NO x-intercept

Worked example Example 6 — the curve

$y = x^{2} + 4$ Forecast: this parabola's lowest point is at $y = 4$ . Can it ever reach the x-axis ( $y = 0$ )? Guess before solving.

Step 1. x-intercepts: set $y = 0$ : $x^{2} + 4 = 0 ⟹ x^{2} = - 4$ . No real $x$ squares to a negative number, so no x-intercept. Why this step? Squaring any real number gives $\geq 0$ , so $x^{2}$ can never equal $- 4$ . The quadratic's discriminant is $0^{2} - 4 (1) (4) = - 16 < 0$ , the algebraic signal for "no real roots" (see Roots of a Quadratic).

Step 2. y-intercept: set $x = 0$ : $y = 0 + 4 = 4$ . Point $(0, 4)$ . Why this step? Constant term = y-intercept, as always.

Verify: the smallest value of $x^{2} + 4$ is $4$ (at $x = 0$ ), so the whole curve floats at least $4$ above the x-axis — it never touches it. The purple curve in the figure sits entirely above the axis. A curve can miss an axis entirely; a line (non-horizontal, non-vertical) never can.

Cell H — reading intercepts backwards

Cell I — real-world word problem (intercepts carry units)

Worked example Example 8 — draining a water tank

A tank holds water $V$ (litres) after $t$ minutes, modelled by $V = 200 - 8 t$ . Find (a) the starting amount and (b) the time it empties. Forecast: which intercept is "start" and which is "empty"?

Step 1. The " $V$ -intercept" (vertical axis is $V$ ): set $t = 0$ : $V = 200 - 0 = 200$ litres. Why this step? $t = 0$ is the start; setting the horizontal variable to zero gives the crossing on the $V$ -axis — the initial volume, $200$ L.

Step 2. The " $t$ -intercept" (horizontal axis is $t$ ): set $V = 0$ : $0 = 200 - 8 t ⟹ 8 t = 200 ⟹ t = 25$ minutes. Why this step? $V = 0$ means the tank is empty; the crossing on the $t$ -axis tells us when. The tank drains after $25$ minutes.

Verify: at $t = 25$ : $V = 200 - 8 (25) = 200 - 200 = 0$ ✓. Units check: $8 L/min \times 25 min = 200 L$ removed = the full tank ✓. Only $0 \leq t \leq 25$ is physically meaningful (you can't have negative water or run time backwards).

Cell J — exam twist: intercept fixed by a parameter

Active-recall

Recall Which cell has NO x-intercept, and why?

Cells D and G. A horizontal line $y = k$ ( $k \neq = 0$ ) is parallel to the x-axis, and $x^{2} + 4 = 0$ has no real root — both never reach $y = 0$ .

Set

y = 0

to find which intercept?

the x-intercept.

For

2 x - 5 y = 10

, the y-intercept is?

- 2

, point

(0, - 2)

Why can't

y = 2 x

use intercept form?

both intercepts are

0

, so

\frac{x}{a} + \frac{y}{b}

divides by zero.

Does the line

y = 3

have an x-intercept?

No — it is parallel to the x-axis.

x-intercepts of

y = x^{2} + x - 6

x = - 3

and

x = 2

How many x-intercepts does

y = x^{2} + 4

have?

Zero — its discriminant

- 16 < 0

For

k x + 3 y = 12

, does the y-intercept depend on

k

No — it is

4

for every

k

Connections

2.3.07 Intercepts — x-intercept, y-intercept (Hinglish) — parent topic, Hinglish version.
General Equation of a Line (Ax + By = C) — the form Examples 1, 2, 7, 9 live in.
Slope-Intercept Form (y = mx + c) — used for the origin line in Example 3.
Roots of a Quadratic — the engine behind Cells F and G.
Graphing Straight Lines — every example's payoff: two points, done.
Distance & Coordinates on the Cartesian Plane — why "on an axis" forces a coordinate to $0$ .

Worked examples — Intercepts — x-intercept, y-intercept