Quarry School

Find where the line reaches the x-axis

Explain it like I am five

Picture a ramp crossing a painted line on the floor. At the crossing, its height is zero. On a coordinate graph, the floor line is the x-axis. An x-intercept is the input where the graph has output 0, and its point looks like (x, 0). To find it, make the output equal to 0 and ask which input produces that output. You are finding where the line reaches the floor, so you do not set the across coordinate to 0. That would find the upright-axis crossing instead. A flat line above or below the floor never reaches it. A line lying on the floor has output 0 everywhere.

−2−1123456789−5−4−3−2−1123(0, −3)(6, 0)(6, 0)
The marked point on the horizontal axis has output 0.
Reminder
  • Solving equations. 0 = 5x − 10 becomes 10 = 5x, then x = 2; 5 × 2 − 10 = 0 checks it.
  • Dividing by a fraction. 3 ÷ 12 = 3 × 2 = 6 because six halves fit in 3.
  • Axis coordinates. (6, 0) has height 0, so it is on the x-axis.
  • Division by zero. 1 ÷ 0 has no value because 0 times any number is 0, never 1.
  • Negative multiplication. (−34) × (−83) = 2; matching signs give a positive product.
Why it works. Every point on the x-axis has second coordinate 0. A point on the graph also obeys y = mx + b, so a crossing must obey 0 = mx + b. Subtract b to get −b = mx, then divide by m to obtain x = −bm. This division requires m ≠ 0. If m = 0, the output stays at b. A nonzero b never becomes 0, while b = 0 makes every input an intercept.
RuleRule: Set f(x) = 0 to find x-intercepts. For f(x) = mx + b with m ≠ 0, x = −bm and the point is (−bm, 0).
The same idea, five ways
Say it

Say: find the input that makes the height zero.

Write it

The x-intercept is where the graph meets the horizontal axis.

In math
  • f(x) = 0
  • mx + b = 0
  • x = −bm, m ≠ 0
  • (x, 0)
Like

The spot where a ramp reaches the floor has no height.

See it
−22468−4−22(0, −3)(6, 0)y-interceptx-intercept
The marked point on the horizontal axis has output 0.
The same idea, other ways
As a picture

For y = x − 1, the point (1, 0) sits on the horizontal axis. Its output is zero even though its input is not.

2−2−112(0, −1)(1, 0)(1, 0)
The marked point on the horizontal axis has output 0.
As a reverse question

Evaluating f(6) asks what comes out of input 6. Solving f(x) = 0 reverses the question: the output is 0, so which input must go in?

which x?multiply by 1/2,subtract 30inputoutput
The desired output is fixed, and the input is unknown.
Strategy: step by step
  1. 1. Replace the output f(x) or y with 0. This selects points lying on the x-axis.
  2. 2. Solve the resulting equation for the input.
  3. 3. Substitute that found input into the original function. Its output must be 0.
  4. 4. State the input value and the coordinate point (x, 0).
  5. 5. If m = 0, inspect the constant output instead of dividing by zero.
Strategy
Strategy: find an axis crossing
1
Are you finding the x-intercept?
YesSet output y or f(x) equal to 0.
NoFor the y-intercept, set input x equal to 0.
↓
2
Is m = 0?
YesIf b ≠ 0 there is no x-intercept; if b = 0 every input has output 0.
NoSolve for x by subtracting b and dividing by m.
  1. Choose the axis: output 0 for x-axis, input 0 for y-axis.
  2. For an x-intercept, solve mx + b = 0.
  3. Check the found input in the original formula.
  4. Include the 0 coordinate in the answer point.
Worked exampleTwo verified x-axis crossings

You need to find where each graph has zero height: f(x) = 12x − 3 and g(x) = 3x − 6.

−22468−4−22(0, −3)(6, 0)(6, 0)
The marked point on the horizontal axis has output 0.
24−8−6−4−2246(0, −6)(2, 0)(2, 0)
The marked point on the horizontal axis has output 0.
  1. For the first graph, write 0 = 12x − 3.This selects the output 0 required on the x-axis.
  2. Add 3 to both sides: 3 = 12x.Undo the downward shift while keeping the equation balanced.
  3. Multiply both sides by 2: x = 6. Plug back in: 12 × 6 − 3 = 3 − 3 = 0.Multiplication by 2 undoes multiplication by one half, and substitution checks the found input.
  4. For the second graph, write 0 = 3x − 6, then add 6: 6 = 3x.Again select zero height, then undo the subtraction.
  5. Divide by 3 to find x = 2. Plug back in: 3 × 2 − 6 = 6 − 6 = 0.Division undoes the multiplication and the original output confirms the answer.
Answer
  • For f(x) = 12x − 3: x = 6.
  • First intercept point: (6, 0).
  • For g(x) = 3x − 6: x = 2.
  • Second intercept point: (2, 0).
Check Using −bm, the first value is −(−3) ÷ 12 = 6; the second is −(−6) ÷ 3 = 2. These match the step-by-step solutions.
Ladder: from easy to exam-hard. Press Try it first on any rung to hide its steps and use them as hints.
Rung 1Rung 1: the intercept is the origin

You need the x-intercept of y = x.

−22−2−112(0, 0)(−0, 0)origin
The marked point on the horizontal axis has output 0.
  1. Set y = 0: 0 = x, so x = 0.The identity output equals its input, so only input 0 gives output 0.
  2. Check the found input: y = 0.Substitution verifies the point's zero height.
Answer
(0, 0)
Check The identity graph visibly passes through the origin.
Rung 2Rung 2: a whole-number slope

You need the x-intercept of f(x) = 3x − 6.

24−8−6−4−2246(0, −6)(2, 0)(2, 0)
The marked point on the horizontal axis has output 0.
  1. Set 0 = 3x − 6 and add 6: 6 = 3x.The x-axis needs output 0; adding 6 undoes the subtraction.
  2. Divide by 3: x = 2. Plug in: f(2) = 6 − 6 = 0.This finds and verifies the required input.
Answer
(2, 0)
Check The intercept formula gives −(−6) ÷ 3 = 2 too.
Rung 3Rung 3: a fractional coefficient

You need the x-intercept of f(x) = 12x − 3.

2468−4−22(0, −3)(6, 0)(6, 0)
The marked point on the horizontal axis has output 0.
  1. Set 0 = 12x − 3 and add 3: 3 = 12x.Select zero output, then undo the shift.
  2. Multiply by 2: x = 6. Substitute: f(6) = 3 − 3 = 0.Doubling undoes halving, and substitution verifies it.
Answer
(6, 0)
Check The intercept formula gives 3 ÷ 12 = 6.
Rung 4Rung 4: a negative slope and negative intercept

You need the x-intercept of y = −34x − 2.

−4−22−4−22(0, −2)(−2.67, 0)([[−8|3]], 0)
The marked point on the horizontal axis has output 0.
  1. Set 0 = −34x − 2, then add 2 to get 2 = −34x.This selects zero height and isolates the product.
  2. Multiply by −43: x = −83.The reciprocal of the coefficient is −43, and their product is 1.
  3. Plug back in: −34 × (−83) − 2 = 2 − 2 = 0.The two negative factors give a positive product; the found input truly reaches the axis.
Answer
(−83, 0)
Check Starting at (0, −2), moving left 83 at slope −34 raises the output by 2, reaching 0.
Counterexamples: what it is not, and the tempting wrong moves
✗ Not this: Counterexample: for y = 3x − 6, input 0 gives the x-intercept (0, −6).
The point has output −6, so it is below the x-axis. It is the y-intercept.
✓ Instead: Set y = 0, solve x = 2, and verify 3 × 2 − 6 = 0. The x-intercept point is (2, 0).
✗ Not this: Counterexample: every linear function must have one x-intercept.
For y = 5 the output never becomes 0. For y = 0 it is already 0 at every input.
✓ Instead: A nonzero slope gives one x-intercept. Nonzero constant functions give none; the zero function gives every point on the x-axis.
Tips and tricks
  • Tip: Memory cue: at an x-intercept, the other letter y is 0; at a y-intercept, the other letter x is 0.
  • Tip: Write both the input and its point if the question's wording is unclear.
  • Put on the cheat sheet: x = −bm requires m ≠ 0. You can rebuild it by solving 0 = mx + b.
Trap. Setting x = 0 to find the x-intercept. The axis name tells you which coordinate is allowed to vary: on the x-axis, x varies while y stays 0.