The y-intercept is the point where a graph crosses the y-axis. Every point on the y-axis has x = 0, so you find the y-intercept by putting x = 0 into the equation. In y = mx + b, the y-intercept is b, written as the point (0, b).
In Spanish
intersección con el eje y (ordenada al origen)
Also called
vertical intercept, y-int, b in y = mx + b, initial value
The y-intercept is on the y-axis, where x = 0. For y = −12x + 3, that is (0, 3). Do not confuse it with the x-intercept, (6, 0).
Formal definition and notation
For a graph in the coordinate plane, a y-intercept is a point (0, b) that lies on the graph; it exists exactly when x = 0 is allowed. For a function y = f(x) defined at 0, there is exactly one y-intercept, (0, f(0)). For a line y = mx + b, it is (0, b); in context it is called the initial value.
Symbol
Read it as
Meaning
b
b
the y-intercept in y = mx + b
(0, b)
zero comma b
the y-intercept written as a point
f(0)
f of zero
the output when the input is 0, the y-intercept of the function f
Key facts
Its x-coordinate is always 0, because it sits on the y-axis.
In y = mx + b, the y-intercept is b. In y = 2x + 5, the y-intercept is 5, the point (0, 5).
A function has at most one y-intercept. A circle that is not a function can cross the y-axis twice.
If b = 0, the line passes through the origin(0, 0), so the y-intercept and x-intercept are the same point.
In a word problem the y-intercept is the starting value: the fixed fee, the first-day amount, the height at time 0.
How to find the y-intercept
How to find the y-intercept
y-intercept = (0, f(0))
0
the x-coordinate of every point on the y-axis
f(0)
the value of y you get when you substitute x = 0
From an equation: substitute x = 0 and solve for y. From a graph: read where the graph touches the y-axis. From a table: find the row with x = 0, or work back to it using the constant rate.
Try it: find where the line crosses
Interactive
Drag point A or B and watch the number b in y = mx + b. It always matches the height where the line crosses the y-axis. Try the presets: a line through the origin has b = 0, and a vertical line has no y-intercept form.
This interactive needs JavaScript. The worked examples below show the same ideas step by step.
Answer: The line crosses the y-axis at (0, −6), so the y-intercept is −6.
Example 3: The starting value in a word problem
Test-style
A phone plan charges a one-time fee plus a fixed amount per month. After 4 months a customer has paid 110 dollars in total; after 10 months, 230 dollars. The total cost is linear in the number of months. What is the y-intercept of the cost line, and what does it mean?
Write the data as points (months, dollars): (4, 110) and (10, 230).
Slope: m = 230 − 11010 − 4 = 1206 = 20 dollars per month.
Use y = mx + b with (4, 110): 110 = 20(4) + b.
Solve: b = 110 − 80 = 30.
Interpret: at 0 months the cost is already 30 dollars. That is the one-time fee.
Answer: The y-intercept is (0, 30): the one-time fee is 30 dollars (and the monthly charge is 20 dollars).
Common mistakes
✗ MistakeFor y = 2x − 6, the y-intercept is (3, 0).
✓ Instead(3, 0) is the x-intercept. The y-intercept is (0, −6).
The y-intercept is not always where the graph crosses the x-axis; it is never there unless the graph passes through the origin. The y-intercept is on the y-axis, where x = 0.
✗ MistakeThe y-intercept of y = 2x − 6 is the point (−6, 0).
✓ InsteadWrite it as (0, −6): x first, then y.
An ordered pair lists x first. On the y-axis the x-coordinate is 0, so 0 goes first.
✗ MistakeFor 3x + 4y = 20, the y-intercept is 20.
✓ InsteadPut x = 0: 4y = 20, so y = 5. The y-intercept is (0, 5).
You can read b directly only when the equation is solved for y. Otherwise substitute x = 0 and solve.
Not to be confused with
Y-Interceptvs.X-intercept
The x-intercept is where a graph crosses the x-axis, so y = 0 there. For y = 2x − 6, the x-intercept is (3, 0) and the y-intercept is (0, −6).
Slope tells how steep a line is (the m); the y-intercept tells where it starts on the y-axis (the b). In y = 4x + 1, the slope is 4 and the y-intercept is 1.
Quick check: 3 questions
Question 1 of 3
What is the y-intercept of y = 4 − 6x?
Show answer and explanation
Answer: C. Put x = 0: y = 4 − 0 = 4, so the point is (0, 4). Choice A uses the slope, B writes the coordinates in the wrong order, and D is the x-intercept.
Question 2 of 3
Where does the graph of y = x2 + 3x − 10 cross the y-axis?
Show answer and explanation
Answer: A. Any graph crosses the y-axis where x = 0: y = 0 + 0 − 10 = −10. Choice B takes the coefficient of x, C is an x-intercept, and D takes the coefficient of x2.
Question 3 of 3
A linear table has the points (1, 7), (2, 10) and (3, 13). What is the y-intercept?
Show answer and explanation
Answer: B. The slope is 3 (y goes up 3 each time x goes up 1). Step back from x = 1 to x = 0: 7 − 3 = 4. Choice A is the first y-value, not the value at x = 0; C is the slope.
Y-Intercept: frequently asked questions
Can a graph have more than one y-intercept?
A function cannot: each input, including x = 0, has only one output, so it crosses the y-axis at most once. Other graphs can. The circle x2 + y2 = 25 crosses the y-axis at (0, 5) and (0, −5). A vertical line such as x = 3 never crosses it at all.
Can the y-intercept be zero?
Yes. If b = 0, the graph passes through the origin, (0, 0). The line y = 3x is an example. Then the origin is both the x-intercept and the y-intercept. In a word problem a zero y-intercept means the quantity starts at nothing, as in a proportional relationship.
Is the y-intercept a number or a point?
Both forms are used. Many textbooks call the point (0, b) the y-intercept, and many teachers and tests also accept just the number b. For y = 2x + 5 you may answer 5 or (0, 5). If a question asks for coordinates, give the point, with 0 written first.
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