Key takeaway: The x-intercept is the point where a graph crosses or touches the x-axis, so its y-coordinate is 0 and its x-coordinate solves the equation when y equals 0.
Grade 9 – Algebra 1F-IF.B.4Functions and Graphs
Definition of X-Intercept
The x-intercept is the location where a graph meets the horizontal axis. Every point on the x-axis has height 0, so it always looks like \((a, 0)\).
In a real situation, it often answers “when does it run out?” or “when does it reach zero?”. A savings balance that falls over time hits zero at that point, and a ball thrown upward lands there. A line can have one, none (if it is horizontal and above or below the axis), or, for a horizontal line sitting on the axis, infinitely many. Curves may have several.
Formula and notation
To locate it, replace \(y\) with 0 and solve for \(x\):
\[y = 0 \ \Longrightarrow\ \text{solve for } x\]
The answer is written as an ordered pair \((a, 0)\), or sometimes just as the number \(a\). For a function \(f\), these points are the inputs where \(f(x) = 0\), also called zeros or roots.
Key facts
- The y-coordinate of this point is always 0.
- Find it algebraically by setting y = 0; find the y-intercept by setting x = 0.
- A line with nonzero slope has exactly one such point.
- The inputs where a function crosses are the solutions of f(x) = 0.
X-Intercept example
A school store sells pencils for \$3 and notebooks for \$4. The equation \(3x + 4y = 60\) shows the ways to spend a \$60 gift card on \(x\) pencils and \(y\) notebooks. Find the x-intercept and say what it means.
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Set \(y = 0\): \(3x + 4(0) = 60\).
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Simplify: \(3x = 60\), so \(x = 20\).
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The point is \((20, 0)\): 20 pencils and 0 notebooks.
The point is \((20, 0)\): the whole \$60 buys 20 pencils and no notebooks.
Common mistake
Watch out
Nadia sets \(x = 0\) in \(3x + 4y = 60\), gets \(y = 15\), and reports the x-intercept as 15.
Why it is wrong
Setting \(x = 0\) finds where the graph crosses the y-axis, the point \((0, 15)\). For this point the variable that becomes 0 is \(y\). Remember: it lies on the x-axis, where the height is zero.
Related terms and lessons
- Y-Intercept
- Zeros of a Function
- Linear Equation
- Standard Form of a Linear Equation
- Coordinate Plane
- Slope-Intercept Form
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