This worksheet covers finding x- and y-intercepts across a range of function types, from GCSE-level polynomials through to A-Level rational, logarithmic, and trigonometric functions. Exercises are graded by difficulty with full worked solutions.
By the end of this worksheet, you should be able to:
- Find x- and y-intercepts of polynomial, rational, exponential, logarithmic, and trigonometric functions
- Recognise when a function has no x-intercept or no y-intercept, and explain why
- Use factorisation, the quadratic formula, and algebraic manipulation to solve for intercepts
- Interpret intercepts in context and relate them to the graph of a function
Theory Recap - X and Y Intercepts
The intercepts of a function are the points where its graph crosses (or touches) the coordinate axes.
| Intercept | What it means | How to find it |
|---|---|---|
| y-intercept | Where the graph crosses the y-axis | Set  and evaluate  |
| x-intercept(s) | Where the graph crosses or touches the x-axis (also called roots or zeros) | Set  and solve for  |
A function can have zero, one, or many x-intercepts, but a function (as opposed to a relation) can have at most one y-intercept — because each input gives exactly one output.
• Students often swap the method — setting 
to find the y-intercept and to find the x-intercept. Remember: y-intercept means set x to zero.
• Not all functions are defined at . For example, 
and 
have no y-intercept because they are undefined at .
• The equation 
has no solution, since 
for all real . Exponential functions of the form 
only have an x-intercept when 
.
Worked Examples
Find the x- and y-intercepts of:
Function:
y-intercept — set :
x-intercept — set :
y-intercept: 
| x-intercept: 
Find the x- and y-intercepts of:
Function:
y-intercept — set :
x-intercepts — set and factorise:
So 
or 
.
y-intercept: 
| x-intercepts: and 
Find the x- and y-intercepts of:
Function:
y-intercept — set :
x-intercept — set and factorise:
The discriminant is zero, so the parabola touches the x-axis at exactly one point (a repeated root). The graph does not cross — it is tangent to the x-axis at .
y-intercept: 
| x-intercept: — repeated root (tangent point)
Find the x- and y-intercepts of:
Function:
y-intercept — set 
:
x-intercepts — check the discriminant:
Since 
| x-intercepts: none (discriminant is negative)
Find the x- and y-intercepts of:
Function:
y-intercept — set :
The y-intercept is the origin.
x-intercepts — set and factorise:
So , , or 
.
y-intercept: 
| x-intercepts: 
, , and
Find the x- and y-intercepts of:
Function: (domain: 
)
y-intercept — set :
is outside the domain (), so there is no y-intercept.
x-intercept — set :
y-intercept: none (function undefined for 
x < 3[/latex]) | x-intercept: [latex](7,\ 0)[/latex]
Find the x- and y-intercepts (if they exist) of:
Function: (undefined at 
)
y-intercept — set :
x-intercepts — set the numerator equal to zero (provided the denominator is non-zero at those points):
Check: neither nor makes the denominator zero, so both are valid intercepts.
y-intercept: | x-intercepts: and
Note: For a rational function 
, the x-intercepts come from solving 
, but only where 
. If both 
and 
at the same point, there is a hole in the graph rather than an intercept.
Find the x- and y-intercepts (if they exist) of:
Function:
y-intercept — set :
x-intercept — set :
y-intercept: 
| x-intercept: 
Find the x- and y-intercepts (if they exist) of:
Function: (domain: 
)
y-intercept — set :
x-intercept — set :
y-intercept: 
| x-intercept: 
Note: 
when the argument equals 1 (since 
), not when it equals 0. A common error is to write 
— this gives the boundary of the domain, not the intercept.
Find the x- and y-intercepts of:
Function:
y-intercept — set :
x-intercepts — set :
The modulus equation gives two cases:
y-intercept: 
| x-intercepts: 
and
Find all x-intercepts of the function on the interval 
, and state the y-intercept:
Function: on
y-intercept — set :
x-intercepts — set :
The principal value is 
. Since sine is also positive in the second quadrant:
Both solutions lie within .
y-intercept: 
| x-intercepts: 
and 
Find the x- and y-intercepts (if they exist) of:
Function: (undefined at )
y-intercept — set :
The denominator is 
when , so the function is undefined at . There is no y-intercept.
x-intercepts — set the numerator equal to zero:
Check: neither value makes the denominator zero, so both are valid.
y-intercept: none (function undefined at ) | x-intercepts: and
Find the x- and y-intercepts (if they exist) of:
Function: (domain: 
)
y-intercept — set :
At , the argument of the logarithm is 
:
Check: 
, so this lies within the domain. ✓
y-intercept: none | x-intercept: 
A curve is defined parametrically by:
Find the coordinates of all points where the curve crosses the x-axis and the y-axis.
Curve: 
x-axis crossings — find values of 
where , then substitute into :
So 
. Substitute each into 
:
The curve crosses the x-axis at 
and touches it at — note both 
and 
give the same Cartesian point, meaning the curve passes through the origin twice (a self-intersection on the x-axis).
y-axis crossings — find values of where , then substitute into 
:
Both values give , confirming the single point is where the curve meets the y-axis.
x-axis crossings: and | y-intercept: — the origin (self-intersection point)
Note: With parametric curves, always work in the parameter — find values first, then convert back to 
. Different values of can produce the same Cartesian point; this signals a self-intersection and is worth noting explicitly in an exam answer.
A curve is defined implicitly by:
Find the x- and y-intercepts of the curve.
Curve:
x-intercepts — set and solve:
y-intercepts — set and solve:
Since 
has no real solutions, the curve does not cross the y-axis.
This makes geometric sense: the curve is a hyperbola-like conic that opens in the horizontal direction and does not reach the y-axis for real .
x-intercepts: 
and 
| y-intercepts: none (gives , no real solutions)
Note: For implicit curves, intercepts are found by the same rules — set the other variable to zero and solve — but the resulting equation may be quadratic or harder. Always check whether solutions are real before claiming an intercept exists.
Find the x- and y-intercepts (if they exist) of:
Function: 
y-intercept — set :
The y-intercept is the origin .
x-intercepts — set :
Step 1 — Factor rather than taking logs immediately. Write 
:
Step 2 — Solve each factor. Since 
for all real , it can never equal zero. So:
The only x-intercept is at , confirming it coincides with the y-intercept — the curve passes through the origin and that is its only intercept on either axis.
y-intercept: | x-intercept: — the origin is the only intercept on both axes
⚠ Common Mistake: Taking 
of both sides of 
and writing 
gives the correct answer here, but misses the factorisation structure entirely and would fail on similar questions where 
is not a common factor. Always try to factorise first when the equation involves sums or differences of exponentials.
Key Techniques For Solving X and Y Intercept Problems
. To find the x-intercepts, set . Always check these rules before starting.
to determine how many real roots exist before attempting to solve (Exercises 2, 3, 4, 5). is in the domain before claiming a y-intercept (Exercises 7, 12).
and 
. Always check the domain of a logarithmic function — it has a y-intercept only if is in the domain (Exercises 8, 9, 13).Always check the domain before claiming an intercept exists. Examiners regularly award marks for correctly stating "no y-intercept because the function is undefined at " — a one-line explanation is enough for full credit.
Summarise with AI:
Did you like this article? Rate it!
Loading...
