To find the y-intercept of a rational function, simply set x=0x equals 0 𝑥=0 and solve for yy
𝑦
. This is because the yy
𝑦
-intercept is the point where the graph crosses the yy
𝑦
-axis, and every point on the yy
𝑦
-axis has an xx
𝑥
-coordinate of 0. After substituting x=0x equals 0
𝑥=0
, the resulting yy
𝑦
-value is the yy
𝑦
-intercept, which can be expressed as the ordered pair (0,y)open paren 0 comma y close paren
(0,𝑦)
.
Special case: What if the denominator is zero at x=0x equals 0
𝑥=0
?
For a rational function, f(x)=p(x)q(x)f of x equals the fraction with numerator p open paren x close paren and denominator q open paren x close paren end-fraction
𝑓(𝑥)=𝑝(𝑥)𝑞(𝑥)
, the yy
𝑦
-intercept exists only if the function is defined at x=0x equals 0
𝑥=0
. If plugging in x=0x equals 0
𝑥=0
results in the denominator being equal to zero, there is a vertical asymptote at x=0x equals 0
𝑥=0
, and therefore, no yy
𝑦
-intercept.
Step-by-step method with examples
Step 1: Identify the rational function
A rational function is a ratio of two polynomial functions, f(x)=p(x)q(x)f of x equals the fraction with numerator p open paren x close paren and denominator q open paren x close paren end-fraction
𝑓(𝑥)=𝑝(𝑥)𝑞(𝑥)
, where q(x)≠0q open paren x close paren is not equal to 0
𝑞(𝑥)≠0
.
Step 2: Substitute x=0x equals 0
𝑥=0
into the function
Replace every instance of xx
𝑥
in the function's equation with 00
0
.
Step 3: Solve for yy
𝑦
Simplify the expression to find the value of yy
𝑦
.
Step 4: Write the intercept as a coordinate pair
The yy
𝑦
-intercept is represented by the coordinate pair (0,y)open paren 0 comma y close paren
(0,𝑦)
.
Example 1: Finding a yy
𝑦
-intercept
**Function:**f(x)=2x−7x−9f of x equals the fraction with numerator 2 x minus 7 and denominator x minus 9 end-fraction
𝑓(𝑥)=2𝑥−7𝑥−9
-
**Set x=0x equals 0
𝑥=0
:**f(0)=2(0)−70−9f of 0 equals the fraction with numerator 2 open paren 0 close paren minus 7 and denominator 0 minus 9 end-fraction
𝑓(0)=2(0)−70−9
-
**Simplify:**f(0)=-7-9=79f of 0 equals negative 7 over negative 9 end-fraction equals seven-nineths
𝑓(0)=−7−9=79
-
**The yy
𝑦
-intercept is (0,79)open paren 0 comma seven-nineths close paren
(𝟎,𝟕𝟗)** .
Example 2: Finding a yy
𝑦
-intercept for a factored function
**Function:**f(x)=(x+1)(x−3)3x−2f of x equals the fraction with numerator open paren x plus 1 close paren open paren x minus 3 close paren and denominator 3 x minus 2 end-fraction
𝑓(𝑥)=(𝑥+1)(𝑥−3)3𝑥−2
-
**Set x=0x equals 0
𝑥=0
:**f(0)=(0+1)(0−3)3(0)−2f of 0 equals the fraction with numerator open paren 0 plus 1 close paren open paren 0 minus 3 close paren and denominator 3 open paren 0 close paren minus 2 end-fraction
𝑓(0)=(0+1)(0−3)3(0)−2
-
**Simplify:**f(0)=(1)(-3)-2=-3-2=32f of 0 equals the fraction with numerator open paren 1 close paren open paren negative 3 close paren and denominator negative 2 end-fraction equals negative 3 over negative 2 end-fraction equals three-halves
𝑓(0)=(1)(−3)−2=−3−2=32
-
**The yy
𝑦
-intercept is (0,32)open paren 0 comma three-halves close paren
(𝟎,𝟑𝟐)** .
Example 3: When no yy
𝑦
-intercept exists
**Function:**f(x)=x2−1x2f of x equals the fraction with numerator x squared minus 1 and denominator x squared end-fraction
𝑓(𝑥)=𝑥2−1𝑥2
-
**Set x=0x equals 0
𝑥=0
:**f(0)=(0)2−1(0)2=-10f of 0 equals the fraction with numerator open paren 0 close paren squared minus 1 and denominator open paren 0 close paren squared end-fraction equals negative 1 over 0 end-fraction
𝑓(0)=(0)2−1(0)2=−10
-
**Analyze the result:**Division by zero is undefined. This indicates that the function is not defined at x=0x equals 0
𝑥=0
.
-
Conclusion: The graph of this rational function has a vertical asymptote at x=0x equals 0
𝑥=0
(the yy
𝑦
-axis) and therefore **has no yy
𝑦
-intercept**.
Why this method works
The Cartesian coordinate system is a grid where points are located by their (x,y)open paren x comma y close paren
(𝑥,𝑦)
coordinates. The yy
𝑦
-axis is the vertical line defined by the equation x=0x equals 0
𝑥=0
. Therefore, any point that lies on the yy
𝑦
-axis must have an xx
𝑥
-coordinate of 0. By substituting x=0x equals 0
𝑥=0
into the function, you are evaluating the function at the exact location where it should cross the yy
𝑦
-axis. The resulting yy
𝑦
-value is the height of that point, which is the yy
𝑦
-intercept.