Intercepts of Rational Functions

An intercept of a rational function is a point where the graph of the rational function intersects the $x$- or $y$-axis.

For example, the function $y = \frac{(x+2)(x-1)}{(x-3)}$ has $x$-intercepts at $x=-2$ and $x=1,$ and a $y$-intercept at $y=\frac{2}{3}.$

Graph of $y = \frac{(x+2)(x-1)}{(x-3)}$

Finding the intercepts of a rational function is helpful for graphing the function and understanding its behavior.

The $y$-intercept of a function is the $y$-coordinate of the point where the function crosses the $y$-axis.

The value of the $y$-intercept of $y = f(x)$ is numerically equal to $f(0)$. A function can have at most one $y$-intercept, as it can have at most one value of $f(0)$.

What is the $y$-intercept in the graph of $y = 3x + 4$?

---

The $y$-intercept of the graph can be obtained by setting $x= 0$, and thus we get $y = 3 \times 0 + 4 = 4.$ $_\square$

What is the $y$-intercept in the graph of $y = e^{2x} + 4$?

---

The $y$-intercept can be obtained by setting $x =0$, and thus we get $y = e^{0} + 4 = 1+ 4 = 5$. $_\square$

What is the $y$-intercept in the graph of $y = \dfrac{x + 9}{x^2 + 3}$?

---

The $y$-intercept can be obtained by setting $x = 0$, and thus we get $y = \dfrac{9}{3} = 3$. $_\square$

What is the $y$-intercept of the graph $x^3 + 3x^2 + 3x + 3 = y?$

---

Setting $x = 0$, we get $y =0^3 + 3×0^2 + 3×0 + 3 = 3$. $_\square$

The $x$-intercept of a function is the $x$-coordinate of the point where the function crosses the $x$-axis.

Since the function can cross the $x$-axis multiple times, it can have multiple $x$-intercepts. The values of $x$-intercepts of $y = f(x)$ can be obtained by setting $f(x) = 0$.

What is the $x$-intercept in the graph of $y = 2x + 5$?

---

The $x$-intercept can be obtained by setting $y = 0$, so we get $2x + 5 = 0 \implies x = -\dfrac{5}{2}$. $_\square$

What is the sum of all the $x$-intercepts in the graph of $y = x^2 - 5x + 6$?

---

We can get the $x$-intercept by setting $y = 0$, so we get $x^2 - 5x + 6= 0$.
It can be factored as $(x-2)(x-3) = 0$, which implies $x = 2, 3$.
Thus, the $x$-intercepts of the graph are $2$ and $3$, so the sum of the $x$-intercepts is $2 + 3 = 5$. $_\square$

How many $x$-intercepts are there in the graph of $y = e^x + 1$?

---

We can get the $x$-intercepts by setting $y = 0$, so we get $e^x = -1$.

However, $e^x$ is a positive quantity for all real values of $x$ and can never be equal to $-1$. Hence, this equation has no solutions, and thus the number of $x$-intercepts in the graph is 0. $_\square$

What is the product of the $x$-intercepts in the graph of $y = \dfrac{(x-1)(x-2)(x-3)}{x(x-1)(x+1)}$?

---

We can get the $x$ intercepts by setting $y=0: \dfrac{(x-1)(x-2)(x-3)}{x(x-1)(x+1)} = 0$.
Hence $(x-1)(x-2)(x-3) =0 \implies x = 1, 2, 3$.
However, $x \neq -1, 0, 1$ because each of these values of $x$ makes the denominator zero.
So the $x$-intercepts are $x = 2, 3$, and thus their product is $2 \times 3 = 6$. $_\square$

• Real world examples, where intercepts represent some meaningful concept
• Applications to graphing, link out to Graphing Rational Equations

• Rational Functions
• Graphing Rational Equations