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 \(y = 3x + 4\)?

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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 \(y = e^{2x} + 4\)?

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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 \(y = \dfrac{x + 9}{x^2 + 3}\)?

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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 \(x^3 + 3x^2 + 3x + 3 = y?\)

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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 \(y = 2x + 5\)?

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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 \(y = x^2 - 5x + 6\)?

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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 \(y = e^x + 1\)?

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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 \(y = \dfrac{(x-1)(x-2)(x-3)}{x(x-1)(x+1)}\)?

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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