Question

Sketch the graph of the function. Label the intercepts, relative extrema, points of inflection, and asymptotes. Then state the domain of the function. \(y=\frac{x^{2}-6 x+12}{x-4}\)

Short answer

The function \(y = \frac{x^{2} - 6x + 12}{x - 4}\) has a domain of \(\mathbb{R}\) - {4}, a y-intercept at \(y = -3\), and no x-intercepts. The vertical asymptote is at \(x = 4\), and the horizontal asymptote is \(y = 1\). Due to the complexity of the function, the relative extrema and points of inflection cannot easily be determined without further computations.

Step-by-step solution

Identify the Domain

The domain of the function is all real numbers except where the denominator equals zero. For the given function \(y = \frac{x^{2} - 6x + 12}{x - 4}\), the domain is \(\mathbb{R}\) - {4}. This is because the denominator becomes zero at \(x = 4\) and the function becomes undefined there.

Find the Intercepts

To find the y-intercept, set \(x = 0\) in the function and find the value of \(y\). To find the x-intercepts, set \(y = 0\), and find the corresponding value of \(x\). For the given function, the y-intercept is \(y = \frac{0^2 - 6*0 + 12}{0 - 4} = -3\) and there are no x-intercepts because the numerator \(x^{2} - 6x + 12\) is always greater than zero.

Find the Asymptotes

The vertical asymptote is found at the value of \(x\) for which the function tends to ±∞, which corresponds to \(x = 4\) in our case. For the horizontal asymptote, as the degree of the numerator and denominator are equal, it is the ratio of the leading coefficients, which is 1.

Find Relative Extrema

Relative extrema are the local minimum and maximum values. They are found by first computing the derivative of the function, setting it equal to zero, and solving for \(x\). Next, the second derivative is used to determine whether each point is a minimum or maximum. For the given function, this step turns out to be complex due to the rational function nature.

Find Points of Inflection

Points of inflection are points on the curve where the concavity changes. Find the second derivative of the function, set it to zero, and solve for \(x\). Inspecting the inflection point for this function is also complex due to it being a rational function.

Sketch the Graph

Combining the information from steps 1 to 5, one can sketch the graph of the function.