Questions: Graph the rational function. h(x) = (3x^2 - 12x) / (x^2 - 5x + 4) Start by drawing the asymptotes (if there are any). Then plot two points on each piece of the graph. Finally, click on the graph-a-function button. Be sure to plot a hollow dot wherever there is a "hole" in the graph.

Graph the rational function.
h(x) = (3x^2 - 12x) / (x^2 - 5x + 4)

Start by drawing the asymptotes (if there are any). Then plot two points on each piece of the graph. Finally, click on the graph-a-function button. Be sure to plot a hollow dot wherever there is a "hole" in the graph.

Transcript text: Graph the rational function. \[ h(x)=\frac{3 x^{2}-12 x}{x^{2}-5 x+4} \] Start by drawing the asymptotes (if there are any). Then plot two points on each piece of the graph. Finally, click on the graph-a-function button. Be sure to plot a hollow dot wherever there is a "hole" in the graph.

Solution

Solution Steps

Step 1: Identify the Asymptotes

To find the vertical asymptotes, set the denominator equal to zero and solve for \( x \): \[ x^2 - 5x + 4 = 0 \] Factoring the quadratic equation: \[ (x - 1)(x - 4) = 0 \] So, the vertical asymptotes are at \( x = 1 \) and \( x = 4 \).

To find the horizontal asymptote, compare the degrees of the numerator and the denominator. Both the numerator and the denominator are of degree 2. Therefore, the horizontal asymptote is the ratio of the leading coefficients: \[ y = \frac{3}{1} = 3 \]

Step 2: Identify Holes in the Graph

Check for common factors in the numerator and the denominator. The numerator is: \[ 3x^2 - 12x = 3x(x - 4) \] The denominator is: \[ x^2 - 5x + 4 = (x - 1)(x - 4) \] There is a common factor of \( x - 4 \), so there is a hole at \( x = 4 \).

Step 3: Plot Points on Each Piece of the Graph

Choose points around the vertical asymptotes and the hole to plot the graph accurately. For example:

For \( x = 0 \): \[ h(0) = \frac{3(0)^2 - 12(0)}{0^2 - 5(0) + 4} = 0 \] So, the point (0, 0) is on the graph.
For \( x = 2 \): \[ h(2) = \frac{3(2)^2 - 12(2)}{2^2 - 5(2) + 4} = \frac{12 - 24}{4 - 10 + 4} = \frac{-12}{-2} = 6 \] So, the point (2, 6) is on the graph.