Question

student: class: date rational expressions and functions student activity sheet 4; exploring “graphing rational functions” 18. reinforce consider the function ( f(x)=\frac{5 x + 12}{x + 2} ). a. write the function in the form ( y=\frac{a}{x - h}+k ). b. describe how ( f(x) ) is a transformation of the parent function ( y=\frac{1}{x} ).

Explanation:

Step1: Perform polynomial long - division

Divide \(5x + 12\) by \(x + 2\).

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Step2: Identify \(a\), \(h\), and \(k\)

In the form \(y=\frac{a}{x - h}+k\), we have \(a = 2\), \(h=-2\), \(k = 5\). So \(y=\frac{2}{x-(-2)}+5\)

Step3: Analyze the transformation for part (b)

For the parent function \(y=\frac{1}{x}\):

• The horizontal shift: The function \(y=\frac{2}{x+2}+5\) is a horizontal shift of the parent function \(y = \frac{1}{x}\). Since \(h=-2\), it is a shift of \(2\) units to the left (because \(x\) is replaced by \(x+2\)).
• The vertical stretch: The coefficient \(a = 2\) causes a vertical stretch by a factor of \(2\).
• The vertical shift: The value \(k = 5\) causes a vertical shift of \(5\) units up.

Answer:

a. \(y=\frac{2}{x+2}+5\)
b. The function \(f(x)\) is a transformation of \(y=\frac{1}{x}\) that is a horizontal shift \(2\) units to the left, a vertical stretch by a factor of \(2\), and a vertical shift \(5\) units up.