Question

learn: summarize transformations of graphs for exercises 55–62, a function g is given. identify the parent function from table 1-2 on pages 92–93. then use the steps for graphing multiple transformations of functions on page 102 to list, in order, the transformations applied to the parent function to obtain the graph of g. 58. $g(x) = \frac{1}{2}\sqrt{x + 4.3} - 8.4$

Explanation:

Step1: Identify the parent function

The parent function of \(g(x)=\frac{1}{2}\sqrt{x + 4.3}-8.4\) is \(f(x)=\sqrt{x}\) (square - root function).

Step2: Analyze the horizontal transformation

For the function \(y = f(x + h)\), when \(h = 4.3\), the graph of \(y = f(x)\) is shifted horizontally. The transformation is a shift to the left by \(4.3\) units (since for \(y=\sqrt{x+4.3}\), if we let \(u=x + 4.3\), when \(u = 0\), \(x=-4.3\) compared to \(y = \sqrt{x}\) where \(x = 0\) when \(y = 0\)).

Step3: Analyze the vertical transformation (stretch/compression)

For the function \(y=af(x)\), when \(a=\frac{1}{2}\), the graph of \(y = f(x)\) is vertically compressed. The transformation is a vertical compression by a factor of \(\frac{1}{2}\) (since \(g(x)=\frac{1}{2}\sqrt{x + 4.3}-8.4\) and for \(y=\frac{1}{2}u\) where \(u=\sqrt{x + 4.3}\), the \(y\) - values of \(y = \sqrt{x+4.3}\) are multiplied by \(\frac{1}{2}\)).

Step4: Analyze the vertical transformation (shift)

For the function \(y=f(x)+k\), when \(k=-8.4\), the graph of \(y = f(x)\) is shifted vertically. The transformation is a shift down by \(8.4\) units (since \(g(x)=\frac{1}{2}\sqrt{x + 4.3}+(- 8.4)\)).

Answer:

Parent function: \(f(x)=\sqrt{x}\). Transformations (in order):

1. Shift the graph of \(y = \sqrt{x}\) to the left by \(4.3\) units.
2. Vertically compress the resulting graph by a factor of \(\frac{1}{2}\).
3. Shift the resulting graph down by \(8.4\) units.