Question

lesson 10.2 checkpoint
once you have completed the above problems and checked your solutions, complete the lesson checkpoint below.
complete the lesson reflection above by circling your current understanding of the learning goal.

1. choose the function that represents the graph to the left.

a. g(x)=−√x + 3
b. g(x)=−√x + 3
c. g(x)=√−x + 3
d. g(x)=√−x + 3

1. what transformations on the graph f(x)=√x result in the graph of the given function?

h(x)=√5x−1
a. translate up by 5 units and left 1 unit.
b. compress vertically by a factor of 5 and translate down by 1 unit.
c. compress horizontally by a factor of 1/5 and translate down by 1 unit.

Explanation:

Step1: Analyze the domain and y - intercept of the first graph

The graph starts at the y - axis (x = 0) and has a y - intercept of 3. For the square - root function, when x = 0, we want to find the function that gives y = 3. Also, the domain of the function seems to be x≥0.
For option A: \(g(x)=-\sqrt{x}+3\), when x = 0, \(g(0)=-\sqrt{0}+3 = 3\), and the domain of \(y = \sqrt{x}\) is \(x\geq0\). The negative sign in front of the square - root flips the graph of \(y=\sqrt{x}\) over the x - axis.
For option B: \(g(x)=-\sqrt{x + 3}\), when x = 0, \(g(0)=-\sqrt{0 + 3}=-\sqrt{3}
eq3\).
For option C: \(g(x)=\sqrt{-x}+3\), the domain of \(y=\sqrt{-x}\) is \(x\leq0\), which does not match the given graph.
For option D: \(g(x)=\sqrt{-x + 3}\), when x = 0, \(g(0)=\sqrt{-0 + 3}=\sqrt{3}
eq3\). So the function for the first graph is \(g(x)=-\sqrt{x}+3\).

Step2: Analyze the transformations of the second function

For the function \(h(x)=\sqrt{5x}-1\) compared to \(f(x)=\sqrt{x}\).
The general form of a horizontal compression of a function \(y = f(x)\) to \(y = f(bx)\) (where \(b>1\)) is a horizontal compression by a factor of \(\frac{1}{b}\). Here, \(b = 5\), so there is a horizontal compression by a factor of \(\frac{1}{5}\).
The subtraction of 1 from the function \(y=\sqrt{5x}\) (i.e., \(y=\sqrt{5x}-1\)) represents a vertical translation down by 1 unit.

Answer:

1. A. \(g(x)=-\sqrt{x}+3\)
2. C. Compress horizontally by a factor of \(\frac{1}{5}\) and translate down by 1 unit.