Question

chapter 1 quiz (1.1,1.2, 1.4, 1.5, 1.7)
14 points possible answered: 11/14
question 12
graph $y = 4\sqrt{x + 2} - 1$.
(grid and graphing interface shown)
clear all draw:

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Explanation:

Step1: Identify the parent function

The parent function here is \( y = \sqrt{x} \), which has a domain \( x \geq 0 \) and starts at the point \( (0, 0) \), with a range \( y \geq 0 \) and a shape that increases slowly.

Step2: Analyze the horizontal shift

The function \( y = 4\sqrt{x + 2}-1 \) has a horizontal shift. For a square - root function of the form \( y=\sqrt{x - h}\), the horizontal shift is \( h \) units. In our function, it is \( y = \sqrt{x+2}=\sqrt{x-(- 2)} \), so the graph of the parent function \( y = \sqrt{x} \) is shifted 2 units to the left. This means the new starting point (the vertex of the square - root graph) will be at \( x=-2 \) (since when \( x + 2=0\), \( x=-2 \)) and \( y = 0\) (before considering the vertical stretch and shift) for the inner square - root part.

Step3: Analyze the vertical stretch

The coefficient 4 in front of the square - root is a vertical stretch. For a function \( y = a\sqrt{x}\), if \( |a|>1 \), the graph is stretched vertically by a factor of \( |a| \). So the graph of \( y=\sqrt{x + 2} \) is stretched vertically by a factor of 4. This means that for each \( x\) - value, the \( y\) - value of the function \( y=\sqrt{x + 2} \) is multiplied by 4.

Step4: Analyze the vertical shift

The - 1 at the end of the function \( y = 4\sqrt{x + 2}-1 \) is a vertical shift. For a function \( y=f(x)+k \), if \( k<0 \), the graph is shifted down by \( |k| \) units. So the graph of \( y = 4\sqrt{x + 2} \) is shifted down by 1 unit.

Step5: Find key points

• Starting point: When \( x=-2 \), \( y=4\sqrt{-2 + 2}-1=4\times0 - 1=-1 \). So the starting point is \( (-2,-1) \).
• Another point: Let's choose \( x = 2\) (since when \( x = 2\), \( x + 2=4\), and \( \sqrt{4}=2 \)). Then \( y=4\sqrt{2 + 2}-1=4\times2-1=8 - 1 = 7\). So the point \( (2,7) \) is on the graph.
• Another point: Let's choose \( x=-1\). Then \( x + 2=1\), \( y=4\sqrt{-1 + 2}-1=4\times1-1 = 3\). So the point \( (-1,3) \) is on the graph.

To graph the function:

1. Plot the starting point \( (-2,-1) \).
2. Plot the other points we found, such as \( (-1,3) \) and \( (2,7) \).
3. Draw a smooth curve starting at \( (-2,-1) \) and passing through the other plotted points, following the shape of a vertically stretched and shifted square - root curve. The domain of the function \( y = 4\sqrt{x + 2}-1 \) is \( x\geq - 2\) (because the expression inside the square root \( x + 2\geq0\)) and the range is \( y\geq - 1\) (since the square - root part \( 4\sqrt{x + 2}\geq0\), so \( 4\sqrt{x + 2}-1\geq - 1\)).

(Note: Since this is a graphing problem, the final answer is the graph constructed by following the above steps. If we were to describe the graph in words, it is a square - root graph that is shifted 2 units to the left, stretched vertically by a factor of 4, and shifted 1 unit down, with the starting point at \( (-2,-1) \) and passing through points like \( (-1,3) \) and \( (2,7) \), and increasing as \( x\) increases.)

Answer:

The graph of \( y = 4\sqrt{x + 2}-1 \) is a transformation of the parent square - root function \( y=\sqrt{x} \). It is shifted 2 units left, stretched vertically by a factor of 4, and shifted 1 unit down. The graph starts at the point \( (-2,-1) \) and passes through points such as \( (-1,3) \) and \( (2,7) \), with a domain \( x\geq - 2 \) and a range \( y\geq - 1 \), and has the shape of a vertically stretched square - root curve.