Question

1 the graph of $y = -sqrt{x}$ is shifted 4 units up and 3 units right. which equation represents the new graph?
options:
$y = -sqrt{x - 3} + 4$
$y = -sqrt{x + 3} + 4$
$y = -sqrt{x - 4} + 3$
$y = -sqrt{x + 4} + 3$

Explanation:

Step 1: Recall transformation rules

For a function \( y = f(x) \):

• Horizontal shift: To shift \( h \) units to the right, the function becomes \( y=f(x - h) \); to shift \( h \) units to the left, it becomes \( y = f(x+h) \).
• Vertical shift: To shift \( k \) units up, the function becomes \( y=f(x)+k \); to shift \( k \) units down, it becomes \( y=f(x)-k \).

Step 2: Apply horizontal shift (3 units right) to \( y = -\sqrt{x} \)

Using the horizontal shift rule, with \( h = 3 \) (shift right), we replace \( x \) with \( x - 3 \) in the function \( y=-\sqrt{x} \). So the function becomes \( y=-\sqrt{x - 3} \).

Step 3: Apply vertical shift (4 units up) to the horizontally shifted function

Using the vertical shift rule, with \( k = 4 \) (shift up), we add 4 to the function \( y=-\sqrt{x - 3} \). So the function becomes \( y=-\sqrt{x - 3}+4 \).

Looking at the options, the fourth option (from the left) is \( y = -\sqrt{x - 3}+4 \), so this is the correct equation.

Answer:

The second option (from the left) \( y = -\sqrt{x - 4}+3 \) (Wait, let's re - solve it properly)

Wait, the original function is \( y=-\sqrt{x} \). Shifting 3 units right: for a horizontal shift, if we shift \( h \) units right, the function becomes \( y = f(x - h) \). So shifting 3 units right, it becomes \( y=-\sqrt{x - 3} \)? Wait, no, the problem says "shifted 4 units up and 3 units right". Wait, maybe I misread. Let's do the transformation step by step.

The original function is \( y = -\sqrt{x} \).

1. Shift 3 units right: the rule for horizontal shift is \( y = f(x - h) \) where \( h>0 \) is a shift to the right. So replacing \( x \) with \( x - 3 \), we get \( y=-\sqrt{x - 3} \).

1. Shift 4 units up: the rule for vertical shift is \( y = f(x)+k \) where \( k > 0 \) is a shift up. So adding 4 to the function, we get \( y=-\sqrt{x - 3}+4 \). Wait, but looking at the options, one of the options is \( y = -\sqrt{x - 3}+4 \) (the fourth option from the left? Wait, the options are:

First: \( y=-\sqrt{x + 4}+3 \)

Second: \( y=-\sqrt{x - 4}+3 \)

Third: \( y=-\sqrt{x + 3}+4 \)

Fourth: \( y=-\sqrt{x - 3}+4 \)

Ah, so the correct transformation for shifting 3 units right (so \( x\to x - 3 \)) and 4 units up (add 4) to \( y = -\sqrt{x} \) gives \( y=-\sqrt{x - 3}+4 \), which is the fourth option (from the left, the one with \( y = -\sqrt{x - 3}+4 \)).