Question

use transformations to graph the function.
\\( r(x) = \sqrt{-x + 4} \\)

Explanation:

Step1: Identify the parent function

The parent function for square root functions is \( y = \sqrt{x} \), which has a domain \( x \geq 0 \) and starts at the point \( (0, 0) \), increasing slowly.

Step2: Rewrite the given function

Rewrite \( r(x) = \sqrt{-x + 4} \) as \( r(x) = \sqrt{-(x - 4)} \). This helps in identifying the transformations.

Step3: Analyze horizontal reflection

The negative sign inside the square root (affecting the \( x \)-term) indicates a horizontal reflection over the \( y \)-axis. For the parent function \( y = \sqrt{x} \), a horizontal reflection would transform it to \( y = \sqrt{-x} \), which reflects the graph over the \( y \)-axis, so the domain becomes \( x \leq 0 \) for \( y = \sqrt{-x} \), and it starts at \( (0, 0) \) but decreases (or is reflected) in the horizontal direction.

Step4: Analyze horizontal shift

The \( -4 \) inside the square root (with the \( x \)) indicates a horizontal shift. The general form for a horizontal shift of a function \( y = f(x) \) is \( y = f(x - h) \), where \( h \) is the shift. If \( h > 0 \), it's a shift to the right; if \( h < 0 \), it's a shift to the left. In our case, the function is \( y = \sqrt{-(x - 4)} \), so comparing to \( y = \sqrt{-x} \), we have \( h = 4 \), which means a shift to the right by 4 units.

Step5: Combine transformations

First, take the parent function \( y = \sqrt{x} \). Reflect it horizontally over the \( y \)-axis to get \( y = \sqrt{-x} \) (domain \( x \leq 0 \), starting at \( (0, 0) \), and the graph is a mirror image over the \( y \)-axis of \( y = \sqrt{x} \)). Then, shift this reflected graph 4 units to the right. Shifting \( y = \sqrt{-x} \) 4 units to the right: to shift a function \( y = f(x) \) 4 units to the right, we replace \( x \) with \( x - 4 \), so \( y = \sqrt{-( (x - 4) )} = \sqrt{-x + 4} \). The domain of \( y = \sqrt{-x + 4} \) is found by setting \( -x + 4 \geq 0 \), which gives \( -x \geq -4 \) or \( x \leq 4 \). So the starting point (the vertex of the square root graph) is when \( -x + 4 = 0 \), so \( x = 4 \), and \( y = 0 \). So the graph should start at \( (4, 0) \) and increase (since after reflection and shift, the direction: let's check a point. For \( y = \sqrt{-x + 4} \), when \( x = 4 \), \( y = 0 \); when \( x = 3 \), \( y = \sqrt{-3 + 4} = \sqrt{1} = 1 \); when \( x = 0 \), \( y = \sqrt{0 + 4} = 2 \); when \( x = -5 \), \( y = \sqrt{5 + 4} = 3 \). So the graph starts at \( (4, 0) \) and moves to the left (since \( x \) decreases) while \( y \) increases? Wait, no, wait: when \( x \) decreases (moves to the left from \( x = 4 \)), \( -x + 4 \) increases, so \( y \) increases. So the graph starts at \( (4, 0) \) and as \( x \) decreases (goes to the left), \( y \) increases. Wait, but the parent function \( y = \sqrt{x} \) starts at \( (0, 0) \) and as \( x \) increases, \( y \) increases. After reflecting over \( y \)-axis, \( y = \sqrt{-x} \) starts at \( (0, 0) \) and as \( x \) decreases (goes to the left), \( y \) increases (since \( -x \) increases). Then shifting 4 units to the right: the starting point \( (0, 0) \) of \( y = \sqrt{-x} \) shifts to \( (0 + 4, 0) = (4, 0) \). So now, the graph of \( y = \sqrt{-x + 4} \) starts at \( (4, 0) \) and as \( x \) decreases (moves left from 4), \( y \) increases, which matches the points we calculated (e.g., \( x = 3 \) (left of 4 by 1 unit) gives \( y = 1 \), \( x = 0 \) (left of 4 by 4 units) gives \( y = 2 \), \( x = -5 \) (left of 4 by 9 units) gives \( y = 3 \)).

To graph it:
1. Start with the parent function \( y = \sqrt{x} \) (starts at \( (0, 0) \), passes through \( (1, 1) \), \( (4, 2) \), \( (9, 3) \), increasing).
2. Reflect it over the \( y \)-axis: this gives \( y = \sqrt{-x} \), which starts at \( (0, 0) \), passes through \( (-1, 1) \), \( (-4, 2) \), \( (-9, 3) \), and as \( x \) decreases (moves left), \( y \) increases.
3. Shift this reflected graph 4 units to the right: to shift a point \( (x, y) \) on \( y = \sqrt{-x} \) 4 units to the right, we add 4 to the \( x \)-coordinate. So the starting point \( (0, 0) \) becomes \( (0 + 4, 0) = (4, 0) \). The point \( (-1, 1) \) becomes \( (-1 + 4, 1) = (3, 1) \). The point \( (-4, 2) \) becomes \( (-4 + 4, 2) = (0, 2) \). The point \( (-9, 3) \) becomes \( (-9 + 4, 3) = (-5, 3) \). So now, the graph of \( r(x) = \sqrt{-x + 4} \) starts at \( (4, 0) \) and passes through \( (3, 1) \), \( (0, 2) \), \( (-5, 3) \), etc., with the domain \( x \leq 4 \).

So to graph \( r(x) = \sqrt{-x + 4} \), we apply a horizontal reflection over the \( y \)-axis to the parent square root function, followed by a horizontal shift 4 units to the right.

(Note: The graph provided in the problem seems to have a starting point at \( (0, 0) \), which might be incorrect based on our analysis. The correct starting point should be \( (4, 0) \). So perhaps the graph in the problem is of a different function, but the process for transforming \( y = \sqrt{x} \) to \( y = \sqrt{-x + 4} \) is as above.)

Answer:

To graph \( r(x) = \sqrt{-x + 4} \) using transformations:

1. Parent Function: Start with \( y = \sqrt{x} \) (domain \( x \geq 0 \), starts at \( (0, 0) \), increases).
2. Horizontal Reflection: Reflect \( y = \sqrt{x} \) over the \( y \)-axis to get \( y = \sqrt{-x} \) (domain \( x \leq 0 \), starts at \( (0, 0) \), reflects horizontally).
3. Horizontal Shift: Shift \( y = \sqrt{-x} \) 4 units to the right (replace \( x \) with \( x - 4 \)) to get \( y = \sqrt{-(x - 4)} = \sqrt{-x + 4} \). The domain becomes \( x \leq 4 \), and the graph starts at \( (4, 0) \), increasing as \( x \) decreases (left from \( x = 4 \)).

The key transformations are a horizontal reflection over the \( y \)-axis and a horizontal shift 4 units to the right of the parent square root function.