question 7 given the function, find the following: f(x)=-√(x + 2)+3 a = -1 h = -2 k = 3 endpoint (-2,3) max…

Explanation:

Question 7

Step1: Identify the form of the function

The function $f(x)=-\sqrt{x + 2}+3$ is in the form $y = a\sqrt{x - h}+k$. Here, $a=- 1$, $h=-2$, $k = 3$. The endpoint of the square - root function $y=\sqrt{x}$ is shifted. For $y=\sqrt{x}$, the endpoint is $(0,0)$. For $y=\sqrt{x + 2}$, it is shifted 2 units to the left to $(-2,0)$, and for $y=-\sqrt{x + 2}+3$, it is shifted 3 units up to $(-2,3)$. Since $a=-1<0$, the function has a maximum value.

Question 8

Step1: Analyze the transformations

For the function $y = f(x)=-\sqrt{x + 2}+3$ compared to the parent - function $y=\sqrt{x}$:

• The $x+2$ inside the square - root means a shift of 2 units to the left.
• The $+3$ outside the square - root means a shift of 3 units up.
• The negative sign in front of the square - root means a reflection over the $x$ - axis.
• There is no coefficient $a$ (other than $a = - 1$ which only causes reflection) multiplying the square - root part to cause a stretch or shrink, so there is no stretch/shrink.

Question 9

Step1: Find the domain

For the square - root function $y =-\sqrt{x + 2}+3$, the expression inside the square - root must be non - negative. So $x+2\geq0$, which gives $x\geq - 2$. The domain is $[-2,\infty)$.

Step2: Find the range

Since the square - root function $\sqrt{x + 2}\geq0$, then $-\sqrt{x + 2}\leq0$, and $-\sqrt{x + 2}+3\leq3$. The range is $(-\infty,3]$.

Step3: Determine increasing or decreasing

The parent function $y = \sqrt{x}$ is increasing. But since $y=-\sqrt{x + 2}+3$ has a negative coefficient in front of the square - root, it is decreasing. So the type is D.

Step4: Find the interval

The function is decreasing for all values in its domain, so the interval is $[-2,\infty)$.

Step5: Analyze the end - behavior

As $x\rightarrow\infty$, $\sqrt{x + 2}\rightarrow\infty$, then $-\sqrt{x + 2}\rightarrow-\infty$, and $y=-\sqrt{x + 2}+3\rightarrow-\infty$.

Step6: Find the x - intercept

Set $y = 0$. Then $0=-\sqrt{x + 2}+3$, so $\sqrt{x + 2}=3$. Squaring both sides gives $x + 2 = 9$, and $x=7$.

Step7: Find the y - intercept

Set $x = 0$. Then $y=-\sqrt{0 + 2}+3=3-\sqrt{2}\approx3 - 1.4=1.6$.

Answer:

Question 7

$a=-1$, $h = - 2$, $k = 3$, endpoint $(-2,3)$, max or min: max

Question 8

shift: left 2 units, shift: up 3 units, reflection: over the x - axis, dilation: no stretch/shrink

Question 9

Domain: $[-2,\infty)$, Range: $(-\infty,3]$, Increasing or Decreasing type: D, Interval: $[-2,\infty)$, end behavior: $x\rightarrow\infty,y\rightarrow-\infty$, x - intercept: 7, y - intercept: 1.6