Question

write an equation for a graphed function by using transformations of the graphs of one of the basic functions.

Explanation:

Step1: Identify basic parent function

The graph matches the shape of the square root function, parent function: $y=\sqrt{x}$

Step2: Identify horizontal shift

The parent function $y=\sqrt{x}$ has a vertex at $(0,0)$. This graph's vertex is at $(-2,0)$, so it is shifted 2 units left. For a left shift of $h$ units, replace $x$ with $x+h$: $y=\sqrt{x+2}$

Step3: Identify vertical shift

The graph crosses the y-axis at $y\approx0.8$, which matches $\sqrt{0+2}=\sqrt{2}\approx1.414$? No, wait, check the y-intercept: when $x=0$, $f(0)$ is ~0.7, wait no, correct: wait the vertex is $(-2,0)$, and when $x=0$, $y=\sqrt{0+2}$ is ~1.4, but the graph shows $y$ ~0.7. Wait, no, vertical compression? Wait no, wait the y-intercept is $\sqrt{2}$ is ~1.4, but the graph shows $y$ ~0.7, so vertical compression by $\frac{1}{\sqrt{2}}$? No, wait no, wait the graph at $x=2$: $y$ ~1.4, which is $\sqrt{2+2}=\sqrt{4}=2$? No, the graph at $x=2$ is ~1.4, which is $\sqrt{2}$. Oh! Wait, $y=\sqrt{\frac{1}{2}(x+2)}$? No, wait no, let's re-express: the parent function $y=\sqrt{x}$, vertex at (0,0). This graph has vertex at (-2,0), and when $x=0$, $y=\sqrt{2}$ is ~1.4, but the graph shows $y$ ~0.7, which is $\sqrt{1}$? No, wait the graph's y-intercept is 0.7, which is $\frac{\sqrt{2}}{2}$. Wait, no, maybe the parent function is $y=\sqrt{x+2}$ scaled vertically by $\frac{1}{\sqrt{2}}$? No, wait no, let's check the point $x=-1$: $y$ ~0.7, which is $\sqrt{1}=1$? No, the graph at $x=-1$ is ~0.7, which is $\sqrt{0.5}$. Oh! Wait, the function is $y=\sqrt{\frac{1}{2}(x+2)}$? No, that's $\sqrt{\frac{x+2}{2}}=\frac{\sqrt{x+2}}{\sqrt{2}}$. But when $x=2$, that gives $\frac{\sqrt{4}}{\sqrt{2}}=\frac{2}{\sqrt{2}}=\sqrt{2}\approx1.4$, which matches the graph. Wait, but maybe simpler: the graph is $y=\sqrt{x+2}$ compressed vertically by a factor of $\frac{1}{\sqrt{2}}$? No, wait no, wait the y-intercept is $\sqrt{2}/2\approx0.7$, which is what the graph shows. Wait, no, wait another way: the parent function is $y=\sqrt{x}$, shifted left 2 units, then vertically compressed by $\frac{1}{\sqrt{2}}$? No, wait maybe the function is $y=\sqrt{\frac{x+2}{2}}$, which can be written as $y=\frac{\sqrt{2x+4}}{2}$. But wait, let's check the vertex: when $x=-2$, $y=0$, which matches. When $x=0$, $y=\sqrt{1}=1$? No, no, the graph at $x=0$ is ~0.7, which is $\sqrt{0.5}$. Wait, I made a mistake: the graph's y-intercept is actually $\sqrt{2}/2$, which is $\frac{1}{\sqrt{2}}$, so $y=\frac{\sqrt{x+2}}{\sqrt{2}}=\sqrt{\frac{x+2}{2}}$. But wait, maybe the graph is $y=\sqrt{x+2}$ scaled by $\frac{1}{\sqrt{2}}$, but that's equivalent. Alternatively, maybe the function is $y=\sqrt{\frac{1}{2}x +1}$.

Wait, no, let's start over:

Step1: Recognize parent function

The graph is a square root function, parent: $y=\sqrt{x}$ (domain $x\geq0$, increasing, vertex at (0,0))

Step2: Horizontal translation

The vertex of the given graph is at $(-2,0)$, so we shift the parent function 2 units to the left: replace $x$ with $x+2$, so $y=\sqrt{x+2}$

Step3: Vertical scaling

The parent shifted function $y=\sqrt{x+2}$ at $x=0$ is $\sqrt{2}\approx1.414$, but the given graph at $x=0$ is approximately $0.707$, which is $\frac{\sqrt{2}}{2}$. So we scale the function vertically by a factor of $\frac{1}{\sqrt{2}}$:
$y=\frac{1}{\sqrt{2}}\sqrt{x+2}$
Simplify the expression:
$y=\sqrt{\frac{x+2}{2}} = \sqrt{\frac{1}{2}x + 1}$

Alternatively, rationalizing the scale factor:
$y=\frac{\sqrt{2x+4}}{2}$

But the simplest form is $y=\sqrt{\frac{x+2}{2}}$ or $y=\frac{\sqrt{2(x+2)}}{2}$.

Wait, but let's check another point: when $x=2$, $y=\sqrt{\frac{2+2}{2}}=\sqrt{2}\approx1.414$, which matches the graph's value at $x=2$ (around 1.4). When $x=-1$, $y=\sqrt{\frac{-1+2}{2}}=\sqrt{\frac{1}{2}}\approx0.707$, which matches the graph. When $x=-2$, $y=0$, which is the vertex.

Answer:

$y=\sqrt{\frac{x+2}{2}}$ or simplified as $y=\frac{\sqrt{2x+4}}{2}$