Question

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Explanation:

Step 1: Identify points on the first graph

The line passes through \((0, 1)\) and \((1, -1)\).

Step 2: Calculate the slope of the line

$$m = \frac{-1 - 1}{1 - 0} = -2$$

Step 3: Write the first equation

Using the slope \(m = -2\) and y-intercept \(b = 1\):

$$y = -2x + 1$$

Step 4: Identify points on the second graph

The curve starts at \((-2, 1)\) and passes through \((-1, 2)\) and \((2, 3)\).

Step 5: Determine the square root function form

The general form is \(y = a\sqrt{x - h} + k\) with vertex \((h, k) = (-2, 1)\):

$$y = a\sqrt{x + 2} + 1$$

Step 6: Solve for the coefficient \(a\)

Substitute \((-1, 2)\) into the equation:

$$2 = a\sqrt{-1 + 2} + 1 \implies a = 1$$

Step 7: Write the second equation

Using \(a = 1\):

$$y = \sqrt{x + 2} + 1$$

Answer:

The equation of the first graph is \(y = -2x + 1\).
The equation of the second graph is \(y = \sqrt{x + 2} + 1\).