Use transformations of \( f(x)=\sqrt{x} \) to graph the following function. \( h(x)=\sqrt{x+4}+3 \)

To graph the function \( h(x) = \sqrt{x+4} + 3 \) using transformations of the base function \( f(x) = \sqrt{x} \), we can break down the transformations step by step. 1. **Start with the base function**: The graph of \( f(x) = \sqrt{x} \) is a curve that starts at the origin (0,0) and increases gradually to the right. 2. **Horizontal Shift**: The term \( x + 4 \) inside the square root indicates a horizontal shift. Specifically, \( \sqrt{x + 4} \) represents a shift of the graph of \( f(x) \) to the left by 4 units. This means that the point (0,0) of \( f(x) \) will now be at (-4,0) for \( \sqrt{x + 4} \). 3. **Vertical Shift**: The \( +3 \) outside the square root indicates a vertical shift. The entire graph of \( \sqrt{x + 4} \) is shifted upwards by 3 units. Therefore, the point (-4,0) will move to (-4,3). 4. **Final Transformation**: After applying both transformations, the new starting point of the graph is at (-4, 3). The shape of the graph remains the same, but it is now shifted left and up. ### Summary of Transformations: - Start with \( f(x) = \sqrt{x} \). - Shift left by 4 units: \( f(x) \to f(x + 4) = \sqrt{x + 4} \). - Shift up by 3 units: \( f(x + 4) \to f(x + 4) + 3 = \sqrt{x + 4} + 3 \). ### Key Points for the Graph of \( h(x) \): - The new starting point is (-4, 3). - As \( x \) increases from -4, the function will continue to increase, similar to the original square root function. ### Graphing: 1. Plot the point (-4, 3). 2. Choose a few values of \( x \) greater than -4 to find corresponding \( h(x) \) values: - For \( x = -4 \): \( h(-4) = \sqrt{-4 + 4} + 3 = \sqrt{0} + 3 = 3 \) (point: (-4, 3)) - For \( x = -3 \): \( h(-3) = \sqrt{-3 + 4} + 3 = \sqrt{1} + 3 = 1 + 3 = 4 \) (point: (-3, 4)) - For \( x = -2 \): \( h(-2) = \sqrt{-2 + 4} + 3 = \sqrt{2} + 3 \approx 4.41 \) (point: (-2, 4.41)) - For \( x = -1 \): \( h(-1) = \sqrt{-1 + 4} + 3 = \sqrt{3} + 3 \approx 4.73 \) (point: (-1, 4.73)) - For \( x = 0 \): \( h(0) = \sqrt{0 + 4} + 3 = \sqrt{4} + 3 = 2 + 3 = 5 \) (point: (0, 5)) 3. Draw the curve starting from (-4, 3) and passing through the points calculated above, extending to the right. This will give you the graph of \( h(x) = \sqrt{x + 4} + 3 \).