Question Statement

Draw the graphs of the given functions and then sketch the graphs of other functions using translation. Verify the results using a graphical calculator:

Base function: $y=\sqrt{x}$

(a) $y=\sqrt{2x}$ (b) $y=2\sqrt{x}$ (c) $y=2\sqrt{x}+3$ (d) $y=\sqrt{2x+5}$

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Background and Explanation

To graph these functions, we start with the parent function $f(x)=\sqrt{x}$. We then apply transformations: horizontal scaling ($f(ax)$), vertical scaling ($af(x)$), vertical shifts ($f(x)+k$), and horizontal shifts ($f(x+h)$). These transformations allow us to determine the new coordinates of points from the original graph.

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Solution

To sketch the graphs, we first calculate a table of values for the base function and each transformed function.

Table of Values

(Note: The raw data table label $y=2\sqrt{x+5}$ has been corrected to $y=\sqrt{2x+5}$ to match the question and the calculated values.)

Part (a)

Function: $y=\sqrt{2x}$ Transformation: This is a horizontal compression of the graph $y=\sqrt{x}$ by a factor of $\frac{1}{2}$. Every $x$-value of the original function is multiplied by $\frac{1}{2}$ to achieve the same $y$-value.

Part (b)

Function: $y=2\sqrt{x}$ Transformation: This is a vertical stretch of the graph $y=\sqrt{x}$ by a factor of $2$. Every $y$-value of the original function is multiplied by $2$.

Part (c)

Function: $y=2\sqrt{x}+3$ Transformation: This involves two steps:

1. A vertical stretch of $y=\sqrt{x}$ by a factor of $2$ (resulting in $y=2\sqrt{x}$).
2. A vertical translation (shift) upwards by $3$ units.

Part (d)

Function: $y=\sqrt{2x+5}$ Transformation: This can be rewritten as $y=\sqrt{2(x+2.5)}$.

1. A horizontal compression by a factor of $\frac{1}{2}$.
2. A horizontal translation (shift) to the left by $2.5$ units.

Graphical Representation

The following plots illustrate the transformations described above:

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Key Formulas or Methods Used

• Horizontal Scaling: $y=f(ax)$ compresses the graph horizontally by a factor of $\frac{1}{a}$ if $a>1$.
• Vertical Scaling: $y=af(x)$ stretches the graph vertically by a factor of $a$ if $a>1$.
• Vertical Translation: $y=f(x)+k$ shifts the graph up by $k$ units.
• Horizontal Translation: $y=f(x+h)$ shifts the graph left by $h$ units.

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Summary of Steps

1. Create a table of values for the parent function $y=\sqrt{x}$.
2. Apply the specific transformation rules (stretching, compressing, or shifting) to the coordinates.
3. Calculate the new $y$-values for specific $x$-inputs for each function (a) through (d).
4. Plot the points on a coordinate plane and draw smooth curves through them.
5. Use a graphical calculator to compare the sketched curves with the calculated functions for accuracy.